Introduction
Flow-matching models [21 , 24 ] generate samples by learning a velocity field that transports noise toward the data distribution. Their simple regression objective has shown strong results in image [10 ] , video [43 ] , 3D [49 ] and robotic action [2 ] . Yet generated content still exhibits structural errors, semantic mismatches and physical inconsistencies [63 , 41 , 28 , 59 ] . Recent approaches improve generation quality through large-scale training [10 ] , representation alignment [57 ] , self-supervised feature learning [5 ] and reward-based optimization [58 ] . They focus on improving the model’s ability by learning only from the final answer, which is limited. This limitation can be attributed, in part, to the fact that the model must predict a single velocity to determine the next sample state, without any opportunity to review and revise its own predictions.
We therefore exploit the property that learning should be “refine-in-loop”, in the sense that if a student makes a mistake, e.g., in an examination, they should revisit the answer they wrote, the question they were asked, and the reasoning they followed. By examining these pieces of context together, the student can trace a mistake and determine how to revise it, rather than struggling with a final answer [27 , 66 ] . Inspired by the success of self-refinement in large language models (LLMs) [27 , 38 ] , we apply this principle to flow matching. A flow-based generator already computes analogous information at each sampling step: an estimate of the clean sample→ \to → → \to → → \to →
In this paper, we introduce CorrFlow, a self-correcting flow-matching framework that attaches a lightweight refiner to the base generator. Our core insight is that the model’s one-step clean prediction provides a concrete object for the review, with a direct mathematical connection to the correction we seek [6 ] . Under rectified-flow interpolation, the error between this prediction and the clean sample is exactly realized velocity residual (eq. (4)) scaled by the noise level. This identity connects a mistake in the predicted sample to a correction in the velocity field. We first theoretically prove the relationship between the realized velocity residual and the clean prediction (section 3.1). In practice, the refiner is a transformer that predicts the velocity residual from three sources of information: i) the current answer , given by the one-step clean prediction, ii) the circumstances , given by the generation condition and noise level, and iii) the internal state , given by hidden states from the base model [9 ] . During joint training, clean samples and their sampled noise provide residual targets for the refiner, while the base model and refiner optimize separate objectives. During generation, the predicted residual adjusts the velocity used by the sampler before each state update. All refiner inputs reuse the same base-model forward pass, and each correction requires one additional refiner evaluation.
We empirically apply CORRFLOW to a wide range of applications, including image generation, text-to-video generation and image-to-video generation. Figure 1 illustrates how this review shifts the model’s clean prediction through a correction of its velocity on image generation. We find that training a refiner improves the base model’s performance on aesthetic quality and physics consistency of generated content. The strong performance across multiple tasks and datasets shows that velocity residual correction is generic. Our contributions are summarized as follows:
Figure 1. CORRFLOW learns from its own imperfect predictions. The base model’s clean prediction (blue) deviates from the clean reference (orange arrow). The resulting correction shifts the prediction closer to the clean reference by − σ κ r ^ -\sigma\kappa\hat{\mathbf{r}} − σ κ r ^ σ \sigma σ κ \kappa κ
We introduce CORRFLOW, a self-correcting flow-matching framework that learns velocity residuals from the base model’s current prediction, and applies the resulting corrections within the sampling loop.
We theoretically establish conditions under which reusing base-model representations outperforms raw-input affine predictors, refiner training achieves greater residual-risk reduction than continued base-model training at matched compute, and deployed correction lowers velocity error under its own sampler-induced state law.
We evaluate CORRFLOW on image and video generation benchmarks, demonstrating improvements in visual quality and physics consistency.
Related Works
Improving Flow-Matching Generative Models. Many approaches have been proposed to improve flow-matching generative models. For representation learning, REPA [57 ] aligns hidden states with pretrained visual features, while iREPA [39 ] emphasizes their spatial structure. VA-VAE [56 ] aligns VAE latents with visual features, REPA-E [18 ] jointly tunes the VAE and diffusion transformer through alignment and RAE [62 ] generates directly in pretrained representation spaces. Other methods reuse features available within the generative model: SRA and LayerSync [17 , 12 ] use deeper features to guide shallower representations, and SRA2 [45 ] draws on existing VAE features. Reward-based methods Flow-GRPO and DanceGRPO [22 , 53 ] use stochastic sampling for online reinforcement learning, while AWM [52 ] and DiffusionNFT [65 ] incorporate rewards into flow-matching objectives through advantage weighting and paired generations, respectively. CORRFLOW takes a complementary approach: it trains a lightweight added-on refiner to predict velocity residuals from the base model’s current prediction and hidden states. These residuals directly correct the velocity within the sampling loop, without requiring external representation teachers or reward models for refiner training.
Physics-Aware Video Generation. Several recent works have explored physics-aware video generation with flow-matching models. WISA [44 ] , PhysVid [31 ] and PhyParam [19 ] inject hierarchical, temporally local, or instance-level physical conditions. PhysGen [23 ] and PSIVG [11 ] use simulated dynamics as guidance. Representation-based methods transfer or jointly learn motion, geometry, semantics and latent dynamics, including VideoREPA [61 ] , PhysAlign [50 ] , VideoJAM [4 ] , PHANTOM [37 ] , DreamWorld [42 ] and MMPhysVideo [20 ] . Other approaches use language or feedback, with PhyT2V [51 ] and DiffPhy [60 ] introducing inferred physical rules, PhysMaster [16 ] and PhyGDPO [3 ] applying pairwise or groupwise preference learning and WMReward [58 ] steering denoising trajectories with a latent-world-model reward at inference. In contrast, CORRFLOW learns to correct the base model’s own velocity residuals from its predictions and hidden states, without explicit physical parameters or external physics evaluators.
CorrFlow
Figure 2. CORRFLOW overview for video generation. (a) At noise level σ \sigma σ r ^ \hat{r} r ^ x ^ 0 = x σ − σ v θ \hat{x}_0 = x_\sigma- \sigma v_\theta x ^ 0 = x σ − σ v θ h L h_L h L c c c e σ e_\sigma e σ v κ = v θ + κ r ^ v^\kappa= v_\theta+ \kappa\hat{r} v κ = v θ + κ r ^ κ \kappa κ
CORRFLOW augments a base flow-matching model f b a s e f_{\mathrm{base}} f base refiner that corrects its velocity v θ v_\theta v θ x ^ 0 = x σ − σ v θ \hat{x}_0 = x_\sigma- \sigma v_\theta x ^ 0 = x σ − σ v θ { h ℓ } ℓ ∈ L \{h_\ell\}_{\ell\in\mathcal{L}} { h ℓ } ℓ ∈ L c c c e σ e_\sigma e σ
r ^ = f r e f i n e r ( x ^ 0 , { h ℓ } ℓ ∈ L , c , e σ ) , v κ = v θ + κ r ^ , (1) \hat{r} = f_{\mathrm{refiner}}\left(\hat{x}_0, \{h_\ell\}_{\ell\in\mathcal{L}}, c, e_\sigma\right), \qquad v^\kappa= v_\theta+ \kappa\hat{r}, \tag*{(1)} r ^ = f refiner ( x ^ 0 , { h ℓ } ℓ ∈ L , c , e σ ) , v κ = v θ + κ r ^ , (1) where κ \kappa κ
Preliminaries: flow matching and the residual it leaves behind
Let x 0 ∈ R D x_0 \in\mathbb{R}^D x 0 ∈ R D c c c D D D ϵ ∼ N ( 0 , I D ) \boldsymbol{\epsilon} \sim\mathcal{N}(0, I_D) ϵ ∼ N ( 0 , I D ) σ ∈ [ 0 , 1 ] \sigma\in[0,1] σ ∈ [ 0 , 1 ]
x σ = ( 1 − σ ) x 0 + σ ϵ , d x σ d σ = ϵ − x 0 . (2) \mathbf{x}_{\sigma} = (1-\sigma)\mathbf{x}_0 + \sigma\boldsymbol{\epsilon}, \qquad\frac{d\mathbf{x}_{\sigma}}{d\sigma} = \boldsymbol{\epsilon} - \mathbf{x}_0. \tag*{(2)} x σ = ( 1 − σ ) x 0 + σ ϵ , d σ d x σ = ϵ − x 0 . (2) The target flow velocity is v = ϵ − x 0 \mathbf{v} = \boldsymbol{\epsilon} - \mathbf{x}_0 v = ϵ − x 0 v ∗ ( x , σ , c ) = E [ ϵ − x 0 ∣ x σ = x , c ] \mathbf{v}^{\ast}(\mathbf{x}, \sigma, c) = \mathbb{E}[\boldsymbol{\epsilon} - \mathbf{x}_0 \mid\mathbf{x}_{\sigma} = \mathbf{x}, c] v ∗ ( x , σ , c ) = E [ ϵ − x 0 ∣ x σ = x , c ] f b a s e f_{\mathrm{base}} f base
L b a s e ( θ ) = E x 0 , c , ϵ , σ w ( σ ) ∥ v θ ( x σ , σ , c ) − v ∥ 2 , (3) \mathcal{L}_{\mathrm{base}}(\theta) = \mathbb{E}_{\mathbf{x}_0,c,\boldsymbol{\epsilon},\sigma} w(\sigma) \left\lVert\mathbf{v}_{\theta}(\mathbf{x}_{\sigma}, \sigma, c) - \mathbf{v} \right\rVert^2, \tag*{(3)} L base ( θ ) = E x 0 , c , ϵ , σ w ( σ ) ∥ v θ ( x σ , σ , c ) − v ∥ 2 , (3) with w ( σ ) w(\sigma) w ( σ ) v θ \mathbf{v}_{\theta} v θ f b a s e f_{\mathrm{base}} f base θ \theta θ d x / d σ = v θ ( x , σ , c ) d\mathbf{x}/d\sigma= \mathbf{v}_{\theta}(\mathbf{x}, \sigma, c) d x / d σ = v θ ( x , σ , c ) σ = 1 \sigma= 1 σ = 1 σ = 0 \sigma= 0 σ = 0 realized velocity residual ,
r = v − v θ ( x σ , σ , c ) = ( ϵ − x 0 ) − v θ ( x σ , σ , c ) . (4) \mathbf{r} = \mathbf{v} - \mathbf{v}_{\theta}(\mathbf{x}_{\sigma}, \sigma, c) = (\boldsymbol{\epsilon} - \mathbf{x}_0) - \mathbf{v}_{\theta}(\mathbf{x}_{\sigma}, \sigma, c). \tag*{(4)} r = v − v θ ( x σ , σ , c ) = ( ϵ − x 0 ) − v θ ( x σ , σ , c ) . (4) This is the mistake we want the refiner f r e f i n e r f_{\mathrm{refiner}} f refiner
Clean-prediction identity. Taking a single Euler step from x σ \mathbf{x}_{\sigma} x σ σ = 0 \sigma= 0 σ = 0 one-step clean prediction , its best current guess at the finished sample,
x ^ 0 ( x σ , σ , c ) = x σ − σ v θ ( x σ , σ , c ) . (5) \hat{\mathbf{x}}_0(\mathbf{x}_{\sigma}, \sigma, c) = \mathbf{x}_{\sigma} - \sigma\mathbf{v}_{\theta}(\mathbf{x}_{\sigma}, \sigma, c). \tag*{(5)} x ^ 0 ( x σ , σ , c ) = x σ − σ v θ ( x σ , σ , c ) . (5) Substituting Equation (2) into Equation (5) and using Equation (4) ,
x ^ 0 − x 0 = σ ( ( ϵ − x 0 ) − v θ ) = σ r , equivalently r = x ^ 0 − x 0 σ , σ > 0. (6) \hat{\mathbf{x}}_0 - \mathbf{x}_0 = \sigma\left((\boldsymbol{\epsilon} - \mathbf{x}_0) - \mathbf{v}_{\theta}\right) = \sigma\mathbf{r}, \qquad\text{equivalently} \qquad\mathbf{r} = \frac{\hat{\mathbf{x}}_0 - \mathbf{x}_0}{\sigma}, \quad\sigma> 0. \tag*{(6)} x ^ 0 − x 0 = σ ( ( ϵ − x 0 ) − v θ ) = σ r , equivalently r = σ x ^ 0 − x 0 , σ > 0. (6) This identity holds for any predictor v θ \mathbf{v}_{\theta} v θ r \mathbf{r} r σ \sigma σ x ^ 0 \hat{\mathbf{x}}_0 x ^ 0 x ^ 0 \hat{\mathbf{x}}_0 x ^ 0 x ^ 0 \hat{\mathbf{x}}_0 x ^ 0 h L h_L h L
Refiner Inputs
Focusing on the current answer in isolation is insufficient, because the same prediction may be correct under one condition and a mistake under another, or an artifact that must be repaired at high noise may be a legitimate detail at low noise. Therefore, the refiner is given three views of the moment the mistake was made:
The current answer. The one-step clean prediction x ^ 0 \hat{\mathbf{x}}_0 x ^ 0 (5) , tokenized according to the sample structure to form the refiner’s trunk tokens. This is the base model’s current estimate of the clean sample.
The circumstances. The conditioning tokens c c c σ \sigma σ
The internal state. The base model’s hidden states { h ℓ } ℓ ∈ L \{h_{\ell}\}_{\ell\in L} { h ℓ } ℓ ∈ L L L L v θ \mathbf{v}_{\theta} v θ
Formally, let W = ( x σ , σ , c ) W = (\mathbf{x}_{\sigma}, \sigma, c) W = ( x σ , σ , c ) Z ( W ) = ( x ^ 0 , { h ℓ } ℓ ∈ L , c , σ ) \mathcal{Z}(W) = (\hat{\mathbf{x}}_0, \{h_{\ell}\}_{\ell\in L}, c, \sigma) Z ( W ) = ( x ^ 0 , { h ℓ } ℓ ∈ L , c , σ ) ϕ \phi ϕ g ϕ g_{\phi} g ϕ r ^ = g ϕ ( Z ( W ) ) ≈ r \hat{\mathbf{r}} = g_{\phi}(\mathcal{Z}(W)) \approx\mathbf{r} r ^ = g ϕ ( Z ( W )) ≈ r Z \mathcal{Z} Z no additional base-model evaluation and no access to the clean target at generation time .
Refiner architecture
CorrFlow is designed to be a lightweight add-on transformer to a base flow-matching model, which estimates the realized velocity residual r r r
Token embeddings. Given the clean prediction x ^ 0 \hat{x}_0 x ^ 0 (5) ), the refiner f r e f i n e r f_{\mathrm{refiner}} f refiner N N N d d d t a n s ∈ R N × d t_{\mathrm{ans}} \in\mathbb{R}^{N \times d} t ans ∈ R N × d e σ e_\sigma e σ σ \sigma σ
We also construct the context tokens t c t x ∈ R ( N c + N ) × d t_{\mathrm{ctx}} \in\mathbb{R}^{(N_c+N) \times d} t ctx ∈ R ( N c + N ) × d c c c { h ℓ } ℓ ∈ L \{h_\ell\}_{\ell\in L} { h ℓ } ℓ ∈ L N c N_c N c c c c t c ∈ R N c × d t_c \in\mathbb{R}^{N_c \times d} t c ∈ R N c × d d d d
t f u s e = MLP ( concat ℓ ∈ L h ℓ ) ∈ R N × d . (7) t_{\mathrm{fuse}} = \operatorname{MLP}\left(\operatorname{concat}_{\ell\in L} h_\ell\right) \in\mathbb{R}^{N \times d}. \tag*{(7)} t fuse = MLP ( concat ℓ ∈ L h ℓ ) ∈ R N × d . (7) This fusion across layers at fixed token positions is a deliberate choice. It keeps the hidden-state tokens at N N N ∣ L ∣ ⋅ N |L| \cdot N ∣ L ∣ ⋅ N
t c t x = LayerNorm ( [ t c + s c o n d , t f u s e + s s t a t e ] ) ∈ R ( N c + N ) × d , (8) t_{\mathrm{ctx}} = \operatorname{LayerNorm}\left(\left[t_c + s_{\mathrm{cond}},\ t_{\mathrm{fuse}} + s_{\mathrm{state}}\right]\right) \in\mathbb{R}^{(N_c+N) \times d}, \tag*{(8)} t ctx = LayerNorm ( [ t c + s cond , t fuse + s state ] ) ∈ R ( N c + N ) × d , (8) where s c o n d s_{\mathrm{cond}} s cond s s t a t e s_{\mathrm{state}} s state
Alternating-Attention blocks. We build our lightweight add-on transformer with a stack of B B B self-attention to the answer tokens t a n s t_{\mathrm{ans}} t ans cross-attention to the context tokens t c t x t_{\mathrm{ctx}} t ctx e σ e_\sigma e σ
Training and Inference
Objective. We train the base model f b a s e f_{\mathrm{base}} f base f r e f i n e r f_{\mathrm{refiner}} f refiner
L r e f ( ϕ ) = E [ ∥ r ^ − sg [ r ] ∥ 2 + λ ( 1 − cos ( r ^ , sg [ r ] ) ) ] , (9) \mathcal{L}_{\mathrm{ref}}(\phi) = \mathbb{E}\left[\left\|\hat{r} - \operatorname{sg}[r]\right\|^2 + \lambda\left(1 - \cos\left(\hat{r}, \operatorname{sg}[r]\right)\right)\right], \tag*{(9)} L ref ( ϕ ) = E [ ∥ r ^ − sg [ r ] ∥ 2 + λ ( 1 − cos ( r ^ , sg [ r ] ) ) ] , (9) where sg [ ⋅ ] \operatorname{sg}[\cdot] sg [ ⋅ ] r ^ = f r e f i n e r ( sg [ z ( W ) ] ) \hat{r} = f_{\mathrm{refiner}}(\operatorname{sg}[z(W)]) r ^ = f refiner ( sg [ z ( W )]) r r r (4) and λ \lambda λ
L = L b a s e ( θ ) + λ r e f L r e f ( ϕ ) , (10) \mathcal{L} = \mathcal{L}_{\mathrm{base}}(\theta) + \lambda_{\mathrm{ref}} \mathcal{L}_{\mathrm{ref}}(\phi), \tag*{(10)} L = L base ( θ ) + λ ref L ref ( ϕ ) , (10) where θ \theta θ ϕ \phi ϕ λ r e f \lambda_{\mathrm{ref}} λ ref
Inference. The refiner is applied inside the ordinary sampling loop rather than as a post-process on a finished sample. At each denoising step the sampler evaluates the base model on the current sample, captures e σ e_\sigma e σ { h ℓ } ℓ ∈ L \{h_\ell\}_{\ell\in\mathcal{L}} { h ℓ } ℓ ∈ L x ^ 0 \hat{x}_0 x ^ 0 (5) and runs the refiner once on Z ( W ) \mathcal{Z}(W) Z ( W ) κ \kappa κ
v κ = v θ ( x σ , σ , c ) + κ f r e f i n e r ( Z ( W ) ) , (11) v^\kappa= v_\theta(x_\sigma,\sigma,c) + \kappa f_{\mathrm{refiner}}(\mathcal{Z}(W)), \tag*{(11)} v κ = v θ ( x σ , σ , c ) + κ f refiner ( Z ( W )) , (11) and the corrected velocity is what the scheduler integrates. Because x ^ 0 \hat{x}_0 x ^ 0 { h ℓ } ℓ ∈ L \{h_\ell\}_{\ell\in\mathcal{L}} { h ℓ } ℓ ∈ L
Theoretical Analysis
We analyze three complementary questions: whether frozen base-model features can improve residual prediction beyond raw-input predictors, whether refiner training can outperform continued base-model training at matched budget and whether a trained correction improves velocity error under the state law it induces at deployment.
Shared residual target. Let X \mathcal{X} X P P P W = ( x σ , σ , c ) W=(x_\sigma,\sigma,c) W = ( x σ , σ , c ) P ~ \widetilde{P} P ( W , r ) (W,r) ( W , r ) r ∈ L 2 ( P ~ ; X ) r\in L^2(\widetilde{P};\mathcal{X}) r ∈ L 2 ( P ; X ) r ˉ ( W ) = E P ~ [ r ∣ W ] \bar{r}(W)=\mathbb{E}_{\widetilde{P}}[r\mid W] r ˉ ( W ) = E P [ r ∣ W ] f ∈ L 2 ( P ; X ) f\in L^2(P;\mathcal{X}) f ∈ L 2 ( P ; X )
R P ( f ) : = E P ~ ∥ r − f ( W ) ∥ 2 , R i r r : = E P ~ ∥ r − r ˉ ( W ) ∥ 2 . \mathcal{R}_P(f):=\mathbb{E}_{\widetilde{P}}\lVert r-f(W)\rVert^2,\qquad\mathcal{R}_{\mathrm{irr}}:=\mathbb{E}_{\widetilde{P}}\lVert r-\bar{r}(W)\rVert^2. R P ( f ) := E P ∥ r − f ( W ) ∥ 2 , R irr := E P ∥ r − r ˉ ( W ) ∥ 2 . Proposition 1 (Same-target risk decomposition). Every such predictor satisfies
R P ( f ) = R i r r + E P ∥ f ( W ) − r ˉ ( W ) ∥ 2 . (12) \mathcal{R}_P(f)=\mathcal{R}_{\mathrm{irr}}+\mathbb{E}_P\lVert f(W)-\bar{r}(W)\rVert^2. \tag*{(12)} R P ( f ) = R irr + E P ∥ f ( W ) − r ˉ ( W ) ∥ 2 . (12) The cross term vanishes because E P ~ [ r − r ˉ ( W ) ∣ W ] = 0 \mathbb{E}_{\widetilde{P}}[r-\bar{r}(W)\mid W]=0 E P [ r − r ˉ ( W ) ∣ W ] = 0 (40) and (41) .
Frozen-feature separation. In the scalar Gaussian witness of Appendix § 7 , let c 1 , c 2 c_1,c_2 c 1 , c 2 p = c 1 c 2 p=c_1c_2 p = c 1 c 2 x 0 = δ p + ξ x_0=\delta p+\xi x 0 = δ p + ξ δ > 0 \delta>0 δ > 0 ξ ∼ N ( 0 , 1 ) \xi\sim\mathcal{N}(0,1) ξ ∼ N ( 0 , 1 ) h L = p h_L=p h L = p τ σ = σ / [ ( 1 − σ ) 2 + σ 2 ] \tau_\sigma=\sigma/[(1-\sigma)^2+\sigma^2] τ σ = σ / [( 1 − σ ) 2 + σ 2 ] g b = − b τ σ p g_b=-b\tau_\sigma p g b = − b τ σ p A \mathcal{A} A ( x σ , c 1 , c 2 ) (x_\sigma,c_1,c_2) ( x σ , c 1 , c 2 ) σ \sigma σ
Theorem 2 (Frozen-feature advantage). For an explicit Δ a f f > 0 \Delta_{\mathrm{aff}}>0 Δ aff > 0 § 7 ,
R P ( g δ ) = R i r r , inf a ∈ A R P ( a ) = R i r r + Δ a f f . \mathcal{R}_P(g_\delta)=\mathcal{R}_{\mathrm{irr}},\qquad\inf_{a\in\mathcal{A}}\mathcal{R}_P(a)=\mathcal{R}_{\mathrm{irr}}+\Delta_{\mathrm{aff}}. R P ( g δ ) = R irr , a ∈ A inf R P ( a ) = R irr + Δ aff . If b ^ \hat{b} b ^ ϵ o p t \epsilon_{\mathrm{opt}} ϵ opt J λ ( b ) = R P ( g b ) + λ E P ~ L cos ( g b , r ) J_\lambda(b)=\mathcal{R}_P(g_b)+\lambda\mathbb{E}_{\widetilde{P}}L_{\cos}(g_b,r) J λ ( b ) = R P ( g b ) + λ E P L c o s ( g b , r ) λ = 0.1 \lambda=0.1 λ = 0.1
2 λ + ϵ o p t < Δ a f f ⟹ R P ( g b ^ ) < inf a ∈ A R P ( a ) . (13) 2\lambda+\epsilon_{\mathrm{opt}}<\Delta_{\mathrm{aff}}\quad\Longrightarrow\quad\mathcal{R}_P(g_{\hat{b}})<\inf_{a\in\mathcal{A}}\mathcal{R}_P(a). \tag*{(13)} 2 λ + ϵ opt < Δ aff ⟹ R P ( g b ^ ) < a ∈ A inf R P ( a ) . (13) Matched-resource separation. Returning to the finite-dimensional space X \mathcal{X} X v 0 v_0 v 0 F r ( α ) = g ϕ 0 + α ( Z ( W ) ) F_r(\alpha)=g_{\phi_0+\alpha}(\mathcal{Z}(W)) F r ( α ) = g ϕ 0 + α ( Z ( W )) F b ( α ) = v θ 0 + α ( W ) − v 0 ( W ) F_b(\alpha)=v_{\theta_0+\alpha}(W)-v_0(W) F b ( α ) = v θ 0 + α ( W ) − v 0 ( W ) F j ( 0 ) = 0 F_j(0)=0 F j ( 0 ) = 0 C \mathcal{C} C j j j T j = [ C f i t / C t r a i n ] T_j=[C_{\mathrm{fit}}/C_{\mathrm{train}}] T j = [ C fit / C train ] f j , η f_{j,\eta} f j , η Δ j = sup η ∈ E j [ R P ( 0 ) − R P ( f j , η ) ] \Delta_j=\sup_{\eta\in\mathcal{E}_j}[\mathcal{R}_P(0)-\mathcal{R}_P(f_{j,\eta})] Δ j = sup η ∈ E j [ R P ( 0 ) − R P ( f j , η )] Φ j = D F j ( 0 ) \Phi_j=DF_j(0) Φ j = D F j ( 0 ) H j = Φ j ∗ Φ j H_j=\Phi_j^*\Phi_j H j = Φ j ∗ Φ j b j = Φ j ∗ r ˉ b_j=\Phi_j^*\bar{r} b j = Φ j ∗ r ˉ
Theorem 3 (Finite-ledger refiner separation). Under the local Jacobian, angular-gradient, radius and spectral-stability conditions of Appendix § 8 .§ 1 , both armwise optima are attained and the actual reduction Δ j \Delta_j Δ j Δ j l i n \Delta_j^{\mathrm{lin}} Δ j lin e j e_j e j
∣ Δ j − Δ j l i n ∣ ≤ e j , Δ j l i n = sup η ∈ E j b j ∗ H j † [ I − ( I − η H j ) 2 T j ] b j . \lvert\Delta_j-\Delta_j^{\mathrm{lin}}\rvert\le e_j,\qquad\Delta_j^{\mathrm{lin}}=\sup_{\eta\in\mathcal{E}_j}b_j^*H_j^\dagger[I-(I-\eta H_j)^{2T_j}]b_j. ∣ Δ j − Δ j lin ∣ ≤ e j , Δ j lin = η ∈ E j sup b j ∗ H j † [ I − ( I − η H j ) 2 T j ] b j . Consequently,
Δ r l i n − e r > Δ b l i n + e b ⟹ min η ∈ E r R P ( f r , η ) < min η ∈ E b R P ( f b , η ) . (14) \Delta_r^{\mathrm{lin}}-e_r>\Delta_b^{\mathrm{lin}}+e_b\quad\Longrightarrow\quad\min_{\eta\in\mathcal{E}_r}\mathcal{R}_P(f_{r,\eta})<\min_{\eta\in\mathcal{E}_b}\mathcal{R}_P(f_{b,\eta}). \tag*{(14)} Δ r lin − e r > Δ b lin + e b ⟹ η ∈ E r min R P ( f r , η ) < η ∈ E b min R P ( f b , η ) . (14) Own-law deployment. Write g ϕ g_{\phi} g ϕ W ↦ g ϕ ( Z ( W ) ) W \mapsto g_{\phi}(\mathcal{Z}(W)) W ↦ g ϕ ( Z ( W )) X κ X^{\kappa} X κ v 0 + κ g ϕ v_{0}+\kappa g_{\phi} v 0 + κ g ϕ Σ \Sigma Σ q q q [ 0 , 1 ] [0,1] [ 0 , 1 ] Q κ = Law ( X κ ( Σ ) , Σ , C ) Q_{\kappa}=\operatorname{Law}(X^{\kappa}(\Sigma),\Sigma,C) Q κ = Law ( X κ ( Σ ) , Σ , C ) V Q ( κ ) = E Q ∥ r ˉ − κ g ϕ ∥ 2 V_{Q}(\kappa)=\mathbb{E}_{Q}\lVert\bar{r}-\kappa g_{\phi}\rVert^{2} V Q ( κ ) = E Q ∥ r ˉ − κ g ϕ ∥ 2 r ˉ \bar{r} r ˉ § 9 .§ 1 .
Theorem 4 (Own-law velocity improvement). Under the deployment assumptions of Appendix § 9 , let M M M N N N § 9 .§ 3 . Then, for every κ ∈ [ 0 , κ ˉ ] \kappa\in[0,\bar{\kappa}] κ ∈ [ 0 , κ ˉ ]
V Q κ ( κ ) − V Q 0 ( 0 ) ≤ − κ M + κ 2 N . V_{Q_{\kappa}}(\kappa)-V_{Q_{0}}(0)\leq-\kappa M+\kappa^{2}N. V Q κ ( κ ) − V Q 0 ( 0 ) ≤ − κ M + κ 2 N . If M > 0 M>0 M > 0 N > 0 N>0 N > 0
0 < κ ≤ κ ˉ , κ < M N ⟹ V Q κ ( κ ) < V Q 0 ( 0 ) . (15) 0<\kappa\leq\bar{\kappa},\quad\kappa<\frac{M}{N}\quad\Longrightarrow\quad V_{Q_{\kappa}}(\kappa)<V_{Q_{0}}(0). \tag*{(15)} 0 < κ ≤ κ ˉ , κ < N M ⟹ V Q κ ( κ ) < V Q 0 ( 0 ) . (15) Experiments
We evaluate CorrFlow on a wide range of tasks, including class-conditional image generation, text-to-video generation and image-to-video generation. The image experiments test whether learned velocity corrections improve an established flow-matching model, while the video experiments examine visual quality, temporal consistency and physical plausibility. Red, orange and yellow highlight the best, second-best and third-best scores in each column, respectively.
Image Generation
Settings. We study class-conditional generation on ImageNet at 256 × 256 256\times256 256 × 256 Table 1 . The refiner has 10 transformer blocks with width 768 and 12 attention heads, and reads hidden states from base transformer blocks 3, 11, 19 and 27, using one-based indexing. We retain the base model’s training objective and train the refiner to predict the realized velocity residual with squared error and a cosine-direction term weighted by 0.1. Both networks use GMuon optimizer with a global batch size of 1,024. The evaluation settings are identical to those in Singh et al. [40 ] .
Table 1. Comparison on ImageNet-256 [8 ] generation. Representation Fréchet Distance (FDr _r r
Results. Table 1 compares CorrFlow with RAEv2 at a matched budget of 160 total epochs, because our 80 + 80 schedule spends 80 epochs on the base checkpoint and 80 epochs on joint training. Under this budget, CorrFlow improves over RAEv2 on IS and on all three FDr _r r
Text-to-Video Generation
Settings. We use Wan2.1-T2V-1.3B [43 ] as the base model and jointly adapt its transformer with the refiner on video–text pairs from WISA-80K [44 ] and Open-
VidHD ([29 ] ). The refiner has 10 transformer blocks with width 1,024 and 16 attention heads, and reads hidden states from base blocks 4, 12, 20 and 30, using one-based indexing. The final training stage uses 66,572 clips for five epochs with AdamW and a global batch size of 16, retaining the base flow-matching objective and a residual objective with cosine-direction weight 0.1. We generate 81-frame videos at 480 × 832 resolution with 50 Euler steps and evaluate visual quality and temporal consistency with VBench ([14 ] ), and physical plausibility and prompt adherence with VideoPhy2 ([1 ] ) and FinePhyEval ([32 ] ). Section D.2 provides the training and evaluation details.
Results. Table 2 reports the VideoPhy2 and FinePhyEval results for physics-consistent text-to-video generation. CORRFLOW achieves the highest VideoPhy2 Joint scores on both ALL and HARD, increasing them from 24.8 to 27.0 and from 5.6 to 6.9, respectively, over Wan2.1-T2V-1.3B. It also achieves the highest FinePhyEval Overall score of 0.60, with improvements in object, action and physics scores over the same base model. On VBench (Table 3 ), CORRFLOW leads the compared methods on Quality Score and Overall Score, reaching 84.69 and 81.86. Figure 3 shows the same advantage in the generated frames, where CORRFLOW carries each prompted interaction to its physical outcome, dropping the domino into the mug and landing the coin in the metal cup, while the baselines lose the object or drift away from the prompted action. These results show that the adapted model improves both general video quality and the joint satisfaction of semantic and physical requirements. Section E.2 analyzes the full metric breakdown.
Table 2. Physics-consistent text-to-video generation on VideoPhy2 ([1 ] ) and FinePhyEval ([32 ] ). Higher is better for all metrics.
Table 3. VBench ([14 ] ) comparison on selected metrics (0–100, higher is better).
Figure 3. Qualitative comparison on text-to-video generation. Each scenario compares VideoREPA-5B, VPT-1.3B and CORRFLOW (Ours) under the same text prompt, with frames sampled at 0%, 25%, 50%, 75% and 100% of each generated video. The prompts are drawn from VBench (amusement park), FinePhyEval (a domino dropped into a mug) and VideoPhy2 (a somersault on a bouncy castle, a coin caught in a metal cup).
Image-to-Video Generation
Settings. We use Wan2.2-TI2V-5B ([43 ] ) as the base model, with the same training data, objectives, optimization and five-epoch schedule as in § 5.2 . We enable first-frame image conditioning for 40% of training examples and use text-only conditioning for the remainder. The refiner has 13 transformer blocks with width 2,048 and 32 attention heads, reading hidden states from the same base blocks as in text-to-video generation. We evaluate on the 198 image–text pairs of Physics-IQ Verified ((Radford et al., 2026); [28 ] ), generating 81-frame videos at 704 × 1248 resolution with the same sampling configuration as in text-to-video generation. Section D.3 details the image conditioning, training and evaluation.
Results. Table 4 reports the Physics-IQ Verified results. CORRFLOW achieves the highest Verified Score of 30.95, and leads on Spatial, Weighted Spatial and MSE Score. These results indicate closer agreement with the ground-truth videos in motion coverage and frame appearance. Figure 4 illustrates shadow and object consistency, with the statuette’s shadow tracking the moving light and the blade separating the mandarin into halves. Section E.3 provides the metric breakdown and qualitative analysis.
Table 4. Physics-IQ Verified ((Radsch et al., 2026)) results on the image-to-video track. All evaluations are without LLM usage. All scores are percentages, and higher is better for every metric.
Figure 4. Qualitative comparison on the Physics-IQ Verified image-to-video track. Each scenario compares WMReward-5B, PhysRAG-5B and CORRFLOW (Ours), which share the Wan2.2-TI2V-5B base model and receive the same settings. Temporal order is from left to right. The left scenario shows a statuette’s shadow under a moving light, and the right scenario shows a knife slicing a mandarin.
Ablation Study
We conduct a number of ablations to nanalyze the contribution of each component. All variants are built on WAN-2.1-T2V-1.3B [43 ] and follow the same training protocol. Specifically, each model is trained for five epochs on the same 1,000 samples from WISA-80K [44 ] . We evaluate on a disjoint set of 50 WISA-80K samples selected to provide approximately uniform coverage over the physical categories. We report visual quality and consistency metrics from VBench, together with its aggregate Quality Score. Results are reported in Table 5 .
Table 5. Component ablations (%, higher is better). Colors rank the four-decimal scores in Table 10 , which also expands the metric names.
The full model achieves the highest Quality Score and leads on all visual quality and consistency metrics. Removing cross-attention causes the largest score drop, closely followed by self-attention, highlighting the value of context and spatiotemporal interactions in the base model’s prediction. Removing either the refiner or hidden-state MLP reduces Quality Score and Overall Consistency, supporting refinement and fusion of hidden states across depths. The remaining ablations produce smaller but consistent score decreases. Removing noise-level conditioning or replacing the clean prediction x ^ 0 \hat{x}_0 x ^ 0 v θ v_\theta v θ
Conclusion
We introduced CORRFLOW, a self-correcting flow-matching framework that learns velocity residuals from the base model’s clean prediction, conditioning and hidden states. A lightweight refiner corrects the velocity within the sampling loop. Our theory identifies conditions under which feature reuse improves residual prediction, refiner training outperforms continued base-model training at matched compute, and the deployed correction lowers own-law velocity error. We demonstrate that CORRFLOW achieves state-of-the-art performance across multiple benchmarks including image generation, text-to-video generation, and image-to-video generation. We hope CORRFLOW will be a useful tool for the community to advance research in flow-matching-based generative modeling.
Frozen-Feature Analysis
This appendix establishes an exact population-risk separation for a linear readout of a frozen nonlinear feature relative to predictors that are affine in the raw inputs.
Setting and assumptions
We work in one active coordinate of the sample space. Samples, velocities, residuals and hidden readouts in this construction are scalar and use ordinary symbols. The conditioning vector is bold. Let c = ( c 1 , c 2 ) \mathbf{c}=(c_1,c_2) c = ( c 1 , c 2 ) p : = ψ ( c ) : = c 1 c 2 p:=\psi(\mathbf{c}):=c_1c_2 p := ψ ( c ) := c 1 c 2 δ > 0 \delta>0 δ > 0 σ ∈ [ 0 , 1 ] \sigma\in[0,1] σ ∈ [ 0 , 1 ]
s σ 2 : = ( 1 − σ ) 2 + σ 2 , ℓ σ : = 2 σ − 1 s σ 2 , τ σ : = σ s σ 2 . s_{\sigma}^{2}:=(1-\sigma)^{2}+\sigma^{2},\qquad\ell_{\sigma}:=\frac{2\sigma-1}{s_{\sigma}^{2}},\qquad\tau_{\sigma}:=\frac{\sigma}{s_{\sigma}^{2}}. s σ 2 := ( 1 − σ ) 2 + σ 2 , ℓ σ := s σ 2 2 σ − 1 , τ σ := s σ 2 σ . Assumption 1 (Conditional-Gaussian XOR witness). The variables c 1 , c 2 c_1,c_2 c 1 , c 2 { − 1 , 1 } \{-1,1\} { − 1 , 1 } ξ , ε ∼ N ( 0 , 1 ) \xi,\varepsilon\sim\mathcal{N}(0,1) ξ , ε ∼ N ( 0 , 1 ) ( c , σ ) (\mathbf{c},\sigma) ( c , σ )
x 0 : = δ p + ξ , x σ : = ( 1 − σ ) x 0 + σ ε , u : = ε − x 0 . x_0:=\delta p+\xi,\qquad x_{\sigma}:=(1-\sigma)x_0+\sigma\varepsilon,\qquad u:=\varepsilon-x_0. x 0 := δ p + ξ , x σ := ( 1 − σ ) x 0 + σ ε , u := ε − x 0 . Assumption 2 (Independent Noise Level). The variable σ ∼ q \sigma\sim q σ ∼ q ( c , ξ , ε ) (\mathbf{c},\xi,\varepsilon) ( c , ξ , ε ) q q q [ 0 , 1 ] [0,1] [ 0 , 1 ]
Let ( Ω , F , P ) (\Omega,\mathcal{F},\mathbb{P}) ( Ω , F , P ) W : = ( x σ , σ , c ) W:=(x_{\sigma},\sigma,\mathbf{c}) W := ( x σ , σ , c ) P W : = P ∘ W − 1 P_W:=\mathbb{P}\circ W^{-1} P W := P ∘ W − 1 (5) and the realized residual label are
v θ ( W ) : = ℓ σ x σ , x ^ 0 ( W ) : = x σ − σ v θ ( W ) , r : = u − v θ ( W ) . v_{\theta}(W):=\ell_{\sigma}x_{\sigma},\qquad\widehat{x}_0(W):=x_{\sigma}-\sigma v_{\theta}(W),\qquad r:=u-v_{\theta}(W). v θ ( W ) := ℓ σ x σ , x 0 ( W ) := x σ − σ v θ ( W ) , r := u − v θ ( W ) . Assumption 3 (Frozen base and product feature). The base is frozen with velocity v θ ( W ) = ℓ σ x σ v_{\theta}(W)=\ell_{\sigma}x_{\sigma} v θ ( W ) = ℓ σ x σ Φ ( W ) : = ( x ^ 0 ( W ) , h L ( W ) , σ , c ) \Phi(W):=(\widehat{x}_0(W),h_L(W),\sigma,\mathbf{c}) Φ ( W ) := ( x 0 ( W ) , h L ( W ) , σ , c ) h L ( W ) = c 1 c 2 h_L(W)=c_1c_2 h L ( W ) = c 1 c 2 G \mathcal{G} G
Define
u ⋆ ( W ) : = E [ u ∣ W ] , r ‾ ( W ) : = E [ r ∣ W ] = u ⋆ ( W ) − v θ ( W ) . u^{\star}(W):=\mathbb{E}[u\mid W],\qquad\overline{r}(W):=\mathbb{E}[r\mid W]=u^{\star}(W)-v_{\theta}(W). u ⋆ ( W ) := E [ u ∣ W ] , r ( W ) := E [ r ∣ W ] = u ⋆ ( W ) − v θ ( W ) . All expectations below are with respect to P \mathbb{P} P W W W f f f
R ( f ) : = E [ ( f ( W ) − r ) 2 ] , R i r r : = E [ ( r − r ‾ ( W ) ) 2 ] . \mathcal{R}(f):=\mathbb{E}[(f(W)-r)^2],\qquad\mathcal{R}_{\mathrm{irr}}:=\mathbb{E}[(r-\overline{r}(W))^2]. R ( f ) := E [( f ( W ) − r ) 2 ] , R irr := E [( r − r ( W ) ) 2 ] . In the notation of Section 4, P = P W P=P_W P = P W P ~ = P ∘ ( W , r ) − 1 \widetilde{P}=\mathbb{P}\circ(W,r)^{-1} P = P ∘ ( W , r ) − 1 v 0 v_0 v 0 v θ v_{\theta} v θ R P = R \mathcal{R}_P=\mathcal{R} R P = R
The one-parameter hidden-feature class, viewed as induced predictors on W W W
G : = { g b : b ∈ R } , g b ( W ) : = − b τ σ h L ( W ) . \mathcal{G}:=\{g_b:b\in\mathbb{R}\},\qquad g_b(W):=-b\tau_{\sigma}h_L(W). G := { g b : b ∈ R } , g b ( W ) := − b τ σ h L ( W ) . Each g b g_b g b Φ ( W ) \Phi(W) Φ ( W )
A : = { a 0 ( σ ) + a x ( σ ) x σ + a 1 ( σ ) c 1 + a 2 ( σ ) c 2 : a 0 , a x , a 1 , a 2 are Borel measurable, and the induced predictor lies in L 2 ( P W ) } . \mathcal{A}:=\left\{a_0(\sigma)+a_x(\sigma)x_{\sigma}+a_1(\sigma)c_1+a_2(\sigma)c_2:\ a_0,a_x,a_1,a_2\ \text{are Borel measurable, and the induced predictor lies in }L^2(P_W)\right\}. A := { a 0 ( σ ) + a x ( σ ) x σ + a 1 ( σ ) c 1 + a 2 ( σ ) c 2 : a 0 , a x , a 1 , a 2 are Borel measurable, and the induced predictor lies in L 2 ( P W ) } . The intended one-scalar raw-input comparator families at the same trainable-readout budget are subclasses of A \mathcal{A} A inf ∅ = + ∞ \inf\varnothing=+\infty inf ∅ = + ∞
For scalars a , b a,b a , b
cos 0 ( a , b ) : = { a b ∣ a ∣ ∣ b ∣ , a b ≠ 0 , 0 , a b = 0 , L cos ( a , b ) : = 1 − cos 0 ( a , b ) . \cos_0(a,b):=
\begin{cases}
\dfrac{ab}{|a||b|}, & ab\ne0,\\
0, & ab=0,
\end{cases}
\qquad L_{\cos}(a,b):=1-\cos_0(a,b). cos 0 ( a , b ) := ⎩ ⎨ ⎧ ∣ a ∣∣ b ∣ ab , 0 , ab = 0 , ab = 0 , L c o s ( a , b ) := 1 − cos 0 ( a , b ) . For λ = 0.1 \lambda=0.1 λ = 0.1
J λ ( b ) : = E [ ( g b ( W ) − r ) 2 + λ L cos ( g b ( W ) , r ) ] . \mathcal{J}_{\lambda}(b):=\mathbb{E}\left[(g_b(W)-r)^2+\lambda L_{\cos}(g_b(W),r)\right]. J λ ( b ) := E [ ( g b ( W ) − r ) 2 + λ L c o s ( g b ( W ) , r ) ] . Assumption 4 (Cosine-regularized population fit). For a stated tolerance ϵ o p t ≥ 0 \epsilon_{\mathrm{opt}}\ge0 ϵ opt ≥ 0 b ^ ∈ R \widehat{b}\in\mathbb{R} b ∈ R
J λ ( b ^ ) ≤ inf b ∈ R J λ ( b ) + ϵ o p t . \mathcal{J}_{\lambda}(\widehat{b})\le\inf_{b\in\mathbb{R}}\mathcal{J}_{\lambda}(b)+\epsilon_{\mathrm{opt}}. J λ ( b ) ≤ b ∈ R inf J λ ( b ) + ϵ opt . Define the structural constants
S : = E σ ∼ q [ τ σ 2 ] , Δ a f f : = δ 2 E σ ∼ q [ τ σ 2 s σ 2 s σ 2 + ( 1 − σ ) 2 δ 2 ] . S:=\mathbb{E}_{\sigma\sim q}[\tau_{\sigma}^{2}],\qquad\Delta_{\mathrm{aff}}:=\delta^{2}\mathbb{E}_{\sigma\sim q}\left[\tau_{\sigma}^{2}\frac{s_{\sigma}^{2}}{s_{\sigma}^{2}+(1-\sigma)^{2}\delta^{2}}\right]. S := E σ ∼ q [ τ σ 2 ] , Δ aff := δ 2 E σ ∼ q [ τ σ 2 s σ 2 + ( 1 − σ ) 2 δ 2 s σ 2 ] . Main result
Theorem 5 (Detailed frozen-feature advantage). Under Assumptions 1–4, the following statements hold.
For every σ < 1 \sigma< 1 σ < 1
x σ = s σ 2 1 − σ x ^ 0 , x_\sigma= \frac{s_\sigma^2}{1-\sigma}\widehat{x}_0, x σ = 1 − σ s σ 2 x 0 , whereas x ^ 0 = 0 \widehat{x}_0 = 0 x 0 = 0 at σ = 1 \sigma= 1 σ = 1 On the full interval,
x ^ 0 − x 0 = σ r . \widehat{x}_0-x_0=\sigma r. x 0 − x 0 = σ r . Thus r = ( x ^ 0 − x 0 ) / σ r=(\widehat{x}_0-x_0)/\sigma r = ( x 0 − x 0 ) / σ for σ > 0 \sigma>0 σ > 0 with the direct continuous endpoint values r = ε r=\varepsilon r = ε at σ = 0 \sigma=0 σ = 0 and r = − x 0 r=-x_0 r = − x 0 at σ = 1 \sigma=1 σ = 1 Moreover,
u ∗ ( W ) = ℓ σ x σ − τ σ δ p , r ‾ ( W ) = − τ σ δ p . u^*(W)=\ell_\sigma x_\sigma-\tau_\sigma\delta p,\qquad\overline{r}(W)=-\tau_\sigma\delta p. u ∗ ( W ) = ℓ σ x σ − τ σ δ p , r ( W ) = − τ σ δ p . The irreducible risk and structural constants satisfy
R i r r = E σ ∼ q [ s σ − 2 ] ∈ [ 1 , 2 ] , 0 < S < ∞ , 0 < Δ a f f < ∞ . \mathcal{R}_{\mathrm{irr}}=\mathbb{E}_{\sigma\sim q}[s_\sigma^{-2}]\in[1,2],\qquad0<S<\infty,\qquad0<\Delta_{\mathrm{aff}}<\infty. R irr = E σ ∼ q [ s σ − 2 ] ∈ [ 1 , 2 ] , 0 < S < ∞ , 0 < Δ aff < ∞. For every b ∈ R b\in\mathbb{R} b ∈ R
R ( g b ) = R i r r + S ( b − δ ) 2 , inf a ∈ A R ( a ) = R i r r + Δ a f f . \mathcal{R}(g_b)=\mathcal{R}_{\mathrm{irr}}+S(b-\delta)^2,\qquad\inf_{a\in\mathcal{A}}\mathcal{R}(a)=\mathcal{R}_{\mathrm{irr}}+\Delta_{\mathrm{aff}}. R ( g b ) = R irr + S ( b − δ ) 2 , a ∈ A inf R ( a ) = R irr + Δ aff . In particular, g δ = r ‾ g_\delta=\overline{r} g δ = r and R ( g δ ) = R i r r \mathcal{R}(g_\delta)=\mathcal{R}_{\mathrm{irr}} R ( g δ ) = R irr
The fitted coefficient satisfies
S ( b ^ − δ ) 2 ≤ 2 λ + ϵ o p t . S(\widehat{b}-\delta)^2\leq2\lambda+\epsilon_{\mathrm{opt}}. S ( b − δ ) 2 ≤ 2 λ + ϵ opt . If the static scalar condition
2 λ + ϵ o p t < Δ a f f 2\lambda+\epsilon_{\mathrm{opt}}<\Delta_{\mathrm{aff}} 2 λ + ϵ opt < Δ aff holds, then
R ( g b ^ ) < inf a ∈ A R ( a ) . \mathcal{R}(g_{\widehat{b}})<\inf_{a\in\mathcal{A}}\mathcal{R}(a). R ( g b ) < a ∈ A inf R ( a ) . For every intended equal-budget raw comparator class B ⊆ A \mathcal{B}\subseteq\mathcal{A} B ⊆ A the same condition also gives
R ( g b ^ ) < inf f ∈ B R ( f ) . \mathcal{R}(g_{\widehat{b}})<\inf_{f\in\mathcal{B}}\mathcal{R}(f). R ( g b ) < f ∈ B inf R ( f ) . For every finite d ≥ 1 d\geq1 d ≥ 1 the embedding ι d ( z ) = ( z , 0 , … , 0 ) \iota_d(z)=(z,0,\ldots,0) ι d ( z ) = ( z , 0 , … , 0 ) preserves the feature-risk curve, the raw-affine infimum, the cosine-regularized objective and the strict trained-risk margin.
These identities give the hidden-feature refiner’s population L 2 ( P ) L^2(\mathbb{P}) L 2 ( P ) δ \delta δ q q q λ \lambda λ ϵ o p t \epsilon_{\mathrm{opt}} ϵ opt S S S Δ a f f \Delta_{\mathrm{aff}} Δ aff
The product feature represents the conditional Bayes residual exactly. The noise law q q q S S S Δ a f f \Delta_{\mathrm{aff}} Δ aff λ \lambda λ ϵ o p t \epsilon_{\mathrm{opt}} ϵ opt
Theorem 5 is the detailed version of the main-text Theorem 2.
Proof
Proof. We prove the scalar conclusions in four steps. The finite-dimensional preservation statement is verified in the embedding paragraph below.
Step 1: Residual interface and conditional Bayes residual. Since
s σ 2 = 2 ( σ − 1 2 ) 2 + 1 2 ≥ 1 2 , s_{\sigma}^{2}=2\left(\sigma-\frac{1}{2}\right)^{2}+\frac{1}{2}\geq\frac{1}{2}, s σ 2 = 2 ( σ − 2 1 ) 2 + 2 1 ≥ 2 1 , all denominators below are nonzero. Moreover,
1 − σ ℓ σ = s σ 2 − σ ( 2 σ − 1 ) s σ 2 = 1 − σ s σ 2 , 1-\sigma\ell_{\sigma}=\frac{s_{\sigma}^{2}-\sigma(2\sigma-1)}{s_{\sigma}^{2}}=\frac{1-\sigma}{s_{\sigma}^{2}}, 1 − σ ℓ σ = s σ 2 s σ 2 − σ ( 2 σ − 1 ) = s σ 2 1 − σ , which proves the displayed clean-prediction formula and its inverse for σ < 1 \sigma<1 σ < 1
x ^ 0 = 1 − σ s σ 2 x σ , x σ = s σ 2 1 − σ x ^ 0 . \widehat{x}_{0}=\frac{1-\sigma}{s_{\sigma}^{2}}x_{\sigma},\qquad x_{\sigma}=\frac{s_{\sigma}^{2}}{1-\sigma}\widehat{x}_{0}. x 0 = s σ 2 1 − σ x σ , x σ = 1 − σ s σ 2 x 0 . Direct substitution gives x ^ 0 = x 0 \widehat{x}_{0}=x_{0} x 0 = x 0 σ = 0 \sigma=0 σ = 0 x ^ 0 = 0 \widehat{x}_{0}=0 x 0 = 0 σ = 1 \sigma=1 σ = 1
x ^ 0 − x 0 = x σ − σ v θ − x 0 = σ ( ε − x 0 − v θ ) = σ r . \widehat{x}_{0}-x_{0}=x_{\sigma}-\sigma v_{\theta}-x_{0}=\sigma(\varepsilon-x_{0}-v_{\theta})=\sigma r. x 0 − x 0 = x σ − σ v θ − x 0 = σ ( ε − x 0 − v θ ) = σ r . Consequently, r = ( x ^ 0 − x 0 ) / σ r=(\widehat{x}_{0}-x_{0})/\sigma r = ( x 0 − x 0 ) / σ σ > 0 \sigma>0 σ > 0 r = ε r=\varepsilon r = ε r = − x 0 r=-x_{0} r = − x 0
To compute the conditional mean, center the observation as
Y σ : = x σ − ( 1 − σ ) δ p = ( 1 − σ ) ξ + σ ε Y_{\sigma}:=x_{\sigma}-(1-\sigma)\delta p=(1-\sigma)\xi+\sigma\varepsilon Y σ := x σ − ( 1 − σ ) δ p = ( 1 − σ ) ξ + σ ε and introduce its orthogonal complement
N σ : = ( 1 − σ ) ε − σ ξ . N_{\sigma}:=(1-\sigma)\varepsilon-\sigma\xi. N σ := ( 1 − σ ) ε − σ ξ . Since p p p c c c
Y σ = x σ − ( 1 − σ ) δ p , x σ = Y σ + ( 1 − σ ) δ p , Y_{\sigma}=x_{\sigma}-(1-\sigma)\delta p,\qquad x_{\sigma}=Y_{\sigma}+(1-\sigma)\delta p, Y σ = x σ − ( 1 − σ ) δ p , x σ = Y σ + ( 1 − σ ) δ p , the tuples W = ( x σ , σ , c ) W=(x_{\sigma},\sigma,c) W = ( x σ , σ , c ) ( Y σ , σ , c ) (Y_{\sigma},\sigma,c) ( Y σ , σ , c ) ( σ , c ) (\sigma,c) ( σ , c ) ( Y σ / s σ , N σ / s σ ) (Y_{\sigma}/s_{\sigma},N_{\sigma}/s_{\sigma}) ( Y σ / s σ , N σ / s σ ) ( ξ , ε ) (\xi,\varepsilon) ( ξ , ε )
( Y σ / s σ N σ / s σ ) = 1 s σ ( 1 − σ σ − σ 1 − σ ) ( ξ ε ) . \begin{pmatrix}
Y_{\sigma}/s_{\sigma}\\
N_{\sigma}/s_{\sigma}
\end{pmatrix}
=
\frac{1}{s_{\sigma}}
\begin{pmatrix}
1-\sigma& \sigma\\
-\sigma& 1-\sigma
\end{pmatrix}
\begin{pmatrix}
\xi\\
\varepsilon
\end{pmatrix}. ( Y σ / s σ N σ / s σ ) = s σ 1 ( 1 − σ − σ σ 1 − σ ) ( ξ ε ) . Because ( ξ , ε ) (\xi,\varepsilon) ( ξ , ε ) ( σ , c ) (\sigma,c) ( σ , c ) t t t ( σ , c ) (\sigma,c) ( σ , c ) s σ 2 ≥ 1 / 2 s_{\sigma}^{2}\geq1/2 s σ 2 ≥ 1/2 W W W ( Y σ , σ , c ) (Y_{\sigma},\sigma,c) ( Y σ , σ , c )
E [ ξ ∣ W ] = 1 − σ s σ 2 Y σ , E [ ε ∣ W ] = σ s σ 2 Y σ . \mathbb{E}[\xi\mid W]=\frac{1-\sigma}{s_{\sigma}^{2}}Y_{\sigma},\qquad\mathbb{E}[\varepsilon\mid W]=\frac{\sigma}{s_{\sigma}^{2}}Y_{\sigma}. E [ ξ ∣ W ] = s σ 2 1 − σ Y σ , E [ ε ∣ W ] = s σ 2 σ Y σ . Using u = ε − ξ − δ p u=\varepsilon-\xi-\delta p u = ε − ξ − δ p 1 + ( 1 − σ ) ℓ σ = τ σ 1+(1-\sigma)\ell_{\sigma}=\tau_{\sigma} 1 + ( 1 − σ ) ℓ σ = τ σ
u ⋆ ( W ) = ℓ σ x σ − τ σ δ p , r ˉ ( W ) = u ⋆ ( W ) − v θ ( W ) = − τ σ δ p . u^{\star}(W)=\ell_{\sigma}x_{\sigma}-\tau_{\sigma}\delta p,\qquad\bar r(W)=u^{\star}(W)-v_{\theta}(W)=-\tau_{\sigma}\delta p. u ⋆ ( W ) = ℓ σ x σ − τ σ δ p , r ˉ ( W ) = u ⋆ ( W ) − v θ ( W ) = − τ σ δ p . Moreover, conditional on ( σ , c ) (\sigma,c) ( σ , c ) N σ / s σ N_{\sigma}/s_{\sigma} N σ / s σ Y σ / s σ Y_{\sigma}/s_{\sigma} Y σ / s σ W W W ( Y σ , σ , c ) (Y_{\sigma},\sigma,c) ( Y σ , σ , c )
r − r ˉ ( W ) = N σ s σ 2 , ( r − r ˉ ( W ) ) ∣ W ∼ N ( 0 , s σ − 2 ) . r-\bar r(W)=\frac{N_{\sigma}}{s_{\sigma}^{2}},\qquad(r-\bar r(W))\mid W\sim\mathcal{N}(0,s_{\sigma}^{-2}). r − r ˉ ( W ) = s σ 2 N σ , ( r − r ˉ ( W )) ∣ W ∼ N ( 0 , s σ − 2 ) . Therefore
R i r r = E σ ∼ q [ s σ − 2 ] . (16) \mathcal{R}_{\mathrm{irr}}=\mathbb{E}_{\sigma\sim q}[s_{\sigma}^{-2}]. \tag*{(16)} R irr = E σ ∼ q [ s σ − 2 ] . (16) Since s σ 2 ∈ [ 1 / 2 , 1 ] s_{\sigma}^{2}\in[1/2,1] s σ 2 ∈ [ 1/2 , 1 ] [ 1 , 2 ] [1,2] [ 1 , 2 ]
Step 2: Same-target decomposition and feature oracle. The following decomposition holds on any probability space with r ∈ L 2 ( P ) r \in L^2(\mathbb{P}) r ∈ L 2 ( P ) f ( W ) ∈ L 2 ( P ) f(W) \in L^2(\mathbb{P}) f ( W ) ∈ L 2 ( P ) r ˉ ( W ) : = E [ r ∣ W ] \bar{r}(W) := \mathbb{E}[r \mid W] r ˉ ( W ) := E [ r ∣ W ] η : = r − r ˉ ( W ) \eta:= r-\bar{r}(W) η := r − r ˉ ( W )
E [ r ˉ ( W ) 2 ] ≤ E [ r 2 ] . \mathbb{E}[\bar{r}(W)^2] \le\mathbb{E}[r^2]. E [ r ˉ ( W ) 2 ] ≤ E [ r 2 ] . Thus η \eta η f ( W ) − r ˉ ( W ) f(W)-\bar{r}(W) f ( W ) − r ˉ ( W ) L 2 L^2 L 2
E [ ( f ( W ) − r ˉ ( W ) ) η ] = E [ ( f ( W ) − r ˉ ( W ) ) E [ η ∣ W ] ] = 0. \mathbb{E}[(f(W)-\bar{r}(W))\eta] = \mathbb{E}[(f(W)-\bar{r}(W))\mathbb{E}[\eta\mid W]] = 0. E [( f ( W ) − r ˉ ( W )) η ] = E [( f ( W ) − r ˉ ( W )) E [ η ∣ W ]] = 0. Expanding the square therefore yields
R ( f ) = R i r r + E [ ( f ( W ) − r ˉ ( W ) ) 2 ] . \mathcal{R}(f) = \mathcal{R}_{\mathrm{irr}} + \mathbb{E}[(f(W)-\bar{r}(W))^2]. R ( f ) = R irr + E [( f ( W ) − r ˉ ( W ) ) 2 ] . This also proves the scalar case of equation 12. For a finite-dimensional X \mathcal{X} X r r r f ( W ) f(W) f ( W ) p 2 = 1 p^2=1 p 2 = 1 g b − r ˉ = ( δ − b ) τ σ p g_b-\bar{r}=(\delta-b)\tau_\sigma p g b − r ˉ = ( δ − b ) τ σ p
R ( g b ) = R i r r + S ( b − δ ) 2 . \mathcal{R}(g_b)=\mathcal{R}_{\mathrm{irr}}+S(b-\delta)^2. R ( g b ) = R irr + S ( b − δ ) 2 . In particular, g δ = r ˉ g_\delta=\bar{r} g δ = r ˉ 0 ≤ τ σ ≤ 2 0\le\tau_\sigma\le2 0 ≤ τ σ ≤ 2 S ≤ 4 S\le4 S ≤ 4 τ σ 2 \tau_\sigma^2 τ σ 2 ( 0 , 1 ] (0,1] ( 0 , 1 ] q q q S > 0 S>0 S > 0 K = [ 1 / 2 , 3 / 4 ] K=[1/2,3/4] K = [ 1/2 , 3/4 ] q ( K ) > 0 q(K)>0 q ( K ) > 0 τ σ 2 ≥ 1 / 4 \tau_\sigma^2\ge1/4 τ σ 2 ≥ 1/4 K K K
S ≥ 1 4 q ( K ) > 0. S\ge\frac{1}{4}q(K)>0. S ≥ 4 1 q ( K ) > 0. Step 3: Exact raw-affine projection. For each t ∈ [ 0 , 1 ] t\in[0,1] t ∈ [ 0 , 1 ] E t \mathbb{E}_t E t ( c , ξ , ε ) (c,\xi,\varepsilon) ( c , ξ , ε ) σ = t \sigma=t σ = t σ \sigma σ H H H
E [ H ] = ∫ [ 0 , 1 ] E t [ H ] q ( d t ) . \mathbb{E}[H]=\int_{[0,1]}\mathbb{E}_t[H]\,q(dt). E [ H ] = ∫ [ 0 , 1 ] E t [ H ] q ( d t ) . Fix t ∈ [ 0 , 1 ] t\in[0,1] t ∈ [ 0 , 1 ] m t : = 1 − t m_t:=1-t m t := 1 − t
V t : = s t 2 + m t 2 δ 2 , d t : = δ 2 τ t 2 s t 2 V t . V_t:=s_t^2+m_t^2\delta^2,\qquad d_t:=\delta^2\tau_t^2\frac{s_t^2}{V_t}. V t := s t 2 + m t 2 δ 2 , d t := δ 2 τ t 2 V t s t 2 . Under the fixed-noise law, the Gram matrix of ( 1 , c 1 , c 2 , x t ) (1,c_1,c_2,x_t) ( 1 , c 1 , c 2 , x t ) ( 1 , 1 , 1 , V t ) (1,1,1,V_t) ( 1 , 1 , 1 , V t )
E t [ x t ] = 0 , E t [ x t 2 ] = V t , E t [ x t c 1 ] = E t [ x t c 2 ] = E t [ c 1 c 2 ] = 0. \mathbb{E}_t[x_t]=0,\qquad\mathbb{E}_t[x_t^2]=V_t,\qquad\mathbb{E}_t[x_tc_1]=\mathbb{E}_t[x_tc_2]=\mathbb{E}_t[c_1c_2]=0. E t [ x t ] = 0 , E t [ x t 2 ] = V t , E t [ x t c 1 ] = E t [ x t c 2 ] = E t [ c 1 c 2 ] = 0. The target-design moments are
E t [ r ˉ ] = E t [ r ˉ c 1 ] = E t [ r ˉ c 2 ] = 0 , E t [ r ˉ x t ] = − m t τ t δ 2 , E t [ r ˉ 2 ] = τ t 2 δ 2 . \mathbb{E}_t[\bar{r}]=\mathbb{E}_t[\bar{r}c_1]=\mathbb{E}_t[\bar{r}c_2]=0,\qquad\mathbb{E}_t[\bar{r}x_t]=-m_t\tau_t\delta^2,\qquad\mathbb{E}_t[\bar{r}^2]=\tau_t^2\delta^2. E t [ r ˉ ] = E t [ r ˉ c 1 ] = E t [ r ˉ c 2 ] = 0 , E t [ r ˉ x t ] = − m t τ t δ 2 , E t [ r ˉ 2 ] = τ t 2 δ 2 . Consequently, for A α : = α 0 + α x x t + α 1 c 1 + α 2 c 2 A_\alpha:=\alpha_0+\alpha_xx_t+\alpha_1c_1+\alpha_2c_2 A α := α 0 + α x x t + α 1 c 1 + α 2 c 2
E t [ ( A α − r ˉ ) 2 ] = α 0 2 + α 1 2 + α 2 2 + V t ( α x + m t τ t δ 2 V t ) 2 + d t . \mathbb{E}_t[(A_\alpha-\bar{r})^2]=\alpha_0^2+\alpha_1^2+\alpha_2^2+V_t\left(\alpha_x+\frac{m_t\tau_t\delta^2}{V_t}\right)^2+d_t. E t [( A α − r ˉ ) 2 ] = α 0 2 + α 1 2 + α 2 2 + V t ( α x + V t m t τ t δ 2 ) 2 + d t . This identity also holds at t = 0 , 1 t=0,1 t = 0 , 1 d 0 = 0 d_0=0 d 0 = 0 d 1 = δ 2 d_1=\delta^2 d 1 = δ 2
a 0 ⋆ = a 1 ⋆ = a 2 ⋆ = 0 , a x ⋆ ( t ) = − m t τ t δ 2 V t . a_0^\star=a_1^\star=a_2^\star=0,\qquad a_x^\star(t)=-\frac{m_t\tau_t\delta^2}{V_t}. a 0 ⋆ = a 1 ⋆ = a 2 ⋆ = 0 , a x ⋆ ( t ) = − V t m t τ t δ 2 . Equivalently, let
a t ⋆ : = − m t τ t δ 2 V t x t , e t ⋆ : = r ˉ − a t ⋆ . a_t^\star:=-\frac{m_t\tau_t\delta^2}{V_t}x_t,\qquad e_t^\star:=\bar{r}-a_t^\star. a t ⋆ := − V t m t τ t δ 2 x t , e t ⋆ := r ˉ − a t ⋆ . The displayed moment identities give
E t [ e t ⋆ ] = E t [ e t ⋆ c 1 ] = E t [ e t ⋆ c 2 ] = E t [ e t ⋆ x t ] = 0. \mathbb{E}_t[e_t^\star]=\mathbb{E}_t[e_t^\star c_1]=\mathbb{E}_t[e_t^\star c_2]=\mathbb{E}_t[e_t^\star x_t]=0. E t [ e t ⋆ ] = E t [ e t ⋆ c 1 ] = E t [ e t ⋆ c 2 ] = E t [ e t ⋆ x t ] = 0. Thus a t ⋆ a_t^\star a t ⋆ L 2 L^2 L 2 r ˉ \bar{r} r ˉ
E t [ ( e t ⋆ ) 2 ] = d t . \mathbb{E}_t[(e_t^\star)^2]=d_t. E t [( e t ⋆ ) 2 ] = d t . Because V t ≥ s t 2 ≥ 1 / 2 V_t \ge s_t^2 \ge1/2 V t ≥ s t 2 ≥ 1/2 [ 0 , 1 ] [0,1] [ 0 , 1 ]
sup t ∈ [ 0 , 1 ] E t [ x t 2 ] = sup t ∈ [ 0 , 1 ] V t ≤ 1 + δ 2 . \sup_{t\in[0,1]} \mathbb{E}_t[x_t^2] = \sup_{t\in[0,1]} V_t \le1+\delta^2. t ∈ [ 0 , 1 ] sup E t [ x t 2 ] = t ∈ [ 0 , 1 ] sup V t ≤ 1 + δ 2 . Thus the induced predictor a ⋆ ( W ) = a x ⋆ ( σ ) x σ a^\star(W)=a_x^\star(\sigma)x_\sigma a ⋆ ( W ) = a x ⋆ ( σ ) x σ A \mathcal{A} A a ∈ A a\in\mathcal{A} a ∈ A a 0 , a x , a 1 , a 2 a_0,a_x,a_1,a_2 a 0 , a x , a 1 , a 2 a ( W ) 2 a(W)^2 a ( W ) 2
∫ [ 0 , 1 ] ( a 0 ( t ) 2 + a 1 ( t ) 2 + a 2 ( t ) 2 + V t a x ( t ) 2 ) q ( d t ) = E [ a ( W ) 2 ] < ∞ . \int_{[0,1]} \left(a_0(t)^2+a_1(t)^2+a_2(t)^2+V_ta_x(t)^2\right)q(dt)=\mathbb{E}[a(W)^2]<\infty. ∫ [ 0 , 1 ] ( a 0 ( t ) 2 + a 1 ( t ) 2 + a 2 ( t ) 2 + V t a x ( t ) 2 ) q ( d t ) = E [ a ( W ) 2 ] < ∞. Hence the conditional square-completion identity may be integrated with respect to q ( d t ) q(dt) q ( d t )
E [ ( a ( W ) − r ˉ ( W ) ) 2 ] = ∫ [ 0 , 1 ] [ a 0 ( t ) 2 + a 1 ( t ) 2 + a 2 ( t ) 2 + V t ( a x ( t ) + ( 1 − t ) τ t δ 2 V t ) 2 + d t ] q ( d t ) . \mathbb{E}\left[(a(W)-\bar r(W))^2\right]=\int_{[0,1]}\left[a_0(t)^2+a_1(t)^2+a_2(t)^2+V_t\left(a_x(t)+\frac{(1-t)\tau_t\delta^2}{V_t}\right)^2+d_t\right]q(dt). E [ ( a ( W ) − r ˉ ( W ) ) 2 ] = ∫ [ 0 , 1 ] [ a 0 ( t ) 2 + a 1 ( t ) 2 + a 2 ( t ) 2 + V t ( a x ( t ) + V t ( 1 − t ) τ t δ 2 ) 2 + d t ] q ( d t ) . Consequently,
E [ ( a ( W ) − r ˉ ( W ) ) 2 ] ≥ ∫ [ 0 , 1 ] d t q ( d t ) . \mathbb{E}\left[(a(W)-\bar r(W))^2\right]\ge\int_{[0,1]}d_tq(dt). E [ ( a ( W ) − r ˉ ( W ) ) 2 ] ≥ ∫ [ 0 , 1 ] d t q ( d t ) . The coefficient choice a ⋆ a^\star a ⋆ t ∈ [ 0 , 1 ] t\in[0,1] t ∈ [ 0 , 1 ]
inf a ∈ A E [ ( a ( W ) − r ˉ ( W ) ) 2 ] = E t ∼ q [ d t ] = Δ a f f . \inf_{a\in\mathcal{A}}\mathbb{E}\left[(a(W)-\bar r(W))^2\right]=\mathbb{E}_{t\sim q}[d_t]=\Delta_{\mathrm{aff}}. a ∈ A inf E [ ( a ( W ) − r ˉ ( W ) ) 2 ] = E t ∼ q [ d t ] = Δ aff . Applying the same-target decomposition gives
inf a ∈ A R ( a ) = R i r r + Δ a f f . \inf_{a\in\mathcal{A}}\mathcal{R}(a)=\mathcal{R}_{\mathrm{irr}}+\Delta_{\mathrm{aff}}. a ∈ A inf R ( a ) = R irr + Δ aff . The bounds 0 ≤ d t ≤ δ 2 τ t 2 ≤ 4 δ 2 0\le d_t\le\delta^2\tau_t^2\le4\delta^2 0 ≤ d t ≤ δ 2 τ t 2 ≤ 4 δ 2 d t d_t d t t > 0 t>0 t > 0 q q q Δ a f f > 0 \Delta_{\mathrm{aff}}>0 Δ aff > 0 K = [ 1 / 2 , 3 / 4 ] K=[1/2,3/4] K = [ 1/2 , 3/4 ]
d t ≥ δ 2 2 ( 4 + δ 2 ) , Δ a f f ≥ δ 2 2 ( 4 + δ 2 ) q ( K ) > 0. d_t\ge\frac{\delta^2}{2(4+\delta^2)},\qquad\Delta_{\mathrm{aff}}\ge\frac{\delta^2}{2(4+\delta^2)}q(K)>0. d t ≥ 2 ( 4 + δ 2 ) δ 2 , Δ aff ≥ 2 ( 4 + δ 2 ) δ 2 q ( K ) > 0. Equivalently, define the squared conditional correlation
ρ t 2 : = Corr t ( p , x t ) 2 = m t 2 δ 2 V t . \rho_t^2:=\operatorname{Corr}_t(p,x_t)^2=\frac{m_t^2\delta^2}{V_t}. ρ t 2 := Corr t ( p , x t ) 2 = V t m t 2 δ 2 . The conditional gap satisfies
d t = Var t ( r ˉ ) ( 1 − ρ t 2 ) , d_t=\operatorname{Var}_t(\bar r)(1-\rho_t^2), d t = Var t ( r ˉ ) ( 1 − ρ t 2 ) , so it is exactly the product-feature energy unexplained by the raw-affine projection. The raw-affine projection explains the fraction ρ t 2 \rho_t^2 ρ t 2
Step 4: Transfer the fitted objective to strict risk separation. The zero-stabilized cosine loss is measurable and lies in [ 0 , 2 ] [0,2] [ 0 , 2 ] b b b
R ( g b ) ≤ J λ ( b ) ≤ R ( g b ) + 2 λ . \mathcal{R}(g_b)\le\mathcal{J}_\lambda(b)\le\mathcal{R}(g_b)+2\lambda. R ( g b ) ≤ J λ ( b ) ≤ R ( g b ) + 2 λ . Using Assumption 4 and comparing with the oracle coefficient b = δ b=\delta b = δ
R ( g b ^ ) ≤ J λ ( b ^ ) ≤ inf b J λ ( b ) + ϵ o p t ≤ J λ ( δ ) + ϵ o p t ≤ R i r r + 2 λ + ϵ o p t . \begin{aligned}
\mathcal{R}(g_{\hat b})\le\mathcal{J}_\lambda(\hat b) \\
&\le\inf_b\mathcal{J}_\lambda(b)+\epsilon_{\mathrm{opt}} \\
&\le\mathcal{J}_\lambda(\delta)+\epsilon_{\mathrm{opt}} \\
&\le\mathcal{R}_{\mathrm{irr}}+2\lambda+\epsilon_{\mathrm{opt}}.
\end{aligned} R ( g b ^ ) ≤ J λ ( b ^ ) ≤ b inf J λ ( b ) + ϵ opt ≤ J λ ( δ ) + ϵ opt ≤ R irr + 2 λ + ϵ opt . Combining this bound with the exact feature-risk curve yields
S ( b ^ − δ ) 2 ≤ 2 λ + ϵ o p t , ∣ b ^ − δ ∣ ≤ 2 λ + ϵ o p t S . (17) S(\hat{b}-\delta)^2 \le2\lambda+\epsilon_{\mathrm{opt}}, \qquad|\hat{b}-\delta| \le\sqrt{\frac{2\lambda+\epsilon_{\mathrm{opt}}}{S}}. \tag*{(17)} S ( b ^ − δ ) 2 ≤ 2 λ + ϵ opt , ∣ b ^ − δ ∣ ≤ S 2 λ + ϵ opt . (17) For every b b b
inf a ∈ A R ( a ) − R ( g b ) = Δ a f f − S ( b − δ ) 2 . (18) \inf_{a\in\mathcal{A}} \mathcal{R}(a)-\mathcal{R}(g_b)=\Delta_{\mathrm{aff}}-S(b-\delta)^2. \tag*{(18)} a ∈ A inf R ( a ) − R ( g b ) = Δ aff − S ( b − δ ) 2 . (18) Since S > 0 S>0 S > 0 Δ a f f > 0 \Delta_{\mathrm{aff}}>0 Δ aff > 0
R ( g b ) < inf a ∈ A R ( a ) ⟺ ∣ b − δ ∣ < Δ a f f S . (19) \mathcal{R}(g_b)<\inf_{a\in\mathcal{A}}\mathcal{R}(a) \quad\Longleftrightarrow\quad|b-\delta|<\sqrt{\frac{\Delta_{\mathrm{aff}}}{S}}. \tag*{(19)} R ( g b ) < a ∈ A inf R ( a ) ⟺ ∣ b − δ ∣ < S Δ aff . (19) The condition 2 λ + ϵ o p t < Δ a f f 2\lambda+\epsilon_{\mathrm{opt}}<\Delta_{\mathrm{aff}} 2 λ + ϵ opt < Δ aff b ^ \hat{b} b ^
inf a ∈ A R ( a ) − R ( g b ^ ) ≥ Δ a f f − ( 2 λ + ϵ o p t ) . (20) \inf_{a\in\mathcal{A}}\mathcal{R}(a)-\mathcal{R}(g_{\hat{b}})\ge\Delta_{\mathrm{aff}}-(2\lambda+\epsilon_{\mathrm{opt}}). \tag*{(20)} a ∈ A inf R ( a ) − R ( g b ^ ) ≥ Δ aff − ( 2 λ + ϵ opt ) . (20) Under the stated scalar condition this margin is strictly positive. If B ⊆ A B\subseteq\mathcal{A} B ⊆ A inf a ∈ A R ( a ) ≤ inf f ∈ B R ( f ) \inf_{a\in\mathcal{A}}\mathcal{R}(a)\le\inf_{f\in B}\mathcal{R}(f) inf a ∈ A R ( a ) ≤ inf f ∈ B R ( f )
Useful specializations
Uniform noise law. If q q q [ 0 , 1 ] [0,1] [ 0 , 1 ]
z : = 2 σ − 1 , s σ 2 = 1 + z 2 2 , d σ = 1 2 d z . z:=2\sigma-1,\qquad s_{\sigma}^{2}=\frac{1+z^{2}}{2},\qquad\mathrm{d}\sigma=\frac{1}{2}\mathrm{d}z. z := 2 σ − 1 , s σ 2 = 2 1 + z 2 , d σ = 2 1 d z . Then
R i r r = ∫ − 1 1 1 1 + z 2 d z = π 2 , \mathcal{R}_{\mathrm{irr}}=\int_{-1}^{1}\frac{1}{1+z^{2}}\,\mathrm{d}z=\frac{\pi}{2}, R irr = ∫ − 1 1 1 + z 2 1 d z = 2 π , while the odd part of ( z + 1 ) 2 (z+1)^2 ( z + 1 ) 2
S = ∫ − 1 1 ( z + 1 ) 2 2 ( 1 + z 2 ) 2 d z = 1 2 ∫ − 1 1 1 1 + z 2 d z = π 4 . S=\int_{-1}^{1}\frac{(z+1)^2}{2(1+z^2)^2}\,\mathrm{d}z=\frac{1}{2}\int_{-1}^{1}\frac{1}{1+z^2}\,\mathrm{d}z=\frac{\pi}{4}. S = ∫ − 1 1 2 ( 1 + z 2 ) 2 ( z + 1 ) 2 d z = 2 1 ∫ − 1 1 1 + z 2 1 d z = 4 π . Thus
R i r r = π 2 , S = π 4 , R ( g b ) = π 2 + π 4 ( b − δ ) 2 . (21) \mathcal{R}_{\mathrm{irr}}=\frac{\pi}{2},\qquad S=\frac{\pi}{4},\qquad\mathcal{R}(g_b)=\frac{\pi}{2}+\frac{\pi}{4}(b-\delta)^2. \tag*{(21)} R irr = 2 π , S = 4 π , R ( g b ) = 2 π + 4 π ( b − δ ) 2 . (21) Consequently,
∣ b ^ − δ ∣ ≤ 2 π 2 λ + ϵ o p t . |\hat{b}-\delta|\le\frac{2}{\sqrt{\pi}}\sqrt{2\lambda+\epsilon_{\mathrm{opt}}}. ∣ b ^ − δ ∣ ≤ π 2 2 λ + ϵ opt . Finite-dimensional embedding. The witness uses one active sample coordinate so that every source of the risk gap is visible. Fix a finite d ≥ 1 d\ge1 d ≥ 1
ι d : R → R d , ι d ( z ) : = ( z , 0 , … , 0 ) , \iota_d:\mathbb{R}\to\mathbb{R}^{d},\qquad\iota_d(z):=(z,0,\ldots,0), ι d : R → R d , ι d ( z ) := ( z , 0 , … , 0 ) , r ( d ) : = ι d ( r ) , r ˉ ( d ) : = ι d ( r ˉ ) . r^{(d)}:=\iota_d(r),\qquad\bar{r}^{(d)}:=\iota_d(\bar{r}). r ( d ) := ι d ( r ) , r ˉ ( d ) := ι d ( r ˉ ) . For a scalar predictor f f f f ( d ) : = ι d ∘ f f^{(d)}:=\iota_d\circ f f ( d ) := ι d ∘ f
R d ( f ( d ) ) : = E [ ∥ f ( d ) ( W ) − r ( d ) ∥ 2 2 ] . \mathcal{R}_d(f^{(d)}):=\mathbb{E}\left[\left\|f^{(d)}(W)-r^{(d)}\right\|_2^2\right]. R d ( f ( d ) ) := E [ f ( d ) ( W ) − r ( d ) 2 2 ] . The embedded classes are
G d : = { g b ( d ) : b ∈ R } , A d : = { a ( d ) : a ∈ A } . \mathcal{G}_d:=\{g_b^{(d)}:b\in\mathbb{R}\},\qquad\mathcal{A}_d:=\{a^{(d)}:a\in\mathcal{A}\}. G d := { g b ( d ) : b ∈ R } , A d := { a ( d ) : a ∈ A } . Since ι d \iota_d ι d
E [ r ( d ) ∣ W ] = r ˉ ( d ) , R d ( g b ( d ) ) = R ( g b ) , R d ( a ( d ) ) = R ( a ) . \mathbb{E}[r^{(d)}\mid W]=\bar{r}^{(d)},\qquad\mathcal{R}_d(g_b^{(d)})=\mathcal{R}(g_b),\qquad\mathcal{R}_d(a^{(d)})=\mathcal{R}(a). E [ r ( d ) ∣ W ] = r ˉ ( d ) , R d ( g b ( d ) ) = R ( g b ) , R d ( a ( d ) ) = R ( a ) . The map a ↦ a ( d ) a \mapsto a^{(d)} a ↦ a ( d ) A \mathcal{A} A A d \mathcal{A}_d A d
inf a ( d ) ∈ A d R d ( a ( d ) ) = R i r r + Δ a f f , g δ ( d ) = r ˉ ( d ) . \inf_{a^{(d)} \in\mathcal{A}_d} R_d(a^{(d)}) = R_{\mathrm{irr}} + \Delta_{\mathrm{aff}}, \qquad g_\delta^{(d)} = \bar{r}^{(d)}. a ( d ) ∈ A d inf R d ( a ( d ) ) = R irr + Δ aff , g δ ( d ) = r ˉ ( d ) . For y , z ∈ R d \mathbf{y}, \mathbf{z} \in\mathbb{R}^d y , z ∈ R d
cos 0 , d ( y , z ) : = { ⟨ y , z ⟩ / ( ∥ y ∥ 2 ∥ z ∥ 2 ) , ∥ y ∥ 2 ∥ z ∥ 2 > 0 , 0 , ∥ y ∥ 2 ∥ z ∥ 2 = 0 , \cos_{0,d}(\mathbf{y},\mathbf{z}) := \begin{cases} \langle\mathbf{y},\mathbf{z}\rangle/(\|\mathbf{y}\|_2\|\mathbf{z}\|_2), & \|\mathbf{y}\|_2\|\mathbf{z}\|_2 > 0, \\ 0, & \|\mathbf{y}\|_2\|\mathbf{z}\|_2 = 0, \end{cases} cos 0 , d ( y , z ) := { ⟨ y , z ⟩ / ( ∥ y ∥ 2 ∥ z ∥ 2 ) , 0 , ∥ y ∥ 2 ∥ z ∥ 2 > 0 , ∥ y ∥ 2 ∥ z ∥ 2 = 0 , and L cos , d : = 1 − cos 0 , d L_{\cos,d} := 1-\cos_{0,d} L c o s , d := 1 − cos 0 , d
J λ , d ( b ) : = E [ ∥ g b ( d ) ( W ) − r ( d ) ∥ 2 2 + λ L cos , d ( g b ( d ) ( W ) , r ( d ) ) ] . \mathcal{J}_{\lambda,d}(b) := \mathbb{E}\left[\|g_b^{(d)}(W)-r^{(d)}\|_2^2+\lambda L_{\cos,d}(g_b^{(d)}(W),r^{(d)})\right]. J λ , d ( b ) := E [ ∥ g b ( d ) ( W ) − r ( d ) ∥ 2 2 + λ L c o s , d ( g b ( d ) ( W ) , r ( d ) ) ] . The same isometry gives J λ , d ( b ) = J λ ( b ) \mathcal{J}_{\lambda,d}(b)=\mathcal{J}_{\lambda}(b) J λ , d ( b ) = J λ ( b ) b b b 2 λ + ϵ o p t < Δ a f f 2\lambda+\epsilon_{\mathrm{opt}}<\Delta_{\mathrm{aff}} 2 λ + ϵ opt < Δ aff
R d ( g b ( d ) ) < inf a ( d ) ∈ A d R d ( a ( d ) ) , R_d(g_b^{(d)})<\inf_{a^{(d)}\in\mathcal{A}_d}R_d(a^{(d)}), R d ( g b ( d ) ) < a ( d ) ∈ A d inf R d ( a ( d ) ) , with margin at least Δ a f f − ( 2 λ + ϵ o p t ) \Delta_{\mathrm{aff}}-(2\lambda+\epsilon_{\mathrm{opt}}) Δ aff − ( 2 λ + ϵ opt ) d d d
Matched-Resource Comparison: Proof
This section proves that a refiner’s tangent risk-reduction advantage transfers to its trained output when the advantage exceeds the two perturbation bounds, as stated in Theorem 3. Unsubscripted norms of random correction maps are those of L 2 ( P ; X ) L^2(P;\mathcal{X}) L 2 ( P ; X )
Full setting and local certificate
Let X \mathcal{X} X W = ( X Σ , Σ , C ) W=(X_\Sigma,\Sigma,C) W = ( X Σ , Σ , C ) P P P U U U v 0 v_0 v 0
r = U − v 0 ( W ) , P ~ = Law ( W , r ) , r ˉ ( W ) = E P ~ [ r ∣ W ] , s = ∥ r ∥ L 2 ( P ~ ; X ) < ∞ . (22) r=U-v_0(W), \qquad\widetilde{P}=\operatorname{Law}(W,r), \qquad\bar{r}(W)=\mathbb{E}_{\widetilde{P}}[r\mid W], \qquad s=\|r\|_{L^2(\widetilde{P};\mathcal{X})}<\infty. \tag*{(22)} r = U − v 0 ( W ) , P = Law ( W , r ) , r ˉ ( W ) = E P [ r ∣ W ] , s = ∥ r ∥ L 2 ( P ; X ) < ∞. (22) For f ∈ L 2 ( P ; X ) f\in L^2(P;\mathcal{X}) f ∈ L 2 ( P ; X )
R P ( f ) = E P ~ ∥ r − f ( W ) ∥ 2 . (23) R_P(f)=\mathbb{E}_{\widetilde{P}}\|r-f(W)\|^2. \tag*{(23)} R P ( f ) = E P ∥ r − f ( W ) ∥ 2 . (23) The refiner and continued-base correction families are
F r ( α ) ( W ) = g ϕ 0 + α ( Z ( W ) ) , F b ( α ) ( W ) = v θ 0 + α ( W ) − v 0 ( W ) , (24) F_r(\alpha)(W)=g_{\phi_0+\alpha}(Z(W)), \qquad F_b(\alpha)(W)=v_{\theta_0+\alpha}(W)-v_0(W), \tag*{(24)} F r ( α ) ( W ) = g ϕ 0 + α ( Z ( W )) , F b ( α ) ( W ) = v θ 0 + α ( W ) − v 0 ( W ) , (24) where F j ( 0 ) = 0 F_j(0)=0 F j ( 0 ) = 0 Z ( W ) Z(W) Z ( W ) § 3.2 .
A componentwise ledger C = ( C t r a i n , C s t a t e , C d e p l o y ) C=(C_{\mathrm{train}},C_{\mathrm{state}},C_{\mathrm{deploy}}) C = ( C train , C state , C deploy ) j ∈ { r , b } j\in\{r,b\} j ∈ { r , b } c j > 0 c_j>0 c j > 0
T j = ⌊ C t r a i n c j ⌋ . (25) T_j=\left\lfloor\frac{C_{\mathrm{train}}}{c_j}\right\rfloor. \tag*{(25)} T j = ⌊ c j C train ⌋ . (25) Both runs must fit the declared optimizer-state and inference budgets. Their population objectives are
L b ( α ) = 1 2 R P ( F b ( α ) ) , L r ( α ) = 1 2 R P ( F r ( α ) ) + λ A ( α ) , λ ∠ = 0.1 , (26) L_b(\alpha)=\tfrac{1}{2}R_P(F_b(\alpha)), \qquad L_r(\alpha)=\tfrac{1}{2}R_P(F_r(\alpha))+\lambda\mathcal{A}(\alpha), \qquad\lambda_{\angle}=0.1, \tag*{(26)} L b ( α ) = 2 1 R P ( F b ( α )) , L r ( α ) = 2 1 R P ( F r ( α )) + λ A ( α ) , λ ∠ = 0.1 , (26) where A \mathcal{A} A η > 0 \eta>0 η > 0
α j , 0 η = 0 , α j , t + 1 η = α j , t η − η ∇ L j ( α j , t η ) , f j , η = F j ( α j , T j η ) . (27) \alpha_{j,0}^{\eta}=0, \qquad\alpha_{j,t+1}^{\eta}=\alpha_{j,t}^{\eta}-\eta\nabla L_j(\alpha_{j,t}^{\eta}), \qquad f_{j,\eta}=F_j(\alpha_{j,T_j}^{\eta}). \tag*{(27)} α j , 0 η = 0 , α j , t + 1 η = α j , t η − η ∇ L j ( α j , t η ) , f j , η = F j ( α j , T j η ) . (27) At the checkpoint, let
Φ j = J j ( 0 ) , H j = Φ j ∗ Φ j , b j = Φ j ∗ r ˉ , (28) \Phi_j=J_j(0), \qquad H_j=\Phi_j^{*}\Phi_j, \qquad b_j=\Phi_j^{*}\bar{r}, \tag*{(28)} Φ j = J j ( 0 ) , H j = Φ j ∗ Φ j , b j = Φ j ∗ r ˉ , (28) where J j ( α ) = D F j ( α ) J_j(\alpha)=DF_j(\alpha) J j ( α ) = D F j ( α )
β j , t + 1 η = ( I − η H j ) β j , t η + η b j , β j , 0 η = 0. (29) \beta_{j,t+1}^{\eta}=(I-\eta H_j)\beta_{j,t}^{\eta}+\eta b_j, \qquad\beta_{j,0}^{\eta}=0. \tag*{(29)} β j , t + 1 η = ( I − η H j ) β j , t η + η b j , β j , 0 η = 0. (29) Assumption 5 (Finite-budget local certificate). For each arm there are finite R j , K j , Λ j ≥ 0 R_j,K_j,\Lambda_j \ge0 R j , K j , Λ j ≥ 0 F j F_j F j ∥ α ∥ ≤ R j \|\alpha\| \le R_j ∥ α ∥ ≤ R j
∥ J j ( α ) ∥ o p ≤ K j , ∥ J j ( α ) − J j ( α ′ ) ∥ o p ≤ Λ j ∥ α − α ′ ∥ . (30) \|J_j(\alpha)\|_{\mathrm{op}} \le K_j,\qquad\|J_j(\alpha)-J_j(\alpha')\|_{\mathrm{op}} \le\Lambda_j\|\alpha-\alpha'\|. \tag*{(30)} ∥ J j ( α ) ∥ op ≤ K j , ∥ J j ( α ) − J j ( α ′ ) ∥ op ≤ Λ j ∥ α − α ′ ∥. (30) On the refiner ball, A \mathcal{A} A ∥ ∇ A ( α ) ∥ ≤ G ∠ < ∞ \|\nabla\mathcal{A}(\alpha)\| \le G_{\angle} < \infty ∥∇ A ( α ) ∥ ≤ G ∠ < ∞
q b = 0 , q r = λ ∠ G ∠ , q_b=0,\qquad q_r=\lambda_{\angle}G_{\angle}, q b = 0 , q r = λ ∠ G ∠ , Γ j = K j ( s + K j R j ) + q j , ω j = Λ j R j ( s + K j R j ) + 1 2 K j Λ j R j 2 + q j . (31) \Gamma_j=K_j(s+K_jR_j)+q_j,\qquad\omega_j=\Lambda_jR_j(s+K_jR_j)+\frac{1}{2}K_j\Lambda_jR_j^2+q_j. \tag*{(31)} Γ j = K j ( s + K j R j ) + q j , ω j = Λ j R j ( s + K j R j ) + 2 1 K j Λ j R j 2 + q j . (31) Each E j ( C ) \mathcal{E}_j(C) E j ( C ) E 0 ( C ) ⊂ ( 0 , ∞ ) \mathcal{E}_0(C)\subset(0,\infty) E 0 ( C ) ⊂ ( 0 , ∞ ) ζ s t a b ∈ ( 0 , 2 ) \zeta_{\mathrm{stab}}\in(0,2) ζ stab ∈ ( 0 , 2 ) η ∈ E j ( C ) \eta\in\mathcal{E}_j(C) η ∈ E j ( C )
η ∥ H j ∥ o p ≤ 2 − ζ s t a b , η T j Γ j ≤ R j . (32) \eta\|H_j\|_{\mathrm{op}}\le2-\zeta_{\mathrm{stab}},\qquad\eta T_j\Gamma_j\le R_j. \tag*{(32)} η ∥ H j ∥ op ≤ 2 − ζ stab , η T j Γ j ≤ R j . (32) The Jacobian bound K j K_j K j Λ j \Lambda_j Λ j q j q_j q j R j R_j R j T j T_j T j
Define the actual and tangent risk reductions by
Δ j ( C , η ) = R P ( 0 ) − R P ( f j , η ) , Δ j ( C ) = sup η ∈ E j ( C ) Δ j ( C , η ) , \Delta_j(C,\eta)=\mathcal{R}_P(0)-\mathcal{R}_P(f_{j,\eta}),\qquad\Delta_j(C)=\sup_{\eta\in\mathcal{E}_j(C)}\Delta_j(C,\eta), Δ j ( C , η ) = R P ( 0 ) − R P ( f j , η ) , Δ j ( C ) = η ∈ E j ( C ) sup Δ j ( C , η ) , Δ j l i n ( C , η ) = b j ∗ T H j T [ I − ( I − η H j ) 2 T j ] b j , Δ j l i n ( C ) = sup η ∈ E j ( C ) Δ j l i n ( C , η ) . (33) \Delta_j^{\mathrm{lin}}(C,\eta)=b_j^{*\mathsf{T}}H_j^{\mathsf{T}}\left[I-(I-\eta H_j)^{2T_j}\right]b_j,\qquad\Delta_j^{\mathrm{lin}}(C)=\sup_{\eta\in\mathcal{E}_j(C)}\Delta_j^{\mathrm{lin}}(C,\eta). \tag*{(33)} Δ j lin ( C , η ) = b j ∗ T H j T [ I − ( I − η H j ) 2 T j ] b j , Δ j lin ( C ) = η ∈ E j ( C ) sup Δ j lin ( C , η ) . (33) Finally, put
ε j ( C , η ) = 1 2 Λ j R j 2 + K j η T j ω j , e j ( C ) = sup η ∈ E j ( C ) 2 ( s + K j R j ) ε j ( C , η ) . (34) \varepsilon_j(C,\eta)=\frac{1}{2}\Lambda_jR_j^2+K_j\eta T_j\omega_j,\qquad e_j(C)=\sup_{\eta\in\mathcal{E}_j(C)}2(s+K_jR_j)\varepsilon_j(C,\eta). \tag*{(34)} ε j ( C , η ) = 2 1 Λ j R j 2 + K j η T j ω j , e j ( C ) = η ∈ E j ( C ) sup 2 ( s + K j R j ) ε j ( C , η ) . (34) Finite-budget perturbation and separation Proposition 6 (Finite-budget tangent certificate). Under Assumption 5, for every j ∈ { r , b } j\in\{r,b\} j ∈ { r , b } η ∈ E j ( C ) \eta\in\mathcal{E}_j(C) η ∈ E j ( C )
∥ α j , t η ∥ , ∥ β j , t η ∥ ≤ t η Γ j ≤ R j ( 0 ≤ t ≤ T j ) , (35) \|\alpha_{j,t}^{\eta}\|,\|\beta_{j,t}^{\eta}\|\le t\eta\Gamma_j\le R_j\qquad(0\le t\le T_j), \tag*{(35)} ∥ α j , t η ∥ , ∥ β j , t η ∥ ≤ t η Γ j ≤ R j ( 0 ≤ t ≤ T j ) , (35) and
∥ F j ( α j , T j η ) − Φ j β j , T j η ∥ ≤ ε j ( C , η ) . (36) \|F_j(\alpha_{j,T_j}^{\eta})-\Phi_j\beta_{j,T_j}^{\eta}\|\le\varepsilon_j(C,\eta). \tag*{(36)} ∥ F j ( α j , T j η ) − Φ j β j , T j η ∥ ≤ ε j ( C , η ) . (36) The tangent reduction is exactly
R P ( 0 ) − R P ( Φ j β j , T j η ) = b j ∗ T H j T [ I − ( I − η H j ) 2 T j ] b j . (37) \mathcal{R}_P(0)-\mathcal{R}_P(\Phi_j\beta_{j,T_j}^{\eta})=b_j^{*\mathsf{T}}H_j^{\mathsf{T}}\left[I-(I-\eta H_j)^{2T_j}\right]b_j. \tag*{(37)} R P ( 0 ) − R P ( Φ j β j , T j η ) = b j ∗ T H j T [ I − ( I − η H j ) 2 T j ] b j . (37) Consequently,
∣ Δ j ( C , η ) − Δ j l i n ( C , η ) ∣ ≤ 2 ( s + K j R j ) ε j ( C , η ) , (38) |\Delta_j(C,\eta)-\Delta_j^{\mathrm{lin}}(C,\eta)|\le2(s+K_jR_j)\varepsilon_j(C,\eta), \tag*{(38)} ∣ Δ j ( C , η ) − Δ j lin ( C , η ) ∣ ≤ 2 ( s + K j R j ) ε j ( C , η ) , (38) and
∣ Δ j ( C ) − Δ j l i n ( C ) ∣ ≤ e j ( C ) . (39) |\Delta_j(C)-\Delta_j^{\mathrm{lin}}(C)|\le e_j(C). \tag*{(39)} ∣ Δ j ( C ) − Δ j lin ( C ) ∣ ≤ e j ( C ) . (39) The actual and tangent optimized reductions are attained on their respective candidate sets.
Proof. We derive the tangent reduction and control the perturbation caused by the actual updates through the full charged horizon.
Step 1: Establish the common risk geometry. We first record the common risk geometry. With η r = r − r ˉ ( W ) \eta_r=r-\bar r(W) η r = r − r ˉ ( W ) E [ η r ∣ W ] = 0 \mathbb{E}[\eta_r\mid W]=0 E [ η r ∣ W ] = 0 h ∈ L 2 ( P ; X ) h\in L^2(P;\mathcal{X}) h ∈ L 2 ( P ; X )
E ⟨ h ( W ) , η r ⟩ = E ⟨ h ( W ) , E [ η r ∣ W ] ⟩ = 0. (40) \mathbb{E}\langle h(W),\eta_r\rangle=\mathbb{E}\langle h(W),\mathbb{E}[\eta_r\mid W]\rangle=0. \tag*{(40)} E ⟨ h ( W ) , η r ⟩ = E ⟨ h ( W ) , E [ η r ∣ W ]⟩ = 0. (40) Expanding the square yields
R P ( h ) = E ∥ r − r ˉ ∥ 2 + ∥ r ˉ − h ∥ 2 , R P ( 0 ) − R P ( h ) = 2 ⟨ r ˉ , h ⟩ − ∥ h ∥ 2 . (41) \mathcal{R}_P(h)=\mathbb{E}\|r-\bar r\|^2+\|\bar r-h\|^2,\qquad\mathcal{R}_P(0)-\mathcal{R}_P(h)=2\langle\bar r,h\rangle-\|h\|^2. \tag*{(41)} R P ( h ) = E ∥ r − r ˉ ∥ 2 + ∥ r ˉ − h ∥ 2 , R P ( 0 ) − R P ( h ) = 2 ⟨ r ˉ , h ⟩ − ∥ h ∥ 2 . (41) Jensen’s inequality also gives ∥ r ˉ ∥ ≤ s \|\bar r\|\le s ∥ r ˉ ∥ ≤ s
Step 2: Keep the actual and tangent iterates in the certified ball. For an in-ball parameter, the line-segment formula and F j ( 0 ) = 0 F_j(0)=0 F j ( 0 ) = 0
F j ( α ) = ∫ 0 1 J j ( u α ) [ α ] d u , ∥ F j ( α ) ∥ ≤ K j ∥ α ∥ . (42) F_j(\alpha)=\int_0^1 J_j(u\alpha)[\alpha]\,du,\qquad\|F_j(\alpha)\|\le K_j\|\alpha\|. \tag*{(42)} F j ( α ) = ∫ 0 1 J j ( uα ) [ α ] d u , ∥ F j ( α ) ∥ ≤ K j ∥ α ∥. (42) Let
ϑ j ( α ) = { λ ∠ ∇ A ( α ) , j = r , 0 , j = b . (43) \vartheta_j(\alpha)=
\begin{cases}
\lambda_{\angle}\nabla A(\alpha), & j=r,\\
0, & j=b.
\end{cases} \tag*{(43)} ϑ j ( α ) = { λ ∠ ∇ A ( α ) , 0 , j = r , j = b . (43) Differentiating equation 41 through F j F_j F j
∇ L j ( α ) = J j ( α ) ∗ ( F j ( α ) − r ˉ ) + ϑ j ( α ) . (44) \nabla L_j(\alpha)=J_j(\alpha)^*(F_j(\alpha)-\bar r)+\vartheta_j(\alpha). \tag*{(44)} ∇ L j ( α ) = J j ( α ) ∗ ( F j ( α ) − r ˉ ) + ϑ j ( α ) . (44) Thus ∥ ∇ L j ( α ) ∥ ≤ Γ j \|\nabla L_j(\alpha)\|\le\Gamma_j ∥∇ L j ( α ) ∥ ≤ Γ j ∥ α ∥ ≤ R j \|\alpha\|\le R_j ∥ α ∥ ≤ R j ∥ Φ j ∥ o p ≤ K j \|\Phi_j\|_{\mathrm{op}}\le K_j ∥ Φ j ∥ op ≤ K j
∥ H j β − b j ∥ = ∥ Φ j ∗ ( Φ j β − r ˉ ) ∥ ≤ K j ( K j R j + s ) ≤ Γ j (45) \|H_j\beta-b_j\|=\|\Phi_j^*(\Phi_j\beta-\bar r)\|\le K_j(K_jR_j+s)\le\Gamma_j \tag*{(45)} ∥ H j β − b j ∥ = ∥ Φ j ∗ ( Φ j β − r ˉ ) ∥ ≤ K j ( K j R j + s ) ≤ Γ j (45) whenever ∥ β ∥ ≤ R j \|\beta\|\le R_j ∥ β ∥ ≤ R j
∥ α j , t + 1 η ∥ , ∥ β j , t + 1 η ∥ ≤ ( t + 1 ) η Γ j . (46) \|\alpha_{j,t+1}^{\eta}\|,\|\beta_{j,t+1}^{\eta}\|\le(t+1)\eta\Gamma_j. \tag*{(46)} ∥ α j , t + 1 η ∥ , ∥ β j , t + 1 η ∥ ≤ ( t + 1 ) η Γ j . (46) The pre-run radius condition makes the right side at most R j R_j R j T j T_j T j
Step 3: Solve the tangent recursion and compute its risk reduction. We next solve the tangent recursion. The operator H j = Φ j ∗ Φ j H_j=\Phi_j^*\Phi_j H j = Φ j ∗ Φ j b j ⊥ ker H j b_j\perp\ker H_j b j ⊥ ker H j ker H j = ker Φ j \ker H_j=\ker\Phi_j ker H j = ker Φ j H j u k = λ k u k H_ju_k=\lambda_ku_k H j u k = λ k u k
∣ 1 − η λ k ∣ ≤ 1 , ∥ ( I − η H j ) t ∥ o p ≤ 1. (47) |1-\eta\lambda_k|\le1,\qquad\|(I-\eta H_j)^t\|_{\mathrm{op}}\le1. \tag*{(47)} ∣1 − η λ k ∣ ≤ 1 , ∥ ( I − η H j ) t ∥ op ≤ 1. (47) The null eigenspaces have zero loading. A finite geometric sum on every positive eigenspace therefore yields
β j , t η = H j † [ I − ( I − η H j ) t ] b j . (48) \beta_{j,t}^{\eta}=H_j^{\dagger}[I-(I-\eta H_j)^t]b_j. \tag*{(48)} β j , t η = H j † [ I − ( I − η H j ) t ] b j . (48) Substitution into equation 41 gives
∑ k : λ k > 0 ∣ ⟨ u k , b j ⟩ ∣ 2 λ k [ 1 − ( 1 − η λ k ) 2 T j ] , (49) \sum_{k:\lambda_k>0}\frac{|\langle u_k,b_j\rangle|^2}{\lambda_k}\left[1-(1-\eta\lambda_k)^{2T_j}\right], \tag*{(49)} k : λ k > 0 ∑ λ k ∣ ⟨ u k , b j ⟩ ∣ 2 [ 1 − ( 1 − η λ k ) 2 T j ] , (49) which is exactly equation 37.
Step 4: Bound the parameter and output perturbations. It remains to transfer the tangent formula to the actual run. The Jacobian Lipschitz condition and the integral formula give
∥ F j ( α ) − Φ j α ∥ ≤ 1 2 Λ j ∥ α ∥ 2 . (50) \|F_j(\alpha)-\Phi_j\alpha\|\le\frac{1}{2}\Lambda_j\|\alpha\|^2. \tag*{(50)} ∥ F j ( α ) − Φ j α ∥ ≤ 2 1 Λ j ∥ α ∥ 2 . (50) At the same parameter define ξ j ( α ) = ∇ L j ( α ) − ( H j α − b j ) \xi_j(\alpha)=\nabla L_j(\alpha)-(H_j\alpha-b_j) ξ j ( α ) = ∇ L j ( α ) − ( H j α − b j )
ξ j ( α ) = [ J j ( α ) − Φ j ] ∗ ( F j ( α ) − r ˉ ) + Φ j ∗ ( F j ( α ) − Φ j α ) + ϑ j ( α ) . (51) \xi_j(\alpha)=[J_j(\alpha)-\Phi_j]^*(F_j(\alpha)-\bar r)
+\Phi_j^*(F_j(\alpha)-\Phi_j\alpha)+\vartheta_j(\alpha). \tag*{(51)} ξ j ( α ) = [ J j ( α ) − Φ j ] ∗ ( F j ( α ) − r ˉ ) + Φ j ∗ ( F j ( α ) − Φ j α ) + ϑ j ( α ) . (51) For ∥ α ∥ ≤ R j \|\alpha\|\le R_j ∥ α ∥ ≤ R j Λ j R j ( s + K j R j ) \Lambda_jR_j(s+K_jR_j) Λ j R j ( s + K j R j ) 1 2 K j Λ j R j 2 \frac{1}{2}K_j\Lambda_jR_j^2 2 1 K j Λ j R j 2 q j q_j q j
∥ ξ j ( α ) ∥ ≤ ω j . (52) \|\xi_j(\alpha)\|\le\omega_j. \tag*{(52)} ∥ ξ j ( α ) ∥ ≤ ω j . (52) Writing d j , t = α j , t η − β j , t η d_{j,t}=\alpha_{j,t}^{\eta}-\beta_{j,t}^{\eta} d j , t = α j , t η − β j , t η
d j , t + 1 = ( I − η H j ) d j , t − η ξ j ( α j , t η ) , d j , 0 = 0. (53) d_{j,t+1}=(I-\eta H_j)d_{j,t}-\eta\xi_j(\alpha_{j,t}^{\eta}),\qquad d_{j,0}=0. \tag*{(53)} d j , t + 1 = ( I − η H j ) d j , t − η ξ j ( α j , t η ) , d j , 0 = 0. (53) Equations 35, 47 and 52 imply
∥ d j , t ∥ ≤ η ∑ ℓ = 0 t − 1 ∥ ξ j ( α j , ℓ η ) ∥ ≤ η t ω j . (54) \|d_{j,t}\|\le\eta\sum_{\ell=0}^{t-1}\|\xi_j(\alpha_{j,\ell}^{\eta})\|\le\eta t\omega_j. \tag*{(54)} ∥ d j , t ∥ ≤ η ℓ = 0 ∑ t − 1 ∥ ξ j ( α j , ℓ η ) ∥ ≤ η t ω j . (54) Adding and subtracting Φ j α j , T j η \Phi_j\alpha_{j,T_j}^{\eta} Φ j α j , T j η
Step 5: Transfer the output bound to risk reductions. Let f = f j , η f=f_{j,\eta} f = f j , η f ~ = Φ j β j η , T j \widetilde{f}=\Phi_j\beta_j^\eta{}_{,T_j} f = Φ j β j η , T j K j R j K_jR_j K j R j (41) give
∣ R P ( f ) − R P ( f ~ ) ∣ ≤ ∥ f − f ~ ∥ ( 2 ∥ r ‾ ∥ + ∥ f ∥ + ∥ f ~ ∥ ) \left|R_P(f)-R_P(\widetilde{f})\right|\le\|f-\widetilde{f}\|\left(2\|\overline{r}\|+\|f\|+\|\widetilde{f}\|\right) R P ( f ) − R P ( f ) ≤ ∥ f − f ∥ ( 2∥ r ∥ + ∥ f ∥ + ∥ f ∥ ) ≤ 2 ( s + K j R j ) ε j ( C , η ) , (55) \le2(s+K_jR_j)\varepsilon_j(C,\eta), \tag*{(55)} ≤ 2 ( s + K j R j ) ε j ( C , η ) , (55) which proves equation (38) . Applying ∣ sup u − sup v ∣ ≤ sup ∣ u − v ∣ \left|\sup u-\sup v\right|\le\sup\left|u-v\right| ∣ sup u − sup v ∣ ≤ sup ∣ u − v ∣ (39) .
Step 6: Establish attainment on the candidate sets. Finally, the actual gradient is continuous on the certified ball. Finite induction in equation (27) makes the terminal actual output continuous in η \eta η η \eta η E j ( C ) \mathcal{E}_j(C) E j ( C )
Proof of Theorem 3. We transfer the certified tangent margin to the actual runs, then construct an instance satisfying all the stated conditions.
Step 1: Transfer the tangent advantage to trained risk. Proposition 6 proves the finite-horizon iterate and output bounds, the exact tangent formula and the optimized error certificate. Its two-sided bounds and the strict margin give the positive chain
Δ r ≥ Δ r l i n − e r > Δ b l i n + e b ≥ Δ b . (56) \Delta_r\ge\Delta_r^{\mathrm{lin}}-e_r>\Delta_b^{\mathrm{lin}}+e_b\ge\Delta_b. \tag*{(56)} Δ r ≥ Δ r lin − e r > Δ b lin + e b ≥ Δ b . (56) Both reductions use the common baseline R P ( 0 ) R_P(0) R P ( 0 ) (56) from that baseline and using attainment proves equation (14) .
Step 2: Construct and bound the angular objective. We finish by constructing the stated nonempty instance. Let ( ζ 1 , ζ 2 ) ∼ N ( 0 , I 2 ) (\zeta_1,\zeta_2)\sim\mathcal{N}(0,I_2) ( ζ 1 , ζ 2 ) ∼ N ( 0 , I 2 ) e ∈ X e\in\mathcal{X} e ∈ X α = ( α 1 , α 2 ) \alpha=(\alpha_1,\alpha_2) α = ( α 1 , α 2 )
s α = α 1 ζ 1 + α 2 ζ 2 . (57) s_\alpha=\alpha_1\zeta_1+\alpha_2\zeta_2. \tag*{(57)} s α = α 1 ζ 1 + α 2 ζ 2 . (57) With prediction s α e s_\alpha e s α e β ζ 2 e \beta\zeta_2e β ζ 2 e τ \tau τ
A β ( α ) = 1 − E [ sgn ( s α ζ 2 ) 1 { ∣ s α ∣ > τ , ∣ ζ 2 ∣ > τ / β } ] . (58) A_\beta(\alpha)=1-\mathbb{E}\left[\operatorname{sgn}(s_\alpha\zeta_2)\mathbf{1}_{\{|s_\alpha|>\tau,\ |\zeta_2|>\tau/\beta\}}\right]. \tag*{(58)} A β ( α ) = 1 − E [ sgn ( s α ζ 2 ) 1 { ∣ s α ∣ > τ , ∣ ζ 2 ∣ > τ / β } ] . (58) We verify the gradient envelope consumed below. Let φ ( t ) = ( 2 π ) − 1 / 2 e − t 2 / 2 \varphi(t)=(2\pi)^{-1/2}e^{-t^2/2} φ ( t ) = ( 2 π ) − 1/2 e − t 2 /2 Φ ‾ ( t ) = ∫ t ∞ φ ( s ) d s \overline{\Phi}(t)=\int_t^\infty\varphi(s)\,\mathrm{d}s Φ ( t ) = ∫ t ∞ φ ( s ) d s α = ρ ( cos ϑ , sin ϑ ) \alpha=\rho(\cos\vartheta,\sin\vartheta) α = ρ ( cos ϑ , sin ϑ ) t = τ / ρ t=\tau/\rho t = τ / ρ Z ϑ = cos ϑ ζ 1 + sin ϑ ζ 2 Z_\vartheta=\cos\vartheta\,\zeta_1+\sin\vartheta\,\zeta_2 Z ϑ = cos ϑ ζ 1 + sin ϑ ζ 2 m ϑ ( z ) m_\vartheta(z) m ϑ ( z ) Z ϑ = z Z_\vartheta=z Z ϑ = z ∣ m ϑ ∣ ≤ 1 |m_\vartheta|\le1 ∣ m ϑ ∣ ≤ 1
∫ ∣ z ∣ > t m ϑ ( z ) φ ( z ) d z (59) \int_{|z|>t}m_\vartheta(z)\varphi(z)\,\mathrm{d}z \tag*{(59)} ∫ ∣ z ∣ > t m ϑ ( z ) φ ( z ) d z (59) at the moving cutoff gives
∣ ∂ ρ A β ∣ ≤ 2 τ ρ 2 φ ( τ / ρ ) ≤ 4 2 π τ . (60) |\partial_\rho A_\beta|\le\frac{2\tau}{\rho^2}\varphi(\tau/\rho)\le\frac{4}{\sqrt{2\pi}\tau}. \tag*{(60)} ∣ ∂ ρ A β ∣ ≤ ρ 2 2 τ φ ( τ / ρ ) ≤ 2 π τ 4 . (60) For the angular derivative, use polar Gaussian coordinates
( ζ 1 , ζ 2 ) = R ( cos ϕ , sin ϕ ) . (61) (\zeta_1,\zeta_2)=R(\cos\phi,\sin\phi). \tag*{(61)} ( ζ 1 , ζ 2 ) = R ( cos ϕ , sin ϕ ) . (61) At each R > τ / ρ R>\tau/\rho R > τ / ρ γ R = arccos ( τ / ( ρ R ) ) \gamma_R=\arccos(\tau/(\rho R)) γ R = arccos ( τ / ( ρR )) q R ( ϕ ) q_R(\phi) q R ( ϕ ) q R ( ϕ + π ) = − q R ( ϕ ) q_R(\phi+\pi)=-q_R(\phi) q R ( ϕ + π ) = − q R ( ϕ )
1 π ∫ ϑ − γ R ϑ + γ R q R ( ϕ ) d ϕ . (62) \frac{1}{\pi}\int_{\vartheta-\gamma_R}^{\vartheta+\gamma_R}q_R(\phi)\,\mathrm{d}\phi. \tag*{(62)} π 1 ∫ ϑ − γ R ϑ + γ R q R ( ϕ ) d ϕ . (62) Translating this interval changes the integral at rate at most 2 / π 2/\pi 2/ π
1 ρ ∣ ∂ ϑ A β ∣ ≤ 2 π ρ e − τ 2 / ( 2 ρ 2 ) ≤ 2 π e τ . (63) \frac{1}{\rho}|\partial_\vartheta A_\beta|\le\frac{2}{\pi\rho}e^{-\tau^2/(2\rho^2)}\le\frac{2}{\pi\sqrt{e}\tau}. \tag*{(63)} ρ 1 ∣ ∂ ϑ A β ∣ ≤ π ρ 2 e − τ 2 / ( 2 ρ 2 ) ≤ π e τ 2 . (63) For fixed ( ρ , ϑ ) (\rho,\vartheta) ( ρ , ϑ ) ρ < β \rho<\beta ρ < β R R R
∣ A β ( α ) − 1 ∣ ≤ 2 Φ ‾ ( τ / ρ ) = o ( ρ ) as ρ ↓ 0 , (64) \lvert A_{\beta}(\alpha)-1\rvert\le2\overline{\Phi}(\tau/\rho)=o(\rho)\quad\text{as }\rho\downarrow0, \tag*{(64)} ∣ A β ( α ) − 1 ∣ ≤ 2 Φ ( τ / ρ ) = o ( ρ ) as ρ ↓ 0 , (64) and both displayed derivative bounds converge uniformly to zero. Away from the two cutoff collisions α = ± β ( 0 , 1 ) \alpha=\pm\beta(0,1) α = ± β ( 0 , 1 ) ∥ α ∥ ≤ R < β \lVert\alpha\rVert\le R<\beta ∥ α ∥ ≤ R < β ∇ A β ( 0 ) = 0 \nabla A_{\beta}(0)=0 ∇ A β ( 0 ) = 0
∥ ∇ A β ( α ) ∥ ≤ ( 4 e 2 π + 2 π e ) τ − 1 < τ − 1 . (65) \lVert\nabla A_{\beta}(\alpha)\rVert\le\left(\frac{4}{e\sqrt{2\pi}}+\frac{2}{\pi\sqrt{e}}\right)\tau^{-1}<\tau^{-1}. \tag*{(65)} ∥ ∇ A β ( α )∥ ≤ ( e 2 π 4 + π e 2 ) τ − 1 < τ − 1 . (65) Step 3: Realize the correction heads on a sampler state. We now realize these coordinates through a state of the linear-interpolant flow. Draw ξ ∼ N ( 0 , I d ) \xi\sim\mathcal{N}(0,I_d) ξ ∼ N ( 0 , I d ) d ≥ 2 d\ge2 d ≥ 2 X 0 = 0 X_0=0 X 0 = 0 C C C Σ \Sigma Σ [ 0 , 1 ] [0,1] [ 0 , 1 ]
X σ = σ ξ , U = ξ . (66) X_{\sigma}=\sigma\xi,\qquad U=\xi. \tag*{(66)} X σ = σ ξ , U = ξ . (66) Let ζ 1 , ζ 2 \zeta_1,\zeta_2 ζ 1 , ζ 2 ξ \xi ξ σ > 0 \sigma>0 σ > 0
v 0 ( X σ , σ , C ) = ξ − β ζ 2 e . (67) v_0(X_{\sigma},\sigma,C)=\xi-\beta\zeta_2e. \tag*{(67)} v 0 ( X σ , σ , C ) = ξ − β ζ 2 e . (67) The time law assigns zero mass to σ = 0 \sigma=0 σ = 0 ξ = X σ / σ \xi=X_{\sigma}/\sigma ξ = X σ / σ ( ζ 1 , ζ 2 ) (\zeta_1,\zeta_2) ( ζ 1 , ζ 2 )
r = r ˉ = β ζ 2 e , s = β . (68) r=\bar r=\beta\zeta_2e,\qquad s=\beta. \tag*{(68)} r = r ˉ = β ζ 2 e , s = β . (68) Use the actual two-coordinate heads
F r ( α ) = ( α 1 ζ 1 + α 2 ζ 2 ) e , F_r(\alpha)=(\alpha_1\zeta_1+\alpha_2\zeta_2)e, F r ( α ) = ( α 1 ζ 1 + α 2 ζ 2 ) e , F b ( α ) = ( α 1 ζ 1 + κ α 2 ζ 2 ) e , 0 < κ < 1. (69) F_b(\alpha)=(\alpha_1\zeta_1+\sqrt{\kappa}\,\alpha_2\zeta_2)e,\qquad0<\kappa<1. \tag*{(69)} F b ( α ) = ( α 1 ζ 1 + κ α 2 ζ 2 ) e , 0 < κ < 1. (69) Their Jacobians are constant, so K j = 1 K_j=1 K j = 1 Λ j = 0 \Lambda_j=0 Λ j = 0
H r = I 2 , b r = ( 0 , β ) T , H b = diag ( 1 , κ ) , b b = ( 0 , κ β ) T . (70) H_r=I_2,\qquad b_r=(0,\beta)^{\mathsf{T}},\qquad H_b=\operatorname{diag}(1,\kappa),\qquad b_b=(0,\sqrt{\kappa}\,\beta)^{\mathsf{T}}. \tag*{(70)} H r = I 2 , b r = ( 0 , β ) T , H b = diag ( 1 , κ ) , b b = ( 0 , κ β ) T . (70) Step 4: Choose admissible parameters and equal resource charges. Set τ = 10 − 8 \tau=10^{-8} τ = 1 0 − 8 G ∠ = τ − 1 G_{\angle}=\tau^{-1} G ∠ = τ − 1 Q ∠ = λ ∠ G ∠ Q_{\angle}=\lambda_{\angle}G_{\angle} Q ∠ = λ ∠ G ∠ β ≥ 4 Q ∠ \beta\ge4Q_{\angle} β ≥ 4 Q ∠
κ = 1 4 , η = 1 4 , ζ s t a b = 1 , R b = β 3 , R r = β + Q ∠ 3 . (71) \kappa=\frac{1}{4},\qquad\eta=\frac{1}{4},\qquad\zeta_{\mathrm{stab}}=1,\qquad R_b=\frac{\beta}{3},\qquad R_r=\frac{\beta+Q_{\angle}}{3}. \tag*{(71)} κ = 4 1 , η = 4 1 , ζ stab = 1 , R b = 3 β , R r = 3 β + Q ∠ . (71) Equation 65 applies on the refiner ball because R r ≤ 5 β / 12 < β R_r\le5\beta/12<\beta R r ≤ 5 β /12 < β c ∗ c_* c ∗ m ∗ m_* m ∗ d ∗ d_* d ∗ C = ( c ∗ , m ∗ , d ∗ ) C=(c_*,m_*,d_*) C = ( c ∗ , m ∗ , d ∗ ) { 1 / 4 } \{1/4\} { 1/4 }
η ∥ H j ∥ o p = 1 4 ≤ 1 , η Γ b = R b , η Γ r = R r . (72) \eta\lVert H_j\rVert_{\mathrm{op}}=\frac{1}{4}\le1,\qquad\eta\Gamma_b=R_b,\qquad\eta\Gamma_r=R_r. \tag*{(72)} η ∥ H j ∥ op = 4 1 ≤ 1 , η Γ b = R b , η Γ r = R r . (72) Thus every local and resource condition is satisfied.
Step 5: Verify the strict risk-reduction margin. Finally, equation 37 on the loaded eigendirection gives
Δ r l i n = β 2 [ 1 − ( 1 − 1 / 4 ) 2 ] = 7 16 β 2 , Δ b l i n = β 2 [ 1 − ( 1 − 1 / 16 ) 2 ] = 31 256 β 2 . (73) \Delta_r^{\mathrm{lin}}=\beta^2[1-(1-1/4)^2]=\frac{7}{16}\beta^2,\qquad\Delta_b^{\mathrm{lin}}=\beta^2[1-(1-1/16)^2]=\frac{31}{256}\beta^2. \tag*{(73)} Δ r lin = β 2 [ 1 − ( 1 − 1/4 ) 2 ] = 16 7 β 2 , Δ b lin = β 2 [ 1 − ( 1 − 1/16 ) 2 ] = 256 31 β 2 . (73) Since Λ j = 0 \Lambda_j=0 Λ j = 0 ω b = 0 \omega_b=0 ω b = 0 ω r = Q ∠ \omega_r=Q_{\angle} ω r = Q ∠ ϵ b = 0 \epsilon_b=0 ϵ b = 0 ϵ r = Q ∠ / 4 \epsilon_r=Q_{\angle}/4 ϵ r = Q ∠ /4
e b = 0 , e r = 2 ( β + R r ) Q ∠ 4 = Q ∠ ( 4 β + Q ∠ ) 6 . (74) e_b=0,\qquad e_r=2(\beta+R_r)\frac{Q_{\angle}}{4}=\frac{Q_{\angle}(4\beta+Q_{\angle})}{6}. \tag*{(74)} e b = 0 , e r = 2 ( β + R r ) 4 Q ∠ = 6 Q ∠ ( 4 β + Q ∠ ) . (74) The tangent gap is 81 β 2 / 256 81\beta^2/256 81 β 2 /256
e r β 2 = 1 6 Q ∠ β ( 4 + Q ∠ β ) ≤ 17 96 < 81 256 . (75) \frac{e_r}{\beta^2}=\frac{1}{6}\frac{Q_{\angle}}{\beta}\left(4+\frac{Q_{\angle}}{\beta}\right)\le\frac{17}{96}<\frac{81}{256}. \tag*{(75)} β 2 e r = 6 1 β Q ∠ ( 4 + β Q ∠ ) ≤ 96 17 < 256 81 . (75) This proves the strict margin and completes the theorem.
Deployment Analysis: Gain, Sampler Law and Endpoint
This section establishes how the correction gain controls deployed velocity error and endpoint distance. We first derive an exact gain quadratic at a fixed state law. We then couple the base and corrected trajectories to certify velocity improvement under each sampler’s own law. Finally, we analyze how velocity corrections accumulate along the trajectory and derive an exact endpoint gain identity for affine–Gaussian flows.
Deployment setting
Fix the trained pair and let D d e p D_{\mathrm{dep}} D dep g ϕ g_{\phi} g ϕ W ↦ g ϕ ( Z ( W ) ) W \mapsto g_{\phi}(Z(W)) W ↦ g ϕ ( Z ( W ))
u ⋆ : D d e p → X u^{\star}:D_{\mathrm{dep}} \to\mathcal{X} u ⋆ : D dep → X such that
u ⋆ ( W ) = E [ U ∣ W ] P -almost surely . u^{\star}(W)=\mathbb{E}[U\mid W]\quad P\text{-almost surely}. u ⋆ ( W ) = E [ U ∣ W ] P -almost surely . Set
r ˉ = u ⋆ − v 0 . \bar{r}=u^{\star}-v_{0}. r ˉ = u ⋆ − v 0 . For a state law Q Q Q κ ≥ 0 \kappa\ge0 κ ≥ 0
V Q ( κ ) = E Q ∥ r ˉ − κ g ϕ ∥ 2 , A Q = E Q ⟨ r ˉ , g ϕ ⟩ , B Q = E Q ∥ g ϕ ∥ 2 . (76) \mathcal{V}_{Q}(\kappa)=\mathbb{E}_{Q}\lVert\bar{r}-\kappa g_{\phi}\rVert^{2},\qquad A_{Q}=\mathbb{E}_{Q}\langle\bar{r},g_{\phi}\rangle,\qquad B_{Q}=\mathbb{E}_{Q}\lVert g_{\phi}\rVert^{2}. \tag*{(76)} V Q ( κ ) = E Q ∥ r ˉ − κ g ϕ ∥ 2 , A Q = E Q ⟨ r ˉ , g ϕ ⟩ , B Q = E Q ∥ g ϕ ∥ 2 . (76) On the training law these connect to the risk used above. Taking h = κ g ϕ h=\kappa g_{\phi} h = κ g ϕ (41) gives
R P ( κ g ϕ ) = E P ~ ∥ r − r ˉ ∥ 2 + V P ( κ ) , (77) \mathcal{R}_{P}(\kappa g_{\phi})=\mathbb{E}_{\widetilde{P}}\lVert r-\bar{r}\rVert^{2}+\mathcal{V}_{P}(\kappa), \tag*{(77)} R P ( κ g ϕ ) = E P ∥ r − r ˉ ∥ 2 + V P ( κ ) , (77) so V P \mathcal{V}_{P} V P
The deployed sampler determines its state law through its trajectory. Running Equation (11) from the prescribed Gaussian source with the composite field v 0 + κ g ϕ v_{0}+\kappa g_{\phi} v 0 + κ g ϕ X κ ( σ ) X^{\kappa}(\sigma) X κ ( σ ) σ : 1 → 0 \sigma:1\to0 σ : 1 → 0
Q κ = Law ( X κ ( Σ ) , Σ , C ) , Σ ∼ q . (78) Q_{\kappa}=\operatorname{Law}(X^{\kappa}(\Sigma),\Sigma,C),\qquad\Sigma\sim q. \tag*{(78)} Q κ = Law ( X κ ( Σ ) , Σ , C ) , Σ ∼ q . (78) Here q q q [ 0 , 1 ] [0,1] [ 0 , 1 ] Σ \Sigma Σ Q 0 Q_{0} Q 0
Assumption 6 (Training-law moments). The realized residual satisfies r ∈ L 2 ( P ~ ; X ) r\in L^{2}(\widetilde{P};\mathcal{X}) r ∈ L 2 ( P ; X ) g ϕ ∈ L 2 ( P ; X ) g_{\phi}\in L^{2}(P;\mathcal{X}) g ϕ ∈ L 2 ( P ; X ) r ˉ \bar{r} r ˉ E P ~ [ r ∣ W ] \mathbb{E}_{\widetilde{P}}[r\mid W] E P [ r ∣ W ] P P P r ˉ ∈ L 2 ( P ; X ) \bar{r}\in L^{2}(P;\mathcal{X}) r ˉ ∈ L 2 ( P ; X )
Assumption 7 (Deployment ODE well-posedness). For a fixed gain radius κ ˉ > 0 \bar{\kappa}>0 κ ˉ > 0
{ v 0 + κ g ϕ : κ ∈ [ 0 , κ ˉ ] } \{v_{0}+\kappa g_{\phi}:\kappa\in[0,\bar{\kappa}]\} { v 0 + κ g ϕ : κ ∈ [ 0 , κ ˉ ]} is measurable in σ \sigma σ [ 0 , 1 ] [0,1] [ 0 , 1 ] Q κ Q_{\kappa} Q κ (78) are well-defined for every κ ∈ [ 0 , κ ˉ ] \kappa\in[0,\bar{\kappa}] κ ∈ [ 0 , κ ˉ ]
Exact gain quadratic and improvement interval
At a fixed state law, the deployed velocity error is an exact quadratic in the gain. Residual–correction alignment A Q A_{Q} A Q B Q B_{Q} B Q
Proposition 7 (Exact gain quadratic). Let Q Q Q E Q ∥ r ∥ 2 < ∞ \mathbb{E}_{Q}\lVert r\rVert^{2}<\infty E Q ∥ r ∥ 2 < ∞ E Q ∥ g ϕ ∥ 2 < ∞ \mathbb{E}_{Q}\lVert g_{\phi}\rVert^{2}<\infty E Q ∥ g ϕ ∥ 2 < ∞ κ ∈ [ 0 , ∞ ) \kappa\in[0,\infty) κ ∈ [ 0 , ∞ )
V Q ( κ ) − V Q ( 0 ) = − 2 κ A Q + κ 2 B Q . (79) \mathcal{V}_{Q}(\kappa)-\mathcal{V}_{Q}(0)=-2\kappa A_{Q}+\kappa^{2}B_{Q}. \tag*{(79)} V Q ( κ ) − V Q ( 0 ) = − 2 κ A Q + κ 2 B Q . (79) Proof. We verify finiteness of every term before expanding the squared velocity error. Step 1: Establish integrability on the deployment law. First, A Q A_Q A Q B Q ∈ [ 0 , ∞ ) B_Q \in[0,\infty) B Q ∈ [ 0 , ∞ ) V Q ( κ ) < ∞ \mathcal{V}_Q(\kappa) < \infty V Q ( κ ) < ∞ κ ≥ 0 \kappa\ge0 κ ≥ 0
∣ A Q ∣ ≤ V Q ( 0 ) 1 / 2 B Q 1 / 2 . (80) |A_Q| \le\mathcal{V}_Q(0)^{1/2}B_Q^{1/2}. \tag*{(80)} ∣ A Q ∣ ≤ V Q ( 0 ) 1/2 B Q 1/2 . (80) By definition V Q ( 0 ) = E Q ∥ r ˉ ∥ 2 < ∞ \mathcal{V}_Q(0) = \mathbb{E}_Q\|\bar{r}\|^2 < \infty V Q ( 0 ) = E Q ∥ r ˉ ∥ 2 < ∞ B Q = E Q ∥ g ϕ ∥ 2 ∈ [ 0 , ∞ ) B_Q = \mathbb{E}_Q\|g_\phi\|^2 \in[0,\infty) B Q = E Q ∥ g ϕ ∥ 2 ∈ [ 0 , ∞ ) L 2 ( Q ; X ) L^2(Q;\mathcal{X}) L 2 ( Q ; X )
E Q ∣ ⟨ r ˉ , g ϕ ⟩ ∣ ≤ ( E Q ∥ r ˉ ∥ 2 ) 1 / 2 ( E Q ∥ g ϕ ∥ 2 ) 1 / 2 < ∞ , \mathbb{E}_Q|\langle\bar{r},g_\phi\rangle| \le\left(\mathbb{E}_Q\|\bar{r}\|^2\right)^{1/2}\left(\mathbb{E}_Q\|g_\phi\|^2\right)^{1/2} < \infty, E Q ∣ ⟨ r ˉ , g ϕ ⟩ ∣ ≤ ( E Q ∥ r ˉ ∥ 2 ) 1/2 ( E Q ∥ g ϕ ∥ 2 ) 1/2 < ∞ , which makes A Q A_Q A Q ∥ r ˉ − κ g ϕ ∥ 2 ≤ 2 ∥ r ˉ ∥ 2 + 2 κ 2 ∥ g ϕ ∥ 2 \|\bar{r}-\kappa g_\phi\|^2 \le2\|\bar{r}\|^2+2\kappa^2\|g_\phi\|^2 ∥ r ˉ − κ g ϕ ∥ 2 ≤ 2∥ r ˉ ∥ 2 + 2 κ 2 ∥ g ϕ ∥ 2 V Q ( κ ) < ∞ \mathcal{V}_Q(\kappa)<\infty V Q ( κ ) < ∞
Step 2: Expand the gain quadratic at the fixed law. Pointwise in the state, ∥ r ˉ − κ g ϕ ∥ 2 = ∥ r ˉ ∥ 2 − 2 κ ⟨ r ˉ , g ϕ ⟩ + κ 2 ∥ g ϕ ∥ 2 \|\bar{r}-\kappa g_\phi\|^2=\|\bar{r}\|^2-2\kappa\langle\bar{r},g_\phi\rangle+\kappa^2\|g_\phi\|^2 ∥ r ˉ − κ g ϕ ∥ 2 = ∥ r ˉ ∥ 2 − 2 κ ⟨ r ˉ , g ϕ ⟩ + κ 2 ∥ g ϕ ∥ 2 Q Q Q V Q ( 0 ) \mathcal{V}_Q(0) V Q ( 0 ) Q Q Q
Proposition 8 (Sharp gain boundaries). Under Proposition 7 , exactly one of the following holds.
(i) If B Q = 0 B_Q=0 B Q = 0 g ϕ = 0 g_\phi=0 g ϕ = 0 Q Q Q A Q = 0 A_Q=0 A Q = 0 V Q ( κ ) = V Q ( 0 ) \mathcal{V}_Q(\kappa)=\mathcal{V}_Q(0) V Q ( κ ) = V Q ( 0 ) κ ≥ 0 \kappa\ge0 κ ≥ 0
(ii) If B Q > 0 B_Q>0 B Q > 0 A Q ≤ 0 A_Q\le0 A Q ≤ 0 V Q ( κ ) > V Q ( 0 ) \mathcal{V}_Q(\kappa)>\mathcal{V}_Q(0) V Q ( κ ) > V Q ( 0 ) κ > 0 \kappa>0 κ > 0
(iii) If B Q > 0 B_Q>0 B Q > 0 A Q > 0 A_Q>0 A Q > 0
V Q ( κ ) < V Q ( 0 ) ⟺ 0 < κ < 2 A Q B Q , (81) \mathcal{V}_Q(\kappa)<\mathcal{V}_Q(0) \quad\Longleftrightarrow\quad0<\kappa<\frac{2A_Q}{B_Q}, \tag*{(81)} V Q ( κ ) < V Q ( 0 ) ⟺ 0 < κ < B Q 2 A Q , (81) with equality exactly at the two endpoints κ = 0 \kappa=0 κ = 0 κ = 2 A Q / B Q \kappa=2A_Q/B_Q κ = 2 A Q / B Q V Q ( κ ) > V Q ( 0 ) \mathcal{V}_Q(\kappa)>\mathcal{V}_Q(0) V Q ( κ ) > V Q ( 0 )
Proof. If B Q = 0 B_Q=0 B Q = 0 ∥ g ϕ ∥ 2 \|g_\phi\|^2 ∥ g ϕ ∥ 2 Q Q Q A Q = 0 A_Q=0 A Q = 0 r ˉ − κ g ϕ = r ˉ \bar{r}-\kappa g_\phi=\bar{r} r ˉ − κ g ϕ = r ˉ Q Q Q κ \kappa κ
Suppose B Q > 0 B_Q>0 B Q > 0
V Q ( κ ) − V Q ( 0 ) = κ ( κ B Q − 2 A Q ) . (82) \mathcal{V}_Q(\kappa)-\mathcal{V}_Q(0)=\kappa(\kappa B_Q-2A_Q). \tag*{(82)} V Q ( κ ) − V Q ( 0 ) = κ ( κ B Q − 2 A Q ) . (82) If A Q ≤ 0 A_Q\le0 A Q ≤ 0 κ > 0 \kappa>0 κ > 0 κ B Q > 0 \kappa B_Q>0 κ B Q > 0 − 2 A Q ≥ 0 -2A_Q\ge0 − 2 A Q ≥ 0 A Q > 0 A_Q>0 A Q > 0 κ > 0 \kappa>0 κ > 0 κ B Q − 2 A Q \kappa B_Q-2A_Q κ B Q − 2 A Q κ \kappa κ 2 A Q / B Q 2A_Q/B_Q 2 A Q / B Q κ \kappa κ κ = 0 \kappa=0 κ = 0
Corollary 9 (Unique optimal gain). If A Q > 0 A_Q>0 A Q > 0 B Q > 0 B_Q>0 B Q > 0 κ ↦ V Q ( κ ) \kappa\mapsto\mathcal{V}_Q(\kappa) κ ↦ V Q ( κ ) [ 0 , ∞ ) [0,\infty) [ 0 , ∞ )
κ ∗ = A Q B Q , V Q ( κ ∗ ) = V Q ( 0 ) − A Q 2 B Q ≥ 0. (83) \kappa^*=\frac{A_Q}{B_Q},\qquad\mathcal{V}_Q(\kappa^*)=\mathcal{V}_Q(0)-\frac{A_Q^2}{B_Q}\ge0. \tag*{(83)} κ ∗ = B Q A Q , V Q ( κ ∗ ) = V Q ( 0 ) − B Q A Q 2 ≥ 0. (83) Proof. Completing the square in equation 79 gives
V Q ( κ ) = V Q ( 0 ) − A Q 2 B Q + B Q ( κ − A Q B Q ) 2 . \mathcal{V}_Q(\kappa)=\mathcal{V}_Q(0)-\frac{A_Q^2}{B_Q}+B_Q\left(\kappa-\frac{A_Q}{B_Q}\right)^2. V Q ( κ ) = V Q ( 0 ) − B Q A Q 2 + B Q ( κ − B Q A Q ) 2 . The last term is nonnegative and vanishes at exactly one scalar, which is positive because A Q > 0 A_Q>0 A Q > 0 A Q 2 / B Q ≤ V Q ( 0 ) A_Q^2/B_Q\le\mathcal{V}_Q(0) A Q 2 / B Q ≤ V Q ( 0 )
Own-law velocity improvement
The next theorem establishes a gain interval over which the corrected sampler achieves lower velocity error than the base sampler, with each field evaluated under its own state law. Coupling the trajectories through the same initial noise and condition lets us control their displacement using spatial regularity. Combining this displacement bound with the fixed-law gain identity gives the comparison between Q κ Q_\kappa Q κ Q 0 Q_0 Q 0
Coupled trajectories. Let X 1 \mathbf{X}_1 X 1 C C C κ ∈ [ 0 , κ ˉ ] \kappa\in[0,\bar{\kappa}] κ ∈ [ 0 , κ ˉ ]
d X 0 ( σ ) d σ = v 0 ( X 0 ( σ ) , σ , C ) , X 0 ( 1 ) = X 1 , (84) \frac{d\mathbf{X}^{0}(\sigma)}{d\sigma} = v_0(\mathbf{X}^{0}(\sigma),\sigma,C), \qquad\mathbf{X}^{0}(1)=\mathbf{X}_1, \tag*{(84)} d σ d X 0 ( σ ) = v 0 ( X 0 ( σ ) , σ , C ) , X 0 ( 1 ) = X 1 , (84) d X κ ( σ ) d σ = v 0 ( X κ ( σ ) , σ , C ) + κ g ϕ ( X κ ( σ ) , σ , C ) , X κ ( 1 ) = X 1 . (85) \frac{d\mathbf{X}^{\kappa}(\sigma)}{d\sigma} = v_0(\mathbf{X}^{\kappa}(\sigma),\sigma,C) + \kappa g_{\phi}(\mathbf{X}^{\kappa}(\sigma),\sigma,C), \qquad\mathbf{X}^{\kappa}(1)=\mathbf{X}_1. \tag*{(85)} d σ d X κ ( σ ) = v 0 ( X κ ( σ ) , σ , C ) + κ g ϕ ( X κ ( σ ) , σ , C ) , X κ ( 1 ) = X 1 . (85) Sample the same independent Σ ∼ q \Sigma\sim q Σ ∼ q Q 0 Q_0 Q 0 Q κ Q_{\kappa} Q κ (78) . Write A 0 : = A Q 0 A_0 := A_{Q_0} A 0 := A Q 0
Assumption 8 (Uniform bounds and spatial regularity). The bounds below control the magnitude and spatial variation of the residual r ˉ \bar{r} r ˉ u ⋆ \mathbf{u}^{\star} u ⋆ r ˉ = u ⋆ − v 0 \bar{r}=\mathbf{u}^{\star}-v_0 r ˉ = u ⋆ − v 0 σ ∈ [ 0 , 1 ] \sigma\in[0,1] σ ∈ [ 0 , 1 ] κ ∈ [ 0 , κ ˉ ] \kappa\in[0,\bar{\kappa}] κ ∈ [ 0 , κ ˉ ] R r R_r R r G G G L r L_r L r L g L_g L g L 0 L_0 L 0
∥ r ˉ ( x , σ , c ) ∥ ≤ R r , ∥ g ϕ ( x , σ , c ) ∥ ≤ G , (86) \lVert\bar{r}(x,\sigma,c)\rVert\le R_r,\qquad\lVert g_{\phi}(x,\sigma,c)\rVert\le G, \tag*{(86)} ∥ r ˉ ( x , σ , c )∥ ≤ R r , ∥ g ϕ ( x , σ , c )∥ ≤ G , (86) and, for any two states x , y x,y x , y ( σ , c ) (\sigma,c) ( σ , c )
∥ r ˉ ( x , σ , c ) − r ˉ ( y , σ , c ) ∥ ≤ L r ∥ x − y ∥ , (87) \lVert\bar{r}(x,\sigma,c)-\bar{r}(y,\sigma,c)\rVert\le L_r\lVert x-y\rVert, \tag*{(87)} ∥ r ˉ ( x , σ , c ) − r ˉ ( y , σ , c )∥ ≤ L r ∥ x − y ∥ , (87) ∥ g ϕ ( x , σ , c ) − g ϕ ( y , σ , c ) ∥ ≤ L g ∥ x − y ∥ , (88) \lVert g_{\phi}(x,\sigma,c)-g_{\phi}(y,\sigma,c)\rVert\le L_g\lVert x-y\rVert, \tag*{(88)} ∥ g ϕ ( x , σ , c ) − g ϕ ( y , σ , c )∥ ≤ L g ∥ x − y ∥ , (88) ∥ v 0 ( x , σ , c ) − v 0 ( y , σ , c ) ∥ ≤ L 0 ∥ x − y ∥ . (89) \lVert v_0(x,\sigma,c)-v_0(y,\sigma,c)\rVert\le L_0\lVert x-y\rVert. \tag*{(89)} ∥ v 0 ( x , σ , c ) − v 0 ( y , σ , c )∥ ≤ L 0 ∥ x − y ∥ . (89) The constants are uniform in the noise level, condition, source realization and gain throughout the stated range.
Assumption 7 supplies existence and uniqueness. Assumption 8 quantifies how the correction moves the trajectory and how this displacement affects residual error and alignment across the common deployment domain.
The following theorem gives the full quantitative certificate summarized in the main-text Theorem 4 .
Theorem 10 (Detailed own-law velocity certificate). Under Assumptions 7 and 8, define
H 0 ( σ ) : = { e L 0 ( 1 − σ ) − 1 L 0 , L 0 > 0 , 1 − σ , L 0 = 0 , K Q : = G E Σ ∼ q [ H 0 ( Σ ) ] . (90) H_0(\sigma) :=
\begin{cases}
\dfrac{e^{L_0(1-\sigma)}-1}{L_0}, & L_0>0,\\
1-\sigma, & L_0=0,
\end{cases}
\qquad K_Q := G\,\mathbb{E}_{\Sigma\sim q}[H_0(\Sigma)]. \tag*{(90)} H 0 ( σ ) := ⎩ ⎨ ⎧ L 0 e L 0 ( 1 − σ ) − 1 , 1 − σ , L 0 > 0 , L 0 = 0 , K Q := G E Σ ∼ q [ H 0 ( Σ )] . (90) Set
L V : = 2 R r L r , L A : = L r G + R r L g , M : = 2 A 0 − L V K Q , N : = G 2 + 2 L A K Q . (91) L_V := 2R_rL_r,\qquad L_A := L_rG+R_rL_g,\qquad M := 2A_0-L_VK_Q,\qquad N := G^2+2L_AK_Q. \tag*{(91)} L V := 2 R r L r , L A := L r G + R r L g , M := 2 A 0 − L V K Q , N := G 2 + 2 L A K Q . (91) Then for every κ ∈ [ 0 , κ ˉ ] \kappa\in[0,\bar{\kappa}] κ ∈ [ 0 , κ ˉ ]
V Q κ ( κ ) − V Q 0 ( 0 ) ≤ − κ M + κ 2 N . (92) V_{Q_{\kappa}}(\kappa)-V_{Q_0}(0)\le-\kappa M+\kappa^2N. \tag*{(92)} V Q κ ( κ ) − V Q 0 ( 0 ) ≤ − κ M + κ 2 N . (92) If M > 0 M>0 M > 0 N > 0 N>0 N > 0
0 < κ ≤ κ ˉ , κ < M N (93) 0<\kappa\le\bar{\kappa},\qquad\kappa<\frac{M}{N} \tag*{(93)} 0 < κ ≤ κ ˉ , κ < N M (93) gives the strict comparison
V Q κ ( κ ) < V Q 0 ( 0 ) . (94) V_{Q_{\kappa}}(\kappa)<V_{Q_0}(0). \tag*{(94)} V Q κ ( κ ) < V Q 0 ( 0 ) . (94) Each velocity field is therefore evaluated under the state law generated by its own sampler.
Proof. Fix κ ∈ [ 0 , κ ˉ ] \kappa\in[0,\bar{\kappa}] κ ∈ [ 0 , κ ˉ ]
Step 1: Bound the trajectory displacement. The integral form of equation (85) , with the common source cancelled, gives
∥ X κ ( σ ) − X 0 ( σ ) ∥ ≤ ∫ σ 1 ∥ v 0 ( X κ ( s ) , s , C ) − v 0 ( X 0 ( s ) , s , C ) ∥ d s + κ ∫ σ 1 ∥ g ϕ ( X κ ( s ) , s , C ) ∥ d s ≤ L 0 ∫ σ 1 ∥ X κ ( s ) − X 0 ( s ) ∥ d s + κ G ( 1 − σ ) . (95) \begin{aligned}
\left\lVert X^\kappa(\sigma)-X^0(\sigma)\right\rVert\le\int_\sigma^1 \left\lVert v_0(X^\kappa(s),s,\mathcal{C})-v_0(X^0(s),s,\mathcal{C})\right\rVert\,\mathrm{d}s
+\kappa\int_\sigma^1\left\lVert g_\phi(X^\kappa(s),s,\mathcal{C})\right\rVert\,\mathrm{d}s \\
&\le L_0\int_\sigma^1\left\lVert X^\kappa(s)-X^0(s)\right\rVert\,\mathrm{d}s+\kappa G(1-\sigma). \tag*{(95)}
\end{aligned} X κ ( σ ) − X 0 ( σ ) ≤ ∫ σ 1 v 0 ( X κ ( s ) , s , C ) − v 0 ( X 0 ( s ) , s , C ) d s + κ ∫ σ 1 ∥ g ϕ ( X κ ( s ) , s , C ) ∥ d s ≤ L 0 ∫ σ 1 X κ ( s ) − X 0 ( s ) d s + κ G ( 1 − σ ) . (95) To handle the descending orientation, put t = 1 − σ t=1-\sigma t = 1 − σ φ ( t ) = ∥ X κ ( 1 − t ) − X 0 ( 1 − t ) ∥ \varphi(t)=\left\lVert X^\kappa(1-t)-X^0(1-t)\right\rVert φ ( t ) = X κ ( 1 − t ) − X 0 ( 1 − t )
φ ( t ) ≤ L 0 ∫ 0 t φ ( s ) d s + κ G t . \varphi(t)\le L_0\int_0^t\varphi(s)\,\mathrm{d}s+\kappa Gt. φ ( t ) ≤ L 0 ∫ 0 t φ ( s ) d s + κ Gt . For L 0 > 0 L_0>0 L 0 > 0
φ ( t ) ≤ κ G ( t + L 0 ∫ 0 t s e L 0 ( t − s ) d s ) = κ G e L 0 t − 1 L 0 . \begin{aligned}
\varphi(t)\le\kappa G\left(t+L_0\int_0^t s e^{L_0(t-s)}\,\mathrm{d}s\right) \\
&=\kappa G\frac{e^{L_0t}-1}{L_0}.
\end{aligned} φ ( t ) ≤ κ G ( t + L 0 ∫ 0 t s e L 0 ( t − s ) d s ) = κ G L 0 e L 0 t − 1 . For L 0 = 0 L_0=0 L 0 = 0 φ ( t ) ≤ κ G t \varphi(t)\le\kappa Gt φ ( t ) ≤ κ Gt
∥ X κ ( σ ) − X 0 ( σ ) ∥ ≤ κ G H 0 ( σ ) , E ∥ X κ ( Σ ) − X 0 ( Σ ) ∥ ≤ κ K Q . (96) \left\lVert X^\kappa(\sigma)-X^0(\sigma)\right\rVert\le\kappa GH_0(\sigma),\qquad\mathbb{E}\left\lVert X^\kappa(\Sigma)-X^0(\Sigma)\right\rVert\le\kappa K_Q. \tag*{(96)} X κ ( σ ) − X 0 ( σ ) ≤ κ G H 0 ( σ ) , E X κ ( Σ ) − X 0 ( Σ ) ≤ κ K Q . (96) The expectation uses the shared source, condition and sampled noise level.
Step 2: Control the change in baseline error. Suppressing the fixed arguments ( σ , c ) (\sigma,c) ( σ , c )
∣ ∥ r ˉ ( x ) ∥ 2 − ∥ r ˉ ( y ) ∥ 2 ∣ ≤ ( ∥ r ˉ ( x ) ∥ + ∥ r ˉ ( y ) ∥ ) ∥ r ˉ ( x ) − r ˉ ( y ) ∥ ≤ 2 R r L r ∥ x − y ∥ = L V ∥ x − y ∥ . \left\lvert\left\lVert\bar r(x)\right\rVert^2-\left\lVert\bar r(y)\right\rVert^2\right\rvert
\le\left(\left\lVert\bar r(x)\right\rVert+\left\lVert\bar r(y)\right\rVert\right)\left\lVert\bar r(x)-\bar r(y)\right\rVert
\le2R_rL_r\left\lVert x-y\right\rVert=L_V\left\lVert x-y\right\rVert. ∥ r ˉ ( x ) ∥ 2 − ∥ r ˉ ( y ) ∥ 2 ≤ ( ∥ r ˉ ( x ) ∥ + ∥ r ˉ ( y ) ∥ ) ∥ r ˉ ( x ) − r ˉ ( y ) ∥ ≤ 2 R r L r ∥ x − y ∥ = L V ∥ x − y ∥ . Applying this inequality to the coupled states in equation (96) gives
V Q κ ( 0 ) − V Q 0 ( 0 ) = E [ ∥ r ˉ ( X κ ( Σ ) , Σ , C ) ∥ 2 − ∥ r ˉ ( X 0 ( Σ ) , Σ , C ) ∥ 2 ] ≤ L V E ∥ X κ ( Σ ) − X 0 ( Σ ) ∥ ≤ κ L V K Q . (97) V_{Q_\kappa}(0)-V_{Q_0}(0)=\mathbb{E}\left[\left\lVert\bar r(X^\kappa(\Sigma),\Sigma,\mathcal{C})\right\rVert^2-\left\lVert\bar r(X^0(\Sigma),\Sigma,\mathcal{C})\right\rVert^2\right]
\le L_V\mathbb{E}\left\lVert X^\kappa(\Sigma)-X^0(\Sigma)\right\rVert\le\kappa L_VK_Q. \tag*{(97)} V Q κ ( 0 ) − V Q 0 ( 0 ) = E [ ∥ r ˉ ( X κ ( Σ ) , Σ , C ) ∥ 2 − r ˉ ( X 0 ( Σ ) , Σ , C ) 2 ] ≤ L V E X κ ( Σ ) − X 0 ( Σ ) ≤ κ L V K Q . (97) Step 3: Control the change in alignment. At the same fixed ( σ , c ) (\sigma,c) ( σ , c )
∣ ⟨ r ˉ ( x ) , g ϕ ( x ) ⟩ − ⟨ r ˉ ( y ) , g ϕ ( y ) ⟩ ∣ ≤ ∣ ⟨ r ˉ ( x ) − r ˉ ( y ) , g ϕ ( x ) ⟩ ∣ + ∣ ⟨ r ˉ ( y ) , g ϕ ( x ) − g ϕ ( y ) ⟩ ∣ ≤ ( L r G + R r L g ) ∥ x − y ∥ = L A ∥ x − y ∥ . \left\lvert\langle\bar r(x),g_\phi(x)\rangle-\langle\bar r(y),g_\phi(y)\rangle\right\rvert
\le\left\lvert\langle\bar r(x)-\bar r(y),g_\phi(x)\rangle\right\rvert+\left\lvert\langle\bar r(y),g_\phi(x)-g_\phi(y)\rangle\right\rvert
\le(L_rG+R_rL_g)\left\lVert x-y\right\rVert=L_A\left\lVert x-y\right\rVert. ∣ ⟨ r ˉ ( x ) , g ϕ ( x )⟩ − ⟨ r ˉ ( y ) , g ϕ ( y )⟩ ∣ ≤ ∣ ⟨ r ˉ ( x ) − r ˉ ( y ) , g ϕ ( x )⟩ ∣ + ∣ ⟨ r ˉ ( y ) , g ϕ ( x ) − g ϕ ( y )⟩ ∣ ≤ ( L r G + R r L g ) ∥ x − y ∥ = L A ∥ x − y ∥ . Taking expectations under the coupling therefore yields
A Q κ ≥ A 0 − L A E ∥ X κ ( Σ ) − X 0 ( Σ ) ∥ ≥ A 0 − κ L A K Q . (98) A_{Q_\kappa}\ge A_0-L_A\mathbb{E}\left\lVert X^\kappa(\Sigma)-X^0(\Sigma)\right\rVert\ge A_0-\kappa L_AK_Q. \tag*{(98)} A Q κ ≥ A 0 − L A E X κ ( Σ ) − X 0 ( Σ ) ≥ A 0 − κ L A K Q . (98) Step 4: Combine the fixed-law gain and the trajectory shift. The uniform bounds give B Q κ ≤ G 2 B_{Q_\kappa}\le G^2 B Q κ ≤ G 2 V Q κ ( 0 ) ≤ R 2 V_{Q_\kappa}(0)\le R^2 V Q κ ( 0 ) ≤ R 2 Q = Q κ Q=Q_\kappa Q = Q κ
V Q κ ( κ ) − V Q 0 ( 0 ) = [ V Q κ ( κ ) − V Q κ ( 0 ) ] + [ V Q κ ( 0 ) − V Q 0 ( 0 ) ] ≤ − 2 κ A Q κ + κ 2 B Q κ + κ L V K Q ≤ − 2 κ ( A 0 − κ L A K Q ) + κ 2 G 2 + κ L V K Q = − κ ( 2 A 0 − L V K Q ) + κ 2 ( G 2 + 2 L A K Q ) = − κ M + κ 2 N . (99) \begin{aligned}
V_{Q_\kappa}(\kappa)-V_{Q_0}(0)
&=[V_{Q_\kappa}(\kappa)-V_{Q_\kappa}(0)]+[V_{Q_\kappa}(0)-V_{Q_0}(0)]\\
&\le-2\kappa A_{Q_\kappa}+\kappa^2B_{Q_\kappa}+\kappa L_VK_Q\\
&\le-2\kappa(A_0-\kappa L_AK_Q)+\kappa^2G^2+\kappa L_VK_Q\\
&=-\kappa(2A_0-L_VK_Q)+\kappa^2(G^2+2L_AK_Q)=-\kappa M+\kappa^2N. \tag*{(99)}
\end{aligned} V Q κ ( κ ) − V Q 0 ( 0 ) = [ V Q κ ( κ ) − V Q κ ( 0 )] + [ V Q κ ( 0 ) − V Q 0 ( 0 )] ≤ − 2 κ A Q κ + κ 2 B Q κ + κ L V K Q ≤ − 2 κ ( A 0 − κ L A K Q ) + κ 2 G 2 + κ L V K Q = − κ ( 2 A 0 − L V K Q ) + κ 2 ( G 2 + 2 L A K Q ) = − κ M + κ 2 N . (99) This proves equation (92) . If M > 0 M>0 M > 0 A 0 > 0 A_0>0 A 0 > 0 G = 0 G=0 G = 0 A 0 = 0 A_0=0 A 0 = 0 G > 0 G>0 G > 0 N ≥ G 2 > 0 N\ge G^2>0 N ≥ G 2 > 0 (93) , − κ M + κ 2 N = κ ( − M + κ N ) < 0 -\kappa M+\kappa^2N=\kappa(-M+\kappa N)<0 − κ M + κ 2 N = κ ( − M + κ N ) < 0 (94) .
Meaning of the certificate. The constant K Q K_Q K Q L V L_V L V L A L_A L A M = 2 A 0 − L V K Q M = 2A_0 - L_VK_Q M = 2 A 0 − L V K Q M > 0 M > 0 M > 0 N = G 2 + 2 L A K Q N = G^2 + 2L_AK_Q N = G 2 + 2 L A K Q κ < M / N \kappa< M/N κ < M / N
From velocity error to endpoint distance
The own-law velocity certificate of Theorem 10 does not by itself imply endpoint improvement. Endpoint displacement depends on how velocity errors accumulate over the trajectory. The following proposition identifies the role of temporal cancellation, which motivates the endpoint gain analysis below.
Proposition 11 (Temporal cancellation and endpoint displacement). There is a one-dimensional instance with state-independent ideal, base and corrected fields, satisfying Assumption 7 on one sampler interval, in which the corrected field has strictly smaller integrated squared velocity error and a strictly smaller one-sided path-length bound than the base field, while its terminal W 2 W_2 W 2
Proof. Reparameterize the sampler interval by t = 1 − σ ∈ [ 0 , 1 ] t = 1 - \sigma\in[0,1] t = 1 − σ ∈ [ 0 , 1 ] t = 0 t = 0 t = 0 t = 1 t = 1 t = 1 μ \mu μ R \mathbb{R} R Z ∼ μ Z \sim\mu Z ∼ μ Z Z Z
b ( t ) = { + 1 , 0 ≤ t < 1 2 , − 1 , 1 2 ≤ t ≤ 1 , (100) b(t) =
\begin{cases}
+1, & 0 \le t < \frac{1}{2}, \\
-1, & \frac{1}{2} \le t \le1,
\end{cases} \tag*{(100)} b ( t ) = { + 1 , − 1 , 0 ≤ t < 2 1 , 2 1 ≤ t ≤ 1 , (100) as the base drift, fix any gain κ c e ∈ ( 0 , κ ˉ ] \kappa_{\mathrm{ce}} \in(0,\bar{\kappa}] κ ce ∈ ( 0 , κ ˉ ] g ( t ) = ( 1 2 − b ( t ) ) / κ c e g(t) = \left(\frac{1}{2} - b(t)\right)/\kappa_{\mathrm{ce}} g ( t ) = ( 2 1 − b ( t ) ) / κ ce 1 2 \frac{1}{2} 2 1
Direct integration of X ˙ t ⋆ = 0 \dot{X}^{\star}_t = 0 X ˙ t ⋆ = 0 X ˙ t 0 = b \dot{X}^{0}_t = b X ˙ t 0 = b X ˙ t c e = 1 2 \dot{X}^{\mathrm{ce}}_t = \frac{1}{2} X ˙ t ce = 2 1
X t ⋆ = Z , X t 0 = { Z + t , 0 ≤ t ≤ 1 2 , Z + 1 − t , 1 2 ≤ t ≤ 1 , X t c e = Z + t 2 , (101) X^{\star}_t = Z,\qquad
X^0_t =
\begin{cases}
Z+t, & 0 \le t \le\frac{1}{2}, \\
Z+1-t, & \frac{1}{2} \le t \le1,
\end{cases}
\qquad
X^{\mathrm{ce}}_t = Z+\frac{t}{2}, \tag*{(101)} X t ⋆ = Z , X t 0 = { Z + t , Z + 1 − t , 0 ≤ t ≤ 2 1 , 2 1 ≤ t ≤ 1 , X t ce = Z + 2 t , (101) so that X 1 ⋆ = X 1 0 = Z X^{\star}_1 = X^0_1 = Z X 1 ⋆ = X 1 0 = Z X 1 c e = Z + 1 2 X^{\mathrm{ce}}_1 = Z+\frac{1}{2} X 1 ce = Z + 2 1
Because both velocity errors are state independent, evaluating them under any common state law gives
∫ 0 1 E ∣ 0 − b ( t ) ∣ 2 d t = 1 , ∫ 0 1 E ∣ 0 − 1 2 ∣ 2 d t = 1 4 , (102) \int_0^1 \mathbb{E}|0-b(t)|^2\,dt = 1,\qquad
\int_0^1 \mathbb{E}\left|0-\frac{1}{2}\right|^2\,dt = \frac{1}{4}, \tag*{(102)} ∫ 0 1 E ∣0 − b ( t ) ∣ 2 d t = 1 , ∫ 0 1 E 0 − 2 1 2 d t = 4 1 , (102) and the one-sided path-length bounds from the common-source coupling are ∫ 0 1 ∣ b ∣ d t = 1 \int_0^1 |b|\,dt = 1 ∫ 0 1 ∣ b ∣ d t = 1 ∫ 0 1 1 2 d t = 1 2 \int_0^1 \frac{1}{2}\,dt = \frac{1}{2} ∫ 0 1 2 1 d t = 2 1 (101) gives π 1 ⋆ = π 1 0 = μ \pi^{\star}_1 = \pi^0_1 = \mu π 1 ⋆ = π 1 0 = μ π 1 c e \pi^{\mathrm{ce}}_1 π 1 ce μ \mu μ 1 2 \frac{1}{2} 2 1 W 2 ( π 1 c e , π 1 ⋆ ) ≤ 1 2 W_2(\pi^{\mathrm{ce}}_1,\pi^{\star}_1) \le\frac{1}{2} W 2 ( π 1 ce , π 1 ⋆ ) ≤ 2 1 ( Y , Y ′ ) (Y,Y') ( Y , Y ′ ) E ∣ Y − Y ′ ∣ 2 ≥ ∣ E Y − E Y ′ ∣ 2 = 1 4 \mathbb{E}|Y-Y'|^2 \ge|\mathbb{E}Y-\mathbb{E}Y'|^2 = \frac{1}{4} E ∣ Y − Y ′ ∣ 2 ≥ ∣ E Y − E Y ′ ∣ 2 = 4 1
W 2 ( π 1 0 , π 1 ⋆ ) = 0 , W 2 ( π 1 c e , π 1 ⋆ ) = 1 2 . (103) W_2(\pi^0_1,\pi^{\star}_1) = 0,\qquad
W_2(\pi^{\mathrm{ce}}_1,\pi^{\star}_1) = \frac{1}{2}. \tag*{(103)} W 2 ( π 1 0 , π 1 ⋆ ) = 0 , W 2 ( π 1 ce , π 1 ⋆ ) = 2 1 . (103) These values prove the stated strict inequalities. The signed drift integral determines endpoint displacement in this construction, while squared velocity error measures the drift’s energy over time. ∎
For affine–Gaussian flows with a common linear part, the correction shifts the endpoint mean by an explicit displacement. We use this structure to derive an exact endpoint gain identity. The common linear part propagates the correction through the fundamental solution, and a state-independent correction preserves covariance by translating each conditional law. The square-integrability conditions make the averaged endpoint quadratic finite.
Assumption 9 (Exact affine–Gaussian endpoint specialization). Conditionally on C = c C=c C = c M ( σ ) x M(\sigma)x M ( σ ) x P C P_C P C c c c σ \sigma σ [ 0 , 1 ] [0,1] [ 0 , 1 ] g ϕ ( σ , c ) g_{\phi}(\sigma,c) g ϕ ( σ , c ) Ψ ( σ , s ) \Psi(\sigma,s) Ψ ( σ , s ) x ˙ = M ( σ ) x \dot{x}=M(\sigma)x x ˙ = M ( σ ) x m ⋆ 0 ( c ) m_{\star}^{0}(c) m ⋆ 0 ( c ) m 0 0 ( c ) m_{0}^{0}(c) m 0 0 ( c ) E 0 ( c ) = m ⋆ 0 ( c ) − m 0 0 ( c ) E_{0}(c)=m_{\star}^{0}(c)-m_{0}^{0}(c) E 0 ( c ) = m ⋆ 0 ( c ) − m 0 0 ( c )
G ( c ) = ∫ 1 0 Ψ ( 0 , s ) g ϕ ( s , c ) d s , (104) G(c)=\int_{1}^{0}\Psi(0,s)g_{\phi}(s,c)\,\mathrm{d}s, \tag*{(104)} G ( c ) = ∫ 1 0 Ψ ( 0 , s ) g ϕ ( s , c ) d s , (104) in the sampler orientation. Both E 0 E_{0} E 0 G G G L 2 ( P C ; X ) L^{2}(P_C;\mathcal{X}) L 2 ( P C ; X )
Proposition 12 (Endpoint displacement and covariance preservation). Under Assumptions 7 and 9, for P C P_C P C c c c κ \kappa κ π 0 κ ( ⋅ ∣ c ) \pi_{0}^{\kappa}(\cdot\mid c) π 0 κ ( ⋅ ∣ c )
m 0 κ ( c ) = m 0 0 ( c ) + κ G ( c ) , Cov ( X 0 ⋆ ( c ) ) = Cov ( X 0 0 ( c ) ) = Cov ( X 0 κ ( c ) ) = Ξ 0 ( c ) . (105) m_{0}^{\kappa}(c)=m_{0}^{0}(c)+\kappa G(c),\qquad\operatorname{Cov}(X_{0}^{\star}(c))=\operatorname{Cov}(X_{0}^{0}(c))=\operatorname{Cov}(X_{0}^{\kappa}(c))=\Xi_{0}(c). \tag*{(105)} m 0 κ ( c ) = m 0 0 ( c ) + κ G ( c ) , Cov ( X 0 ⋆ ( c )) = Cov ( X 0 0 ( c )) = Cov ( X 0 κ ( c )) = Ξ 0 ( c ) . (105) Proof. Fix a condition c c c x ↦ M ( σ ) x + b 0 ( σ ) x\mapsto M(\sigma)x+b_{0}(\sigma) x ↦ M ( σ ) x + b 0 ( σ ) x ↦ M ( σ ) x + b 0 ( σ ) + κ g ϕ ( σ , c ) x\mapsto M(\sigma)x+b_{0}(\sigma)+\kappa g_{\phi}(\sigma,c) x ↦ M ( σ ) x + b 0 ( σ ) + κ g ϕ ( σ , c ) X 1 X_{1} X 1
X 0 0 = Ψ ( 0 , 1 ) X 1 + ∫ 1 0 Ψ ( 0 , s ) b 0 ( s ) d s , X_{0}^{0}=\Psi(0,1)X_{1}+\int_{1}^{0}\Psi(0,s)b_{0}(s)\,\mathrm{d}s, X 0 0 = Ψ ( 0 , 1 ) X 1 + ∫ 1 0 Ψ ( 0 , s ) b 0 ( s ) d s , X 0 κ = Ψ ( 0 , 1 ) X 1 + ∫ 1 0 Ψ ( 0 , s ) [ b 0 ( s ) + κ g ϕ ( s , c ) ] d s = X 0 0 + κ G ( c ) . X_{0}^{\kappa}=\Psi(0,1)X_{1}+\int_{1}^{0}\Psi(0,s)\left[b_{0}(s)+\kappa g_{\phi}(s,c)\right]\,\mathrm{d}s=X_{0}^{0}+\kappa G(c). X 0 κ = Ψ ( 0 , 1 ) X 1 + ∫ 1 0 Ψ ( 0 , s ) [ b 0 ( s ) + κ g ϕ ( s , c ) ] d s = X 0 0 + κ G ( c ) . Taking conditional expectations gives the first display of equation 105. The same identity is a pathwise translation under the common-source coupling, and κ G ( c ) \kappa G(c) κ G ( c ) c c c X 0 κ − m 0 κ ( c ) = X 0 0 − m 0 0 ( c ) X_{0}^{\kappa}-m_{0}^{\kappa}(c)=X_{0}^{0}-m_{0}^{0}(c) X 0 κ − m 0 κ ( c ) = X 0 0 − m 0 0 ( c ) Ξ 0 ( c ) \Xi_{0}(c) Ξ 0 ( c )
The terminal-value problem fixes the integration direction from one to zero. For example, if M ≡ 0 M\equiv0 M ≡ 0 g ϕ ( s , c ) ≡ g g_{\phi}(s,c)\equiv g g ϕ ( s , c ) ≡ g G ( c ) = − g G(c)=-g G ( c ) = − g X 0 0 − κ g X_{0}^{0}-\kappa g X 0 0 − κ g
Proposition 13 (Exact endpoint gain quadratic). Under Assumption 9 and Proposition 12, put
A e n d = E C ⟨ E 0 , G ⟩ , B e n d = E C ∥ G ∥ 2 ≥ 0. (106) A_{\mathrm{end}}=\mathbb{E}_{C}\langle E_{0},G\rangle,\qquad B_{\mathrm{end}}=\mathbb{E}_{C}\lVert G\rVert^{2}\ge0. \tag*{(106)} A end = E C ⟨ E 0 , G ⟩ , B end = E C ∥ G ∥ 2 ≥ 0. (106) Then for every finite gain κ \kappa κ
E C W 2 2 ( π 0 κ , π 0 ⋆ ) − E C W 2 2 ( π 0 0 , π 0 ⋆ ) = − 2 κ A e n d + κ 2 B e n d . (107) \mathbb{E}_{C}W_{2}^{2}(\pi_{0}^{\kappa},\pi_{0}^{\star})-\mathbb{E}_{C}W_{2}^{2}(\pi_{0}^{0},\pi_{0}^{\star})=-2\kappa A_{\mathrm{end}}+\kappa^{2}B_{\mathrm{end}}. \tag*{(107)} E C W 2 2 ( π 0 κ , π 0 ⋆ ) − E C W 2 2 ( π 0 0 , π 0 ⋆ ) = − 2 κ A end + κ 2 B end . (107) Proof. We first compute the transport distance for a common covariance, then apply the mean displacement and average over conditions.
Step 1: Compute the Wasserstein distance for a common covariance. For any m , n ∈ X m,n\in\mathcal{X} m , n ∈ X Ξ \Xi Ξ X \mathcal{X} X
W 2 2 ( N ( m , Ξ ) , N ( n , Ξ ) ) = ∥ m − n ∥ 2 . (108) W_{2}^{2}\bigl(\mathcal{N}(m,\Xi),\mathcal{N}(n,\Xi)\bigr)=\lVert m-n\rVert^{2}. \tag*{(108)} W 2 2 ( N ( m , Ξ ) , N ( n , Ξ ) ) = ∥ m − n ∥ 2 . (108) The identity includes singular Ξ \Xi Ξ Ξ = 0 \Xi=0 Ξ = 0
Let Z ∼ N ( 0 , Ξ ) Z\sim\mathcal{N}(0,\Xi) Z ∼ N ( 0 , Ξ ) Ξ \Xi Ξ X = m + Z X=m+Z X = m + Z Y = n + Z Y=n+Z Y = n + Z E ∥ X − Y ∥ 2 = ∥ m − n ∥ 2 \mathbb{E}\lVert X-Y\rVert^{2}=\lVert m-n\rVert^{2} E ∥ X − Y ∥ 2 = ∥ m − n ∥ 2 ( X ~ , Y ~ ) (\widetilde{X},\widetilde{Y}) ( X , Y ) z ↦ ∥ z ∥ 2 z\mapsto\lVert z\rVert^{2} z ↦ ∥ z ∥ 2 E ∥ X ~ − Y ~ ∥ 2 ≥ ∥ E ( X ~ − Y ~ ) ∥ 2 = ∥ m − n ∥ 2 \mathbb{E}\lVert\widetilde{X}-\widetilde{Y}\rVert^{2}\ge\lVert\mathbb{E}(\widetilde{X}-\widetilde{Y})\rVert^{2}=\lVert m-n\rVert^{2} E ∥ X − Y ∥ 2 ≥ ∥ E ( X − Y ) ∥ 2 = ∥ m − n ∥ 2 Ξ \Xi Ξ
Step 2: Apply the conditional endpoint mean displacement. Proposition 12 gives m 0 ⋆ ( c ) − m 0 κ ( c ) = E 0 ( c ) − κ G ( c ) m_0^{\star}(c)-m_0^{\kappa}(c)=E_0(c)-\kappa G(c) m 0 ⋆ ( c ) − m 0 κ ( c ) = E 0 ( c ) − κ G ( c ) Ξ 0 ( c ) \Xi_0(c) Ξ 0 ( c ) P C P_C P C c c c
W 2 2 ( π 0 κ ( ⋅ ∣ c ) , π 0 ⋆ ( ⋅ ∣ c ) ) = ∥ E 0 ( c ) − κ G ( c ) ∥ 2 , W 2 2 ( π 0 0 ( ⋅ ∣ c ) , π 0 ⋆ ( ⋅ ∣ c ) ) = ∥ E 0 ( c ) ∥ 2 (109) W_2^2\left(\pi_0^{\kappa}(\cdot\mid c),\pi_0^{\star}(\cdot\mid c)\right)=\left\|E_0(c)-\kappa G(c)\right\|^2,\qquad W_2^2\left(\pi_0^0(\cdot\mid c),\pi_0^{\star}(\cdot\mid c)\right)=\left\|E_0(c)\right\|^2 \tag*{(109)} W 2 2 ( π 0 κ ( ⋅ ∣ c ) , π 0 ⋆ ( ⋅ ∣ c ) ) = ∥ E 0 ( c ) − κ G ( c ) ∥ 2 , W 2 2 ( π 0 0 ( ⋅ ∣ c ) , π 0 ⋆ ( ⋅ ∣ c ) ) = ∥ E 0 ( c ) ∥ 2 (109) These identities express the conditional endpoint distances as squared norms of mean displacements.
Step 3: Average the exact quadratic over conditions. We now average over conditions. Assumption 9 makes E 0 E_0 E 0 G G G ∥ E 0 − κ G ∥ 2 ≤ 2 ∥ E 0 ∥ 2 + 2 κ 2 ∥ G ∥ 2 \|E_0-\kappa G\|^2\le2\|E_0\|^2+2\kappa^2\|G\|^2 ∥ E 0 − κ G ∥ 2 ≤ 2∥ E 0 ∥ 2 + 2 κ 2 ∥ G ∥ 2 E C ∣ ⟨ E 0 , G ⟩ ∣ ≤ ( E C ∥ E 0 ∥ 2 ) 1 / 2 ( E C ∥ G ∥ 2 ) 1 / 2 < ∞ \mathbb{E}_C|\langle E_0,G\rangle|\le(\mathbb{E}_C\|E_0\|^2)^{1/2}(\mathbb{E}_C\|G\|^2)^{1/2}<\infty E C ∣ ⟨ E 0 , G ⟩ ∣ ≤ ( E C ∥ E 0 ∥ 2 ) 1/2 ( E C ∥ G ∥ 2 ) 1/2 < ∞ ∥ E 0 − κ G ∥ 2 − ∥ E 0 ∥ 2 = − 2 κ ⟨ E 0 , G ⟩ + κ 2 ∥ G ∥ 2 \|E_0-\kappa G\|^2-\|E_0\|^2=-2\kappa\langle E_0,G\rangle+\kappa^2\|G\|^2 ∥ E 0 − κ G ∥ 2 − ∥ E 0 ∥ 2 = − 2 κ ⟨ E 0 , G ⟩ + κ 2 ∥ G ∥ 2
Corollary 14 (Endpoint gain window). Under Proposition 13 , the corrected sampler has strictly smaller averaged squared endpoint distance than the base sampler exactly when B e n d > 0 B_{\mathrm{end}}>0 B end > 0 A e n d > 0 A_{\mathrm{end}}>0 A end > 0
0 < κ < 2 A e n d B e n d , (110) 0<\kappa<\frac{2A_{\mathrm{end}}}{B_{\mathrm{end}}}, \tag*{(110)} 0 < κ < B end 2 A end , (110) with equality at both endpoints of that interval and strict degradation beyond it. The unique optimal gain is A e n d / B e n d A_{\mathrm{end}}/B_{\mathrm{end}} A end / B end B e n d = 0 B_{\mathrm{end}}=0 B end = 0 G = 0 G=0 G = 0 E 0 = 0 E_0=0 E 0 = 0 A e n d = 0 A_{\mathrm{end}}=0 A end = 0 B e n d > 0 B_{\mathrm{end}}>0 B end > 0
Proof. (107) has the same shape as equation 79 with A e n d A_{\mathrm{end}} A end B e n d B_{\mathrm{end}} B end A Q A_Q A Q B Q B_Q B Q κ ↦ E C W 2 2 ( π 0 κ , π 0 ⋆ ) \kappa\mapsto\mathbb{E}_C W_2^2(\pi_0^{\kappa},\pi_0^{\star}) κ ↦ E C W 2 2 ( π 0 κ , π 0 ⋆ ) E 0 = 0 E_0=0 E 0 = 0 ⟨ E 0 , G ⟩ = 0 \langle E_0,G\rangle=0 ⟨ E 0 , G ⟩ = 0 A e n d = 0 A_{\mathrm{end}}=0 A end = 0 Proposition 13 covers every positive-semidefinite conditional covariance Ξ 0 \Xi_0 Ξ 0
Implementation Details
We expand the settings in Sections 5.1–5.3, covering model configurations, optimization, inference and evaluation.
Image Generation
Model configuration. We build on the released ImageNet-256 RAEv2 checkpoint ([40 ] ) trained for 80 epochs. The representation autoencoder uses the DINOv3-L encoder with seven-layer feature aggregation and a pretrained ViT-XL decoder. We keep the encoder, decoder and latent normalization statistics fixed throughout adaptation. Each image is represented by a 16 × 16 16\times16 16 × 16
Training. The base retains its flow-matching objective, including supervision of the internal guidance branch. Training noise levels follow the base configuration: we draw a logit-normal variable with underlying Gaussian mean 0 and standard deviation 1, then apply the same factor-8 shift used for sampling below. The refiner minimizes the mean squared error to the realized velocity residual, together with a cosine-direction term of weight 0.1. Its overall loss weight is 1. We detach the residual target and all base-model inputs to the refiner, so the refiner loss updates only the refiner and the base is updated by its own objective. Both networks use GMuon with momentum 0.95, Nesterov updates, zero weight decay and gradient-norm clipping at 1. The base and refiner learning rates warm up from zero over 0.2 epochs to 2 × 10 − 5 2\times10^{-5} 2 × 1 0 − 5 4 × 10 − 4 4\times10^{-4} 4 × 1 0 − 4 2 × 10 − 6 2\times10^{-6} 2 × 1 0 − 6 4 × 10 − 5 4 \times10^{-5} 4 × 1 0 − 5
Sampling and deployment. At test time, we use the EMA weights and generate 50,000 images, with 50 images for each of the 1,000 classes. We start from standard Gaussian latent noise and take 100 Euler steps on the base model’s shifted noise schedule. Specifically, a uniform grid s i = 1 − i / 100 s_i = 1 - i/100 s i = 1 − i /100 i = 0 , … , 100 i = 0, \ldots, 100 i = 0 , … , 100 σ i = 8 s i / ( 1 + 7 s i ) \sigma_i = 8s_i/(1 + 7s_i) σ i = 8 s i / ( 1 + 7 s i ) 1024 ⋅ 16 ⋅ 16 / 4096 \sqrt{1024 \cdot16 \cdot16}/4096 1024 ⋅ 16 ⋅ 16 /4096 max ( σ , 0.05 ) \max(\sigma, 0.05) max ( σ , 0.05 ) σ ∈ [ 0.1 , 1 ] \sigma\in[0.1, 1] σ ∈ [ 0.1 , 1 ] κ = 1 \kappa= 1 κ = 1 σ ∈ [ 0.15 , 0.92 ] \sigma\in[0.15, 0.92] σ ∈ [ 0.15 , 0.92 ]
Baselines. The published-baseline rows in Table 7 cover SiT-XL/2, DDT-XL, SiT-XL/2-REPA, LightningDiT, REG, REPA-E and RAE-XL. We use their reported evaluations from FD-loss (Yang et al., [54 ] ), whose representation-distance results are also reproduced by RAEv2 (Singh et al., [40 ] ). That evaluation generates 50,000 images from official checkpoints with the corresponding official sampling code, balancing the sample count across classes and using method-specific inference settings. Table 6 summarizes the sampling budgets and guidance mechanisms. The six VAE-based methods use 250 sampling steps with classifier-free guidance (CFG), while RAE-XL uses 50 Euler steps and AutoGuidance from a separately trained weaker model. These published results therefore retain their original sampling budgets and guidance configurations.
Table 6. Evaluation configurations for all seven published image-generation baselines in Table 7. Steps count sampler updates. CFG combines conditional and unconditional predictions, while AutoGuidance combines a main model with a weaker model.
Table 7. Comparison on ImageNet-256 (Deng et al., [8 ] ) generation. For published baselines, the FID column reports guided FID (gFID) as in the original papers. Models are evaluated with and without guidance as indicated. For our method, 80 + 40 and 80 + 80 indicate that we start from the 80-epoch RAEv2 checkpoint and train our refiner for an additional 40 and 80 epochs, respectively. “Ours (base only)” evaluates the corresponding base model without the refiner at inference and reports the total base-model training epochs. Representation Fréchet Distance (FDr) is reported in the feature spaces of ConvNeXt (Liu et al., [25 ] ), DINOv2 (Oquab et al., [30 ] ) and CLIP (Radford et al., [34 ] ).
The RAEv2 models with different training epochs use the same frozen autoencoder, EMA weights, 100-step sampler, class-balanced sample count and guidance configuration as CorrFlow. This comparison isolates the effect of deploying the refiner at a fixed base checkpoint. For SelfFlow (Chefer et al., [5 ] ), we evaluate a local adaptation of its self-supervised flow-matching objective to the same frozen DINOv3-L representation autoencoder. This baseline uses a 30-block transformer of width 1,280 with 20 attention heads, a dual-noise mixing ratio of 0.25 and representation alignment of weight 0.8 between student block 9 and EMA-target block 21. We use EMA weights, the same 100-step shifted Euler grid and 50,000 class-balanced images. The unguided runs disable guidance, and the guided runs use CFG at scale 1.5 over σ ∈ [ 0 , 0.7 ] \sigma\in[0, 0.7] σ ∈ [ 0 , 0.7 ]
Inception-based metrics. We report Inception Score (IS, higher is better) (Salimans et al., [36 ] ) and Fréchet Inception Distance (FID, lower is better) (Heusel et al., [13 ] ). IS measures the confidence and marginal diversity of Inception-v3 class predictions on generated images. For a generated image x x x p ( y ∣ x ) p(y \mid x) p ( y ∣ x ) y y y p ( y ) p(y) p ( y ) = exp ( E x K L ( p ( y ∣ x ) ∥ p ( y ) ) ) = \exp(\mathbb{E}_{x}\mathrm{KL}(p(y \mid x)\mathbin{\|}p(y))) = exp ( E x KL ( p ( y ∣ x ) ∥ p ( y )))
Representation-based metrics. We also report Representation Fréchet Distance (F D r \mathrm{FD}_{r} FD r [54 ] ), each table entry is a ratio rather than a raw Fréchet distance. For a frozen feature extractor ϕ \phi ϕ ( μ G , Σ G ) (\mu_G, \Sigma_G) ( μ G , Σ G ) ( μ R , Σ R ) (\mu_R, \Sigma_R) ( μ R , Σ R ) G G G R R R
F D ϕ ( G , R ) = ∥ μ G − μ R ∥ 2 2 + Tr ( Σ G + Σ R − 2 ( Σ R 1 / 2 Σ G Σ R 1 / 2 ) 1 / 2 ) (111) \mathrm{FD}_{\phi}(G,R)=\lVert\mu_G-\mu_R\rVert_2^2+\operatorname{Tr}\left(\Sigma_G+\Sigma_R-2\left(\Sigma_R^{1/2}\Sigma_G\Sigma_R^{1/2}\right)^{1/2}\right) \tag*{(111)} FD ϕ ( G , R ) = ∥ μ G − μ R ∥ 2 2 + Tr ( Σ G + Σ R − 2 ( Σ R 1/2 Σ G Σ R 1/2 ) 1/2 ) (111) and normalize it by the distance from real ImageNet validation images V V V
F D r , ϕ ( G ) = F D ϕ ( G , R ) F D ϕ ( V , R ) . (112) \mathrm{FD}_{r,\phi}(G)=\frac{\mathrm{FD}_{\phi}(G,R)}{\mathrm{FD}_{\phi}(V,R)}. \tag*{(112)} FD r , ϕ ( G ) = FD ϕ ( V , R ) FD ϕ ( G , R ) . (112) The ConvNeXt column uses spatially averaged ConvNeXt V2-Base features (Woo et al. [47 ] ) at input resolution 224 × 224 224 \times224 224 × 224 [30 ] ) and OpenAI CLIP ViT-L/14 (Radford et al. [34 ] ), respectively, both evaluated at 256 × 256 256 \times256 256 × 256
Text-to-Video Generation
Model configuration. We build on Wan2.1-T2V-1.3B (Wan et al. [43 ] ), keeping its pretrained video variational autoencoder (VAE) and UMT5-XXL text encoder fixed while updating the full base transformer and the refiner. The base transformer has 30 blocks of width 1,536 with 12 attention heads. The VAE maps an 81-frame video at 480 × 832 480 \times832 480 × 832 60 × 104 60 \times104 60 × 104 1 × 2 × 2 1 \times2 \times2 1 × 2 × 2
We read hidden states after base blocks 4, 12, 20 and 30, corresponding to zero-based code indices 3, 11, 19 and 29. At each token position, we concatenate the four 1,536-dimensional states and fuse them with a 6144 → 1024 → 1024 MLP with a GELU activation. We project the 4,096-dimensional text embeddings to the same width, add learned segment embeddings to distinguish text from hidden-state features and concatenate the two token sequences before layer normalization. The refiner processes the one-step clean prediction and maps its output back to a velocity residual of the original latent shape. It contains 146.58 million parameters, approximately 10.3% of the 1.419-billion-parameter base transformer. The combined transformer and refiner contain 1.566 billion parameters, excluding the frozen VAE and text encoder.
Data and preprocessing. The final training mixture contains 46,572 WISA-80K clips and 20,000 curated OpenVidHD clips, for 66,572 video–text pairs. The WISA subset retains clips with visual-quality scores of at least 4.0 and motion scores of at least 0.1. For WISA, we concatenate the video caption with its physical-law annotation, following the dataset’s level-B conditioning. For OpenVidHD (Nan et al. [29 ] ), we use the supplied caption. We shuffle the concatenated datasets without additional source reweighting. Each training example contains 81 frames sampled uniformly across the complete clip, preserving the temporal extent of the depicted event. Short clips reuse sampled frame indices to obtain the same length. We resize frames while preserving their aspect ratio and center-crop them to 480 × 832 before VAE encoding.
Training. We jointly train the base model and refiner for five epochs on the processed 66,572-clip dataset using AdamW, with learning rates of 5 × 10 − 6 5 \times10^{-6} 5 × 1 0 − 6 10 − 4 10^{-4} 1 0 − 4 ( 0.9 , 0.999 ) (0.9, 0.999) ( 0.9 , 0.999 )
Sampling and deployment. We generate 81-frame videos at 480 × 832 resolution and save them at 16 frames per second. Sampling starts from standard Gaussian latent noise and uses 50 Euler updates with noise shift 5 and classifier-free guidance (CFG) scale 5. For s i = 1 − i / 50 s_i = 1 - i/50 s i = 1 − i /50 i = 0 , … , 50 i = 0, \ldots, 50 i = 0 , … , 50 σ i = 5 s i / ( 1 + 4 s i ) \sigma_i = 5s_i/(1 + 4s_i) σ i = 5 s i / ( 1 + 4 s i ) κ = 0.2 \kappa= 0.2 κ = 0.2 σ ∈ [ 0.15 , 0.92 ] \sigma\in[0.15, 0.92] σ ∈ [ 0.15 , 0.92 ] v + v_+ v + v − v_- v − r ^ + \hat{r}_+ r ^ +
v s a m p l e = v − + 5 ( v + + 0.2 r ^ + − v − ) = v − + 5 ( v + − v − ) + r ^ + . (113) v_{\mathrm{sample}} = v_- + 5(v_+ + 0.2\hat{r}_+ - v_-) = v_- + 5(v_+ - v_-) + \hat{r}_+. \tag*{(113)} v sample = v − + 5 ( v + + 0.2 r ^ + − v − ) = v − + 5 ( v + − v − ) + r ^ + . (113) Thus the branch-level gain yields a unit-strength residual contribution to the guided velocity. We recompute the clean prediction and hidden states from the current latent at every active step. The base runs in bfloat16 and the refiner in float32, and the frozen VAE decodes the final latent to RGB frames.
VBench. We use the released VBench prompt suite and evaluators [14 ] to measure 16 dimensions of video quality and text alignment. Seven dimensions measure visual quality and temporal behavior, and nine measure semantic consistency. Table 8 and Table 9 display dimension scores on a 0–100 scale. Quality Score and Semantic Score use the benchmark’s dimension-specific normalization and weights, and Overall Score combines them with weights 0.8 and 0.2, respectively. Temporal Flickering measures resistance to frame-to-frame flicker, so a larger score indicates better temporal stability. Dynamic Degree measures the presence of motion through optical flow and complements Motion Smoothness, which evaluates temporal regularity.
Table 8. VBench results (Part 1/2): consistency, motion and object-level metrics. All scores are reported on a 0–100 scale.
Table 9. VBench results (Part 2/2): appearance, style and overall scores. All scores are reported on a 0–100 scale.
VideoPhy2. We evaluate the 590-prompt test set, including its 180-prompt HARD subset, with the released VideoPhy2 automatic evaluator [1 ] . Generation uses the benchmark’s expanded captions, while scoring uses the original action captions. The evaluator assigns ratings from 1 to 5 for physical commonsense (PC) and semantic adherence (SA). The PC and SA columns report the percentage of ratings at least 4, and Joint reports the percentage satisfying both thresholds. Joint therefore measures whether the requested event is depicted with plausible physics, combining two requirements that the separate scores assess individually.
FinePhyEval. We evaluate all 65 FinePhyEval prompts using their released question graphs [32 ] . These graphs decompose each event into object existence, action execution and physical behavior. Our local evaluation uses GPT-5.5 to answer the fixed questions from sampled video frames in the two-step PQSG question-answering procedure. A failed prerequisite propagates failure to dependent questions, so a physics judgment also requires the relevant objects and actions to be present. Object, Action and Physics are the fractions of satisfied questions in each category after this propagation, with all questions retained in their respective denominators. Overall is the corresponding fraction across all questions, weighting categories by their question counts. All four scores lie between 0 and 1.
Comparison methods. We compare with the released Wan2.1 models and methods that adapt video generators for motion or physical consistency, including VideoJAM [4 ] , DreamWorld [42 ] , VPT [64 ] , VideoRPE [61 ] and PhysRAG [7 ] , as available for each benchmark. The model-size suffixes identify the base-generator family, while the refiner parameter count is reported above. The tables combine published benchmark results with local evaluations of released implementations, retaining method-specific generation settings.
Image-to-Video Generation
Model configuration. We build on Wan2.2-TI2V-5B [43 ] , whose base transformer has 30 blocks of width 3,072 with 24 attention heads. As in Appendix D.2, we freeze the video VAE and text encoder and update the full base transformer together with the refiner. The Wan2.2 VAE encodes an 81-frame training video at 704 × 1248 704 \times1248 704 × 1248 30 × 52 30 \times52 30 × 52 12288 → 2048 → 2048 12288 \to2048 \to2048 12288 → 2048 → 2048
Training. The training mixture, caption construction, frame sampling and spatial preprocessing are the same as in Appendix D.2. We also use the same five-epoch schedule, global batch size, AdamW configuration, learning rates, loss weights, mixed precision and gradient checkpointing. For each training example, we enable image conditioning with probability 0.4 by encoding the first sampled frame with the frozen VAE. For these examples, we replace the first temporal latent with this clean image latent and assign its tokens noise level zero in the base transformer. Both the base flow-matching loss and the refiner residual loss are computed over the remaining temporal latents. The other examples use text conditioning alone. We detach the residual target and base-model inputs to the refiner, so each network is updated by its own objective. Training begins on 16 NVIDIA H200 GPUs with one clip per GPU and resumes after 4,200 optimizer steps on four H200 GPUs with four gradient-accumulation steps, preserving the global batch size of 16. We evaluate the paired base and refiner weights at the end of the fifth epoch.
Sampling and image conditioning. We generate at 704 × 1248 704 \times1248 704 × 1248 (113) , and the base and refiner use the same inference precisions as in text-to-video generation. For each benchmark example, we supply the released image at the start of the continuation and its corresponding text prompt, including the benchmark’s fixed-camera instructions and action description. We resize the image to the generation resolution, encode it with the frozen VAE and keep its latent fixed at the first temporal position. We restore this condition in the refiner’s clean prediction and after every sampling update.
Physics-IQ Verified. The image-to-video evaluation covers 66 physical scenarios from three camera viewpoints, yielding 198 image–text pairs. We retain the first 80 frames of each generated video for the benchmark’s five-second evaluation at 16 frames per second. The released evaluator compares generated continuations with recorded reference videos at 540 × 960 540 \times960 540 × 960
For each scenario and viewpoint, the evaluator normalizes these measurements by the corresponding comparison between two real recordings of the same event. The three overlap metrics are divided by this real-to-real reference, while MSE uses the inverse ratio so that higher is better. Each normalized component is clipped to [ 0 , 1 ] [0, 1] [ 0 , 1 ] Table 4 reports all scores on a 0–100 scale. All rows in Table 4 use the benchmark’s setting without LLM support.
Yuan et al. [58 ] describe two inference-time uses of the WMReward signal: gradient guidance, which steers each denoising trajectory with the reward gradient, and Best-of-N selection, which draws multiple guided samples and keeps the one with the highest reward. We enable only reward guidance and generate a single sample per condition, without Best-of-N selection. Because Yuan et al. [58 ] attribute most of their Physics-IQ improvement to the selection step, the WMReward row is expected to score below their guidance-plus-selection results. This choice keeps the comparison at a matched sampling budget. The other baselines are run using their default configurations as released.
More Results
More Results on ImageNet Generation
Quantitative Results
Table 7 extends the comparison in Table 1 with additional methods, training budgets, base-only evaluations and FID scores.
Qualitative Results
Figures 5 and 6 compare selected samples from Self-Flow, RAEv2 and CorrFlow on class-conditional ImageNet generation at 256 × 256 resolution across eight categories. In these examples, CorrFlow produces clearer object structures and more recognizable category-specific details, including the lattice of shoji screens, the heads and handles of hatchets and the gills of axolotls.
At the same 160-epoch base-model training budget, CorrFlow improves all three FDr metrics over RAEv2 both with and without guidance, consistent with the clearer shoji lattices, hatchet geometry and axolotl gills in Figures 5 and 6. With guidance, IS increases from 264.97 to 266.60 and DINOv2 FDr decreases from 1.0797 to 1.0339, although FID worsens from 1.0337 to 1.0593 (Table 7 ). The base model exhibits a similar discrepancy: continuing RAEv2 fine-tuning from 120 to 160 total epochs raises guided FID from 1.0235 to 1.0337, while ConvNeXt and CLIP FDr continue to decrease. These discrepancies suggest practical saturation of FID as a perceptual ranking signal in this regime near 1, where small score differences are an unreliable standalone indicator of generation quality. FID compares only the means and covariances of Inception-v3 features, so improvements in object structure and fine detail need not yield a lower score. Jayasumana et al. [15 ] document disagreements between FID and human judgments and show that FID can fail to track progressive visual improvements, while Singh et al. [40 ] observe that incremental gFID gains provide little practical signal for strong RAEv2 models. The consistent improvements across complementary representation spaces, together with the visual examples, support the benefit of learned residual corrections beyond what FID alone captures.
Figure 5. Qualitative results on ImageNet at 256 × 256 resolution. Columns show shoji, plate rack, hatchet and marmoset. Row groups show Self-Flow (Selfflow), RAEv2 and CORRFLOW (Ours). Each method–category block contains four samples in a 2 × 2 grid, giving 48 images in total. All three base DiTs are trained for 160 epochs. The CORRFLOW refiner is trained for 80 epochs.
More Results on Text-to-Video Generation
Quantitative Results
Tables 8 and 9 report the full VBench results for the methods in Table 3 , covering all 16 evaluation dimensions and the quality, semantic and overall scores.
Visual quality and temporal consistency. CORRFLOW achieves the highest VBench Quality Score (84.69) and Overall Score (81.86) among the compared methods. Relative to Wan2.1-T2V-1.3B, these gains are 4.88 and 4.93 points, respectively. The improvements span both individual-frame quality and consistency over time. Imaging Quality increases from 59.14 to 68.66, Aesthetic Quality from 57.24 to 61.65 and Subject Consistency from 91.83 to 96.68. Temporal Flickering and Motion
Figure 6. Qualitative results on ImageNet at 256 × 256 256 \times256 256 × 256 2 × 2 2 \times2 2 × 2
Smoothness also improve, reaching 99.81 and 98.85. Together, these metrics indicate that the adapted model improves the appearance of generated frames while maintaining more coherent subjects and smoother temporal evolution. Compared with DreamWorld, the strongest alternative by Overall Score in this table, CORRFLOW improves Quality Score by 1.20 points and Overall Score by 0.89 points.
Semantic alignment and composition. CORRFLOW improves prompt alignment across object identity, composition and action. Relative to Wan2.1-T2V-1.3B, Object Class rises from 76.09 to 82.81, Multiple Objects from 53.66 to 67.72 and Spatial Relationship from 62.37 to 77.88. These improvements support more accurate rendering of the requested entities and their arrangement, and Semantic Score increases from 65.43 to 70.55. Human Action increases from 74.60 to 78.00, and Temporal Style from 22.23 to 24.05. The gains therefore extend from the identity and spatial arrangement of objects to the actions and temporal style specified by the prompt.
Physical plausibility with semantic adherence. On VideoPhy2 (Table 2 ), CORRFLOW improves the ALL-set PC score from 57.8 to 65.7 and SA from 30.0 to 31.3 relative to Wan2.1-T2V-1.3B. Joint increases by 2.2 percentage points to 27.0, … also exceeds those of Wan2.1-14B (24.7) and VideoREPA (23.6). On HARD, CORRFLOW raises PC from 36.7 to 51.9 and Joint from 5.6 to 6.9, corresponding to gains of 15.2 and 1.3 percentage points. The improvements on both test splits show that more generated videos depict the requested event while satisfying physical commonsense.
Object, action and physics requirements. FinePhyEval provides a complementary decomposition of these gains. Relative to Wan2.1-T2V-1.3B, CORRFLOW improves Object from 0.84 to 0.90, Action from 0.49 to 0.55, Physics from 0.36 to 0.44 and Overall from 0.54 to 0.60. It ties DreamWorld on Object and obtains the highest Action and Overall scores among the compared methods. Its Overall score of 0.60 also exceeds the 0.59 achieved by Wan2.1-14B. The question-graph metric rewards physical behavior together with the objects and actions on which it depends. The gains across all three categories therefore support more complete realization of the requested physical events, complementing the visual and temporal improvements measured by VBench.
Qualitative Results
Figures 7, 8 and 9 provide additional qualitative comparisons of VideoREPA, VPT, PhysRAG and CORRFLOW (Ours). The five examples cover domino interactions, multiple-object composition, canoe paddling, jetski racing and parasailing. For each case, rows follow this method order from top to bottom, and columns show five frames sampled at evenly spaced relative positions across each video, progressing from left to right. The text below each case gives the corresponding prompt description.
More Results on Image-to-Video Generation
Quantitative Results
Aggregate physical consistency. CORRFLOW achieves the highest Verified Score in Table 4 , reaching 30.95 compared with 27.71 for Wan2.2-5B. The gain is 3.24 points over the base model and 4.47, 3.52 and 5.41 points over WMReward, VLIPP and PhysRAG, respectively. This score combines motion-region agreement, temporal overlap and frame reconstruction after normalization by variation between real recordings.
Motion coverage and frame appearance. Relative to Wan2.2-5B, CORRFLOW increases Spatial from 35.89 to 43.43 and Weighted Spatial from 22.96 to 24.74. These gains indicate closer agreement with the reference in both the regions traversed by moving objects and the fraction of time each region exhibits motion. MSE Score increases from 25.54 to 29.24, indicating improved agreement in RGB frame appearance under the benchmark’s normalization. CORRFLOW leads the compared methods on all three metrics, supporting the aggregate improvement across complementary aspects of the generated continuation.
Qualitative Results
Figure 4 compares CORRFLOW with WMReward and PhysRAG under the same image and text conditions. In the moving-light example, CORRFLOW retains a recognizable statuette silhouette across the displayed frames as its shadow changes position. The mandarin example shows the knife and fruit remaining identifiable through the slicing motion. These examples illustrate object and shadow consistency alongside the motion and appearance metrics above.
Figure 10 and Figure 11 extend this comparison to three additional scenarios, with ground-truth videos (GT) alongside WAN, VLIPP, WMReward, PhysRAG and CORRFLOW. The paper-in-water example shows how each method depicts the paper after release and preserves the surrounding scene, while the mirror examples show the relationship between object motion and its reflection. Each row presents five video frames in temporal order.
Figure 10. Additional image-to-video comparison on Physics-IQ Verified: paper falling into water. A grabber releases crumpled paper into a bowl of water. Rows show GT, WAN, VLIPP, WMReward, PhysRAG and CORRFLOW (Ours). Each row contains five video frames in temporal order.
Figure 11. Additional image-to-video comparisons on Physics-IQ Verified: motion and reflections. The upper case shows a suspended tennis ball being lowered in front of a mirror, and the lower case shows a tennis ball rolling toward a kettlebell beside a mirror. Within each case, rows show GT, WAN, VLIPP, WMReward, PhysRAG and CorrFlow (Ours). Each row contains five video frames in temporal order.
Figure 12 adds a bouncing tennis ball and a blue ball rolling from a table onto glass, with the same six rows per case. These examples show how each model continues object motion from the initial frame. Red boxes mark the visible balls and adjacent reflections in the four baseline methods, while GT and CORRFLOW remain unmarked.
Figure 12. Additional image-to-video comparisons on Physics-IQ Verified: ball motion. The upper case shows a bouncing tennis ball near a ramp, and the lower case shows a blue ball rolling from a table onto glass. Within each case, rows show GT, WAN, VLIPP, WMReward, PhysRAG and CORRFLOW (Ours). Each row contains five video frames in temporal order. Red boxes highlight visible balls and adjacent reflections in the four baselines after the initial frame.
Figure 7. Qualitative results on text-to-video generation. The upper case shows a rubber duck among dominoes, and the lower case shows a skateboard and a surfboard. Within each case, rows show VideoREPA, VPT, PhysRAG and Ours from top to bottom. Each row contains five frames sampled at evenly spaced relative positions across the video, ordered from left to right.
Full-Precision Ablation Results
Table 10 reports all ablations from § 5.4 on the original 0–1 scale with four decimal places. Table 5 displays the same scores as percentages with three significant digits, while retaining the ranking colors determined at four-decimal precision. Rounding can make distinct scores appear tied. For example, the Imaging Quality scores of 0.6369 and 0.6368 both display as 63.7%, and the Quality Scores of 0.7980 and 0.7979 both display as 79.8%.
Table 10. Component ablations at four-decimal precision (0–1, higher is better). Column abbreviations match Table 5 . Colors indicate the top three scores in each column.
Reproducibility Statement
To support reproducibility, § 3 specifies the refiner architecture, residual training objective and inference procedure of CORRFLOW. Appendix § 10 documents the pretrained models, training data, preprocessing, optimization hyperparameters, hardware and sampling settings for image, text-to-video and image-to-video generation, together with the benchmark protocols and metric definitions. § 5.4 describes the ablation setup, and Appendix § 11 .§ 4 provides the corresponding scores at full precision. The assumptions and complete proofs underlying the theoretical results in § 4 are provided in Appendices § 7 , § 8 and § 9 .
Figure 8. Qualitative results on text-to-video generation. The upper case shows a canoeist paddling across a lake, and the lower case shows two jetskis racing side by side. Within each case, rows show VideoREPA, VPT, PhysRAG and Ours from top to bottom. Each row contains five frames sampled at evenly spaced relative positions across the video, ordered from left to right. A canoeist uses a single-bladed paddle to propel their canoe across a lake, the paddle’s movement visible against the still water. Two jetskis race side-by-side, their wakes colliding in a visible spray of water.
Figure 9. A parasailer, harnessed to a bright orange parachute, is pulled along a calm ocean surface by a speedboat. Qualitative results on text-to-video generation. The prompt describes a parasailer with an orange parachute being pulled across the ocean by a speedboat. Rows show VideoREPA, VPT, PhysRAG and Ours from top to bottom. Each row contains five frames sampled at evenly spaced relative positions across the video, ordered from left to right.
