Introduction
Many inverse problems in signal processing, imaging, and scientific inference become difficult when the observation map removes the necessary information for stable reconstruction. In particular, this becomes more severe when the forward operator smooths the unknown signal. In this case, the fine-scale components for accurate reconstruction are also the components that the data reveals least, and accurate recovery must rely on prior information that is matched to the geometry of the forward map. Traditional regularization methods give prior information through analytic structure, i.e., smoothness, sparsity, total variation, Sobolev penalties, or Gaussian measures [9 , 28 ] . These constructions describe many inverse problems with interpretable assumptions. However, modern signal classes often contain statistical structure that is not well captured by a fixed analytic prior.
Learned priors, on the other hand, have changed reconstruction techniques by coupling the forward model with a learned model of the signal class. Plug-and-play and regularization-by-denoising methods use denoisers as implicit priors and have produced strong empirical results in many imaging and signal recovery problems [16 , 22 , 31 ] . Generative priors constrain the reconstruction to the low-dimensional manifold of a trained generator [5 ] . In contrast to a fixed regularity class, this manifold captures the statistical structure of the training data. Score matching and denoising score matching estimate gradients of log densities without access to normalizing constants [13 , 32 ] . Score-based diffusion models then employ the estimated scores to drive a reverse-time stochastic process from noisy marginals back to the data distribution [1 , 12 , 17 , 20 , 25 , 26 , 27 ] . For inverse problems, the prior no longer has to be an explicit penalty or a low-dimensional generator since a score network can model a rich distribution directly on the ambient signal space.
Diffusion priors have therefore become ubiquitous for posterior reconstruction. Existing work has studied posterior samplers, plug-and-play score priors, hard data-consistency updates, filtering interpretations, robustness guarantees, and frequency-aware guidance rules [7 , 10 , 11 , 14 , 15 , 21 , 24 , 29 , 30 , 33 , 34 ] . These methods show that diffusion priors can produce accurate reconstructions and, in several cases, posterior uncertainty. However, the current analysis mainly considers finite dimensions or measures reconstruction quality in an ambient norm. It therefore does not explain how operator smoothing, truth regularity, score regularity, and the reconstruction norm jointly determine the range in which posterior diffusion is stable.
This paper studies posterior diffusion stability for linear inverse problems on Hilbert scales. We consider a smoothing forward operator, a truth with regularity x † ∈ H β x^\dagger\in\mathcal{H}^{\beta} x † ∈ H β H − q \mathcal{H}^{-q} H − q H − q \mathcal{H}^{-q} H − q H s \mathcal{H}^{s} H s H s \mathcal{H}^{s} H s
Contributions. Our main result is a Hilbert-scale stability theorem for posterior diffusion in smoothing linear inverse problems. We prove a non-asymptotic recovery bound in H s \mathcal{H}^{s} H s H − q \mathcal{H}^{-q} H − q H s \mathcal{H}^{s} H s H s \mathcal{H}^{s} H s
Background and Problem Statement
Let H \mathcal{H} H ⟨ ⋅ , ⋅ ⟩ H \langle\cdot,\cdot\rangle_{\mathcal{H}} ⟨ ⋅ , ⋅ ⟩ H Y \mathcal{Y} Y
y = A x † + ϵ , ϵ ∼ N ( 0 , Σ ϵ ) , (1) y = \mathcal{A}x^\dagger+ \epsilon,\qquad\epsilon\sim\mathcal{N}(0,\Sigma_\epsilon), \tag*{(1)} y = A x † + ϵ , ϵ ∼ N ( 0 , Σ ϵ ) , (1) where x † ∈ H x^\dagger\in\mathcal{H} x † ∈ H A : H → Y \mathcal{A}:\mathcal{H}\to\mathcal{Y} A : H → Y ϵ \epsilon ϵ Σ ϵ = σ 2 Σ 0 \Sigma_\epsilon=\sigma^2\Sigma_0 Σ ϵ = σ 2 Σ 0 σ > 0 \sigma>0 σ > 0 Σ 0 \Sigma_0 Σ 0 Y \mathcal{Y} Y ∇ x log p ( y ∣ x ) = − σ − 2 A ∗ Σ 0 − 1 ( y − A x ) \nabla_x\log p(y\mid x)=-\sigma^{-2}\mathcal{A}^*\Sigma_0^{-1}(y-\mathcal{A}x) ∇ x log p ( y ∣ x ) = − σ − 2 A ∗ Σ 0 − 1 ( y − A x ) A \mathcal{A} A
Let L : D ( L ) ⊂ H → H L:\mathcal{D}(L)\subset\mathcal{H}\to\mathcal{H} L : D ( L ) ⊂ H → H D ( L ) \mathcal{D}(L) D ( L ) { e j } j ≥ 1 ⊂ H \{e_j\}_{j\geq1}\subset\mathcal{H} { e j } j ≥ 1 ⊂ H 0 < λ 1 ≤ λ 2 ≤ ⋯ 0<\lambda_1\leq\lambda_2\leq\cdots 0 < λ 1 ≤ λ 2 ≤ ⋯ λ j → ∞ \lambda_j\to\infty λ j → ∞ L e j = λ j e j Le_j=\lambda_je_j L e j = λ j e j s ≥ 0 s\geq0 s ≥ 0 H s \mathcal{H}^{s} H s
H s = D ( L s / 2 ) = { u = ∑ j ≥ 1 u j e j ∈ H : ∑ j ≥ 1 λ j s ∣ u j ∣ 2 < ∞ } , ∥ u ∥ H s 2 = ∑ j ≥ 1 λ j s ∣ u j ∣ 2 . (2) \mathcal{H}^{s}=\mathcal{D}(L^{s/2})=\left\{u=\sum_{j\geq1}u_je_j\in\mathcal{H}:\sum_{j\geq1}\lambda_j^{s}|u_j|^2<\infty\right\},\qquad\|u\|_{\mathcal{H}^{s}}^2=\sum_{j\geq1}\lambda_j^{s}|u_j|^2. \tag*{(2)} H s = D ( L s /2 ) = { u = j ≥ 1 ∑ u j e j ∈ H : j ≥ 1 ∑ λ j s ∣ u j ∣ 2 < ∞ } , ∥ u ∥ H s 2 = j ≥ 1 ∑ λ j s ∣ u j ∣ 2 . (2) The negative-order spaces are defined by duality with H \mathcal{H} H H − s = ( H s ) ∗ \mathcal{H}^{-s}=(\mathcal{H}^{s})^* H − s = ( H s ) ∗ s > 0 s>0 s > 0 H − s \mathcal{H}^{-s} H − s H \mathcal{H} H
∥ u ∥ H − s 2 = ∑ j ≥ 1 λ j − s ∣ u j ∣ 2 . \|u\|_{\mathcal{H}^{-s}}^2=\sum_{j\geq1}\lambda_j^{-s}|u_j|^2. ∥ u ∥ H − s 2 = j ≥ 1 ∑ λ j − s ∣ u j ∣ 2 . We assume that the true signal satisfies the source condition x † ∈ H β x^\dagger\in\mathcal{H}^{\beta} x † ∈ H β β > 0 \beta>0 β > 0 A \mathcal{A} A r > 0 r>0 r > 0 0 < c A ≤ C A < ∞ 0<c_{\mathcal{A}}\leq C_{\mathcal{A}}<\infty 0 < c A ≤ C A < ∞
c A ∥ u ∥ H − r ≤ ∥ A u ∥ Y ≤ C A ∥ u ∥ H − r c_{\mathcal{A}}\|u\|_{\mathcal{H}^{-r}}\leq\|\mathcal{A}u\|_{\mathcal{Y}}\leq C_{\mathcal{A}}\|u\|_{\mathcal{H}^{-r}} c A ∥ u ∥ H − r ≤ ∥ A u ∥ Y ≤ C A ∥ u ∥ H − r for all u ∈ H u\in\mathcal{H} u ∈ H A \mathcal{A} A H s \mathcal{H}^{s} H s s s s
Let μ d a t a \mu_{\mathrm{data}} μ data H \mathcal{H} H μ d a t a \mu_{\mathrm{data}} μ data
s θ ( x , t ) ≈ ∇ x log p t ( x ) , x ∈ H , t ∈ [ 0 , 1 ] , s_\theta(x,t)\approx\nabla_x\log p_t(x),\qquad x\in\mathcal{H},\quad t\in[0,1], s θ ( x , t ) ≈ ∇ x log p t ( x ) , x ∈ H , t ∈ [ 0 , 1 ] , where p t p_t p t t t t s ⋆ ( x , t ) = ∇ x log p t ( x ) s^\star(x,t) = \nabla_x \log p_t(x) s ⋆ ( x , t ) = ∇ x log p t ( x ) y y y t t t
p t ( x ∣ y ) ∝ p t ( y ∣ x ) p t ( x ) , p t ( y ∣ x ) = ∫ H p ( y ∣ x 0 ) μ 0 ∣ t ( d x 0 ∣ x ) , (3) p_t(x \mid y) \propto p_t(y \mid x)\,p_t(x), \qquad p_t(y \mid x) = \int_{\mathcal{H}} p(y \mid x_0)\,\mu_{0\mid t}(\mathrm{d}x_0 \mid x), \tag*{(3)} p t ( x ∣ y ) ∝ p t ( y ∣ x ) p t ( x ) , p t ( y ∣ x ) = ∫ H p ( y ∣ x 0 ) μ 0 ∣ t ( d x 0 ∣ x ) , (3) where μ 0 ∣ t ( ⋅ ∣ x ) \mu_{0\mid t}(\cdot\mid x) μ 0 ∣ t ( ⋅ ∣ x ) x x x t t t (3) does not depend on x x x ∇ x log p t ( x ∣ y ) = s ⋆ ( x , t ) + ∇ x log p t ( y ∣ x ) \nabla_x \log p_t(x \mid y) = s^\star(x,t) + \nabla_x \log p_t(y \mid x) ∇ x log p t ( x ∣ y ) = s ⋆ ( x , t ) + ∇ x log p t ( y ∣ x ) t = 0 t = 0 t = 0 X 0 X_0 X 0 p 0 ( y ∣ x ) = p ( y ∣ x ) p_0(y \mid x) = p(y \mid x) p 0 ( y ∣ x ) = p ( y ∣ x ) t > 0 t > 0 t > 0 p t ( y ∣ x ) p_t(y \mid x) p t ( y ∣ x ) p ( y ∣ x 0 ) p(y \mid x_0) p ( y ∣ x 0 ) μ 0 ∣ t ( ⋅ ∣ x ) \mu_{0\mid t}(\cdot\mid x) μ 0 ∣ t ( ⋅ ∣ x ) ∇ x log p ( y ∣ x ) \nabla_x \log p(y \mid x) ∇ x log p ( y ∣ x ) t t t H \mathcal{H} H H \mathcal{H} H p t p_t p t p t ( ⋅ ∣ y ) p_t(\cdot\mid y) p t ( ⋅ ∣ y ) d ≥ 1 d \ge1 d ≥ 1 P d P_d P d span { e 1 , … , e d } \operatorname{span}\{e_1,\ldots,e_d\} span { e 1 , … , e d } H \mathcal{H} H P d H ≅ R d P_d\mathcal{H} \cong\mathbb{R}^d P d H ≅ R d p t p_t p t p ( y ∣ x ) p(y \mid x) p ( y ∣ x ) s θ s_\theta s θ s ⋆ = ∇ x log p t s^\star= \nabla_x \log p_t s ⋆ = ∇ x log p t d d d d d d H \mathcal{H} H d d d d ∈ { 256 , 512 , 1024 } d \in\{256, 512, 1024\} d ∈ { 256 , 512 , 1024 } p t p_t p t ∇ x log p t \nabla_x \log p_t ∇ x log p t H \mathcal{H} H p t p_t p t X t X_t X t log p t \log p_t log p t [2 , 19 ] .
The posterior sampler is a reverse diffusion whose drift combines the learned prior score with the likelihood information from the observation. Its dynamics are
d X t = ( f ( X t , t ) − g ( t ) 2 s θ ( X t , t ) − g ( t ) 2 Γ t ∇ x log p ( y ∣ X t ) ) d t + g ( t ) d W ‾ t , (4) \mathrm{d}X_t = \left(f(X_t,t) - g(t)^2s_\theta(X_t,t) - g(t)^2\Gamma_t\nabla_x \log p(y \mid X_t)\right)\mathrm{d}t + g(t)\,\mathrm{d}\overline{W}_t, \tag*{(4)} d X t = ( f ( X t , t ) − g ( t ) 2 s θ ( X t , t ) − g ( t ) 2 Γ t ∇ x log p ( y ∣ X t ) ) d t + g ( t ) d W t , (4) where f f f g g g W ‾ t \overline{W}_t W t Γ t \Gamma_t Γ t W ‾ t \overline{W}_t W t (4) is a Q Q Q H s \mathcal{H}^s H s Q Q Q Tr H s ( Q ) = 1 \operatorname{Tr}_{\mathcal{H}^s}(Q) = 1 Tr H s ( Q ) = 1 Q Q Q ∫ t 0 1 g ( u ) d W ‾ u \int_{t_0}^1 g(u)\,\mathrm{d}\overline{W}_u ∫ t 0 1 g ( u ) d W u H s \mathcal{H}^s H s t = 1 t = 1 t = 1 t 0 ∈ ( 0 , 1 ) t_0 \in(0,1) t 0 ∈ ( 0 , 1 ) X t 0 X_{t_0} X t 0 0 0 0 t 0 t_0 t 0 X 0 = X t 0 X_0 = X_{t_0} X 0 = X t 0 Z 0 = Z t 0 Z_0 = Z_{t_0} Z 0 = Z t 0 U 0 , t = U t 0 , t U_{0,t} = U_{t_0,t} U 0 , t = U t 0 , t Γ t = I \Gamma_t = I Γ t = I Γ t \Gamma_t Γ t A A A Γ t i s o = I \Gamma_t^{\mathrm{iso}} = I Γ t iso = I Γ t o p = ( A ∗ A + λ t L ρ ) − 1 \Gamma_t^\mathrm{op} = (A^\ast A + \lambda_t L^\rho)^{-1} Γ t op = ( A ∗ A + λ t L ρ ) − 1 λ t > 0 \lambda_t > 0 λ t > 0 ρ > 0 \rho> 0 ρ > 0 A A A L L L
To measure learned-score error along the posterior reverse process in a weak Hilbert-scale norm, we impose the weak-norm condition. For some q ≥ 0 q \ge0 q ≥ 0
sup t ∈ [ t 0 , 1 ] E ∥ s θ ( X t θ , t ) − ∇ x log p t ( X t θ ) ∥ H − q 2 ≤ δ θ 2 , (5) \sup_{t \in[t_0,1]} \mathbb{E}\left\|s_\theta(X_t^\theta,t) - \nabla_x \log p_t(X_t^\theta)\right\|_{\mathcal{H}^{-q}}^2 \le\delta_\theta^2, \tag*{(5)} t ∈ [ t 0 , 1 ] sup E s θ ( X t θ , t ) − ∇ x log p t ( X t θ ) H − q 2 ≤ δ θ 2 , (5) where X t θ X_t^\theta X t θ s θ s_\theta s θ q q q (5) gives score accuracy only in H − q \mathcal{H}^{-q} H − q H s \mathcal{H}^s H s s s s δ θ \delta_\theta δ θ r r r β \beta β q q q s s s Γ t \Gamma_t Γ t
Literature Review
Traditional inverse-problem theory describes stability using the interaction between the forward operator, the noise model, and analytic prior information. Bayesian inverse problems, on the other hand, combine observational noise with prior structure, and Hilbert or Sobolev regularity assumptions and give rates and stability guarantees for smoothing operators [9 , 28 ] . Learned reconstruction methods reduce the dependence on fixed analytic priors by coupling the forward model with a data-driven representation of the signal class. Plug-and-play methods use an external denoiser within the reconstruction process [31 ] , regularization by denoising gives a variational interpretation of denoiser-driven reconstruction [22 ] , and recent work develops plug-and-play reconstruction as a broad framework for learned computational imaging [16 ] . A parallel line of work uses learned generators as priors, with compressed sensing under generative models as a representative example [5 ] . These methods, however, do not identify the Hilbert-scale range in which a learned prior remains stable under a smoothing forward operator.
Score matching estimates gradients of log densities without normalizing constants [13 ] , and denoising score matching connects score estimation to denoiser training [32 ] . Score-based generative models and diffusion models extend these ideas to stochastic samplers through noisy marginals and reverse-time dynamics [1 , 12 , 17 , 20 , 25 , 26 , 27 ] . For inverse problems, the learned score parameterizes the prior directly in the signal space, which makes diffusion models operate on complex data distributions. Early diffusion-prior methods utilized the generative process itself to solve inverse problems [15 ] , and robust compressed sensing with deep generative priors showed that learned priors can improve recovery under structured measurements [14 ] . Diffusion posterior sampling introduced a widely used approach for noisy inverse problems [7 ] , followed by work on score priors for inverse imaging [11 ] , probabilistic imaging with score-based priors [29 ] , filtering interpretations of diffusion posterior sampling [10 ] , plug-and-play posterior sampling with diffusion priors [6 , 8 ] , robust plug-and-play posterior sampling [34 ] , asymptotically exact plug-and-play diffusion sampling [33 ] , and hard data-consistency or resampling strategies [24 ] . Measurement-score models and frequency-aware posterior sampling further show that the measurement process and the score prior interact strongly in reconstruction quality [21 , 30 ] . Despite establishing foundational frameworks and finite-dimensional bounds, these studies lack a Hilbert-scale recovery condition that quantifies the propagation of weak-norm score discrepancies through the forward operator’s smoothing kernel and the resultant reconstruction norm.
Recent work has extended the analysis of diffusion priors to infinite-dimensional spaces. Function-space score models define diffusion directly on infinite-dimensional objects [19 ] . Conditional score constructions study Bayesian inference in infinite dimensions [2 ] , and scalable posterior samplers extend this direction to infinite-dimensional inverse problems [23 ] . Score-based diffusion priors have been analyzed for infinite-dimensional nonlinear inverse problems with explicit score approximation error [3 ] . The closest result to our work studies preconditioned Langevin dynamics with score-based priors for linear Bayesian inverse problems in function space, where convergence depends on score approximation error and on preconditioning adapted to the forward operator [4 ] . Their analysis explains how score error affects sampler convergence in function space.
In this work, we study smoothing linear inverse problems in which the learned score satisfies a weak Hilbert-scale error bound in H − q \mathcal{H}^{-q} H − q H − q \mathcal{H}^{-q} H − q H s \mathcal{H}^{s} H s A \mathcal{A} A β \beta β q q q s s s Γ t \Gamma_t Γ t
Hilbert-Scale Stability Theory
The reconstruction error in posterior diffusion has three distinct sources, observational noise in the data, approximation error in the learned score, and discretization error in the reverse dynamics. An ill-posed inverse problem does not transmit these contributions uniformly across the Hilbert scale. To understand the effect of each source, we first study the well-posedness of the exact and learned reverse processes, then derive a recovery estimate that separates the three error sources, and finally use the score-propagation factor in that estimate to define the recoverable range.
Proposition 4.1. Fix s ≥ 0 s \ge0 s ≥ 0
b ⋆ ( x , t ) = f ( x , t ) − g ( t ) 2 s ⋆ ( x , t ) − g ( t ) 2 Γ t ∇ x log p ( y ∣ x ) , b^\star(x,t) = f(x,t) - g(t)^2s^\star(x,t) - g(t)^2\Gamma_t\nabla_x \log p(y\mid x), b ⋆ ( x , t ) = f ( x , t ) − g ( t ) 2 s ⋆ ( x , t ) − g ( t ) 2 Γ t ∇ x log p ( y ∣ x ) , b θ ( x , t ) = f ( x , t ) − g ( t ) 2 s θ ( x , t ) − g ( t ) 2 Γ t ∇ x log p ( y ∣ x ) . b^\theta(x,t) = f(x,t) - g(t)^2s^\theta(x,t) - g(t)^2\Gamma_t\nabla_x \log p(y\mid x). b θ ( x , t ) = f ( x , t ) − g ( t ) 2 s θ ( x , t ) − g ( t ) 2 Γ t ∇ x log p ( y ∣ x ) . Assume that, for ♯ ∈ { ⋆ , θ } \sharp\in\{\star,\theta\} ♯ ∈ { ⋆ , θ } ( x , t ) ↦ b ♯ ( x , t ) (x,t) \mapsto b^\sharp(x,t) ( x , t ) ↦ b ♯ ( x , t ) t t t x x x L s > 0 L_s > 0 L s > 0 K s > 0 K_s > 0 K s > 0 x , z ∈ H s x,z \in\mathcal{H}^{s} x , z ∈ H s t ∈ [ t 0 , 1 ] t \in[t_0,1] t ∈ [ t 0 , 1 ]
∥ b ♯ ( x , t ) − b ♯ ( z , t ) ∥ H s ≤ L s ∥ x − z ∥ H s , (6) \left\|b^\sharp(x,t)-b^\sharp(z,t)\right\|_{\mathcal{H}^{s}} \le L_s\left\|x-z\right\|_{\mathcal{H}^{s}}, \tag*{(6)} b ♯ ( x , t ) − b ♯ ( z , t ) H s ≤ L s ∥ x − z ∥ H s , (6) and
∥ b ♯ ( x , t ) ∥ H s ≤ K s ( 1 + ∥ x ∥ H s ) . (7) \left\|b^\sharp(x,t)\right\|_{\mathcal{H}^{s}} \le K_s\left(1+\left\|x\right\|_{\mathcal{H}^{s}}\right). \tag*{(7)} b ♯ ( x , t ) H s ≤ K s ( 1 + ∥ x ∥ H s ) . (7) If g ∈ L 2 ( [ t 0 , 1 ] ) g \in L^2([t_0,1]) g ∈ L 2 ([ t 0 , 1 ]) Γ t ∈ L ( H s ) \Gamma_t \in\mathcal{L}(H^s) Γ t ∈ L ( H s ) t ∈ [ t 0 , 1 ] t \in[t_0,1] t ∈ [ t 0 , 1 ]
sup t ∈ [ t 0 , 1 ] ∥ Γ t ∥ L ( H s ) < ∞ , (8) \sup_{t\in[t_0,1]}\|\Gamma_t\|_{\mathcal{L}(H^s)}<\infty, \tag*{(8)} t ∈ [ t 0 , 1 ] sup ∥ Γ t ∥ L ( H s ) < ∞ , (8) then, for every square-integrable terminal condition ξ ∈ L 2 ( Ω ; H s ) \xi\in L^2(\Omega;H^s) ξ ∈ L 2 ( Ω ; H s )
X t ♯ = ξ − ∫ t 1 b ♯ ( X u ♯ , u ) d u − ∫ t 1 g ( u ) d W ‾ u , ♯ ∈ { ⋆ , θ } , (9) X_t^\sharp=\xi-\int_t^1 b^\sharp(X_u^\sharp,u)\,\mathrm{d}u-\int_t^1 g(u)\,\mathrm{d}\overline{W}_u,\qquad\sharp\in\{\star,\theta\}, \tag*{(9)} X t ♯ = ξ − ∫ t 1 b ♯ ( X u ♯ , u ) d u − ∫ t 1 g ( u ) d W u , ♯ ∈ { ⋆ , θ } , (9) have unique adapted continuous H s H^s H s [ t 0 , 1 ] [t_0,1] [ t 0 , 1 ] C s > 0 C_s>0 C s > 0 L s L_s L s K s K_s K s g g g t 0 t_0 t 0
E sup t ∈ [ t 0 , 1 ] ∥ X t ♯ ∥ H s 2 ≤ C s ( 1 + E ∥ ξ ∥ H s 2 ) . (10) \mathbb{E}\sup_{t\in[t_0,1]}\|X_t^\sharp\|_{H^s}^2\leq C_s\left(1+\mathbb{E}\|\xi\|_{H^s}^2\right). \tag*{(10)} E t ∈ [ t 0 , 1 ] sup ∥ X t ♯ ∥ H s 2 ≤ C s ( 1 + E ∥ ξ ∥ H s 2 ) . (10) If x ↦ b ⋆ ( x , t ) x\mapsto b^\star(x,t) x ↦ b ⋆ ( x , t ) H s H^s H s t ∈ [ t 0 , 1 ] t\in[t_0,1] t ∈ [ t 0 , 1 ] L s ′ > 0 L'_s>0 L s ′ > 0
∥ D x b ⋆ ( x , t ) ∥ L ( H s ) ≤ L s ′ for all x ∈ H s , t ∈ [ t 0 , 1 ] , (11) \|D_xb^\star(x,t)\|_{\mathcal{L}(H^s)}\leq L'_s\qquad\text{for all }x\in H^s,\ t\in[t_0,1], \tag*{(11)} ∥ D x b ⋆ ( x , t ) ∥ L ( H s ) ≤ L s ′ for all x ∈ H s , t ∈ [ t 0 , 1 ] , (11) then, for almost every realization of X ⋆ X^\star X ⋆
d d t Z t = B t Γ Z t , B t Γ = D x b ⋆ ( X t ⋆ , t ) , (12) \frac{\mathrm{d}}{\mathrm{d}t}Z_t=B_t^\Gamma Z_t,\qquad B_t^\Gamma=D_xb^\star(X_t^\star,t), \tag*{(12)} d t d Z t = B t Γ Z t , B t Γ = D x b ⋆ ( X t ⋆ , t ) , (12) generates an evolution family U τ , t Γ U_{\tau,t}^\Gamma U τ , t Γ H s H^s H s
∥ U τ , t Γ ∥ L ( H s ) ≤ e L s ′ ∣ t − τ ∣ for t , τ ∈ [ t 0 , 1 ] . (13) \|U_{\tau,t}^\Gamma\|_{\mathcal{L}(H^s)}\leq e^{L'_s|t-\tau|}\qquad\text{for }t,\tau\in[t_0,1]. \tag*{(13)} ∥ U τ , t Γ ∥ L ( H s ) ≤ e L s ′ ∣ t − τ ∣ for t , τ ∈ [ t 0 , 1 ] . (13) The proof of Proposition 4.1 is given in Appendix A. Proposition 4.1 establishes existence, uniqueness, and moment bounds in H s H^s H s
Definition 4.2. Fix q ≥ 0 q\geq0 q ≥ 0 s ≥ 0 s\geq0 s ≥ 0 Γ = ( Γ t ) t ∈ [ t 0 , 1 ] \Gamma=(\Gamma_t)_{t\in[t_0,1]} Γ = ( Γ t ) t ∈ [ t 0 , 1 ] H s H^s H s ζ \zeta ζ ζ ′ \zeta' ζ ′ U τ , t ζ , ζ ′ U_{\tau,t}^{\zeta,\zeta'} U τ , t ζ , ζ ′ d d t Z t = B ‾ t Z t \frac{\mathrm{d}}{\mathrm{d}t}Z_t=\overline{B}_tZ_t d t d Z t = B t Z t
B ‾ t = ∫ 0 1 D x b ⋆ ( ζ t + λ ( ζ t ′ − ζ t ) , t ) d λ . \overline{B}_t=\int_0^1D_xb^\star(\zeta_t+\lambda(\zeta'_t-\zeta_t),t)\,\mathrm{d}\lambda. B t = ∫ 0 1 D x b ⋆ ( ζ t + λ ( ζ t ′ − ζ t ) , t ) d λ . The propagation envelope M r , q , s , Γ : [ t 0 , 1 ] → [ 0 , ∞ ] M_{r,q,s,\Gamma}:[t_0,1]\to[0,\infty] M r , q , s , Γ : [ t 0 , 1 ] → [ 0 , ∞ ] ζ , ζ ′ \zeta,\zeta' ζ , ζ ′
∥ U 0 , t ζ , ζ ′ ∥ L ( H − q , H s ) ≤ M r , q , s , Γ ( t ) , t ∈ [ t 0 , 1 ] (14) \|U_{0,t}^{\zeta,\zeta'}\|_{\mathcal{L}(H^{-q},H^s)}\leq M_{r,q,s,\Gamma}(t),\qquad t\in[t_0,1] \tag*{(14)} ∥ U 0 , t ζ , ζ ′ ∥ L ( H − q , H s ) ≤ M r , q , s , Γ ( t ) , t ∈ [ t 0 , 1 ] (14) (M r , q , s , Γ ( t ) = ∞ M_{r,q,s,\Gamma}(t)=\infty M r , q , s , Γ ( t ) = ∞
The score-propagation functional is
A ( r , q , s , Γ ) = ∫ t 0 1 g ( t ) 2 M r , q , s , Γ ( t ) d t . (15) \mathcal{A}(r,q,s,\Gamma)=\int_{t_0}^1g(t)^2M_{r,q,s,\Gamma}(t)\,\mathrm{d}t. \tag*{(15)} A ( r , q , s , Γ ) = ∫ t 0 1 g ( t ) 2 M r , q , s , Γ ( t ) d t . (15) An index s ≤ β s\leq\beta s ≤ β recoverable if A ( r , q , s , Γ ) < ∞ \mathcal{A}(r,q,s,\Gamma)<\infty A ( r , q , s , Γ ) < ∞
Assumption 4.4. Let X ⋆ X^\star X ⋆ ♯ = ⋆ \sharp=\star ♯ = ⋆ X 0 ⋆ X_0^\star X 0 ⋆ m y = m y ( Γ ) m_y=m_y(\Gamma) m y = m y ( Γ ) C y = C y ( Γ ) C_y=C_y(\Gamma) C y = C y ( Γ ) y y y η ( r , β , s ) ≥ 0 \eta(r,\beta,s)\geq0 η ( r , β , s ) ≥ 0 R > 0 R>0 R > 0 C b i a s , C v a r < ∞ C_{\mathrm{bias}},C_{\mathrm{var}}<\infty C bias , C var < ∞
E ∥ m y − x † ∥ H s 2 ≤ C b i a s σ 2 η ( r , β , s ) , E Tr ( L s / 2 C y L s / 2 ) ≤ C v a r σ 2 η ( r , β , s ) \mathbb{E}\|m_y-x^\dagger\|_{H^s}^2\leq C_{\mathrm{bias}}\sigma^{2\eta(r,\beta,s)},\qquad\mathbb{E}\operatorname{Tr}\left(L^{s/2}C_yL^{s/2}\right)\leq C_{\mathrm{var}}\sigma^{2\eta(r,\beta,s)} E ∥ m y − x † ∥ H s 2 ≤ C bias σ 2 η ( r , β , s ) , E Tr ( L s /2 C y L s /2 ) ≤ C var σ 2 η ( r , β , s ) for every x † x^\dagger x † ∥ x † ∥ H β ≤ R \|x^\dagger\|_{H^\beta}\leq R ∥ x † ∥ H β ≤ R C n o i s e = C b i a s + C v a r C_{\mathrm{noise}}=C_{\mathrm{bias}}+C_{\mathrm{var}} C noise = C bias + C var
The law of X 0 ⋆ X_0^\star X 0 ⋆ t 0 t_0 t 0 Γ t = I \Gamma_t = I Γ t = I b ⋆ b^\star b ⋆ t t t p t ( y ∣ x ) p_t(y \mid x) p t ( y ∣ x ) Assumption 4.4 holds, for instance, for a diagonal Gaussian model whose eigenvalues satisfy λ j = j 2 \lambda_j = j^2 λ j = j 2 τ j 2 ∝ j − ( 2 β + 1 ) \tau_j^2 \propto j^{-(2\beta+1)} τ j 2 ∝ j − ( 2 β + 1 ) [18 ] considers s = 0 s = 0 s = 0 σ 2 = 1 / n \sigma^2 = 1/n σ 2 = 1/ n H s \mathcal{H}^s H s
η ( r , β , s ) = 2 ( β − s ) 2 β + 2 r + 1 , 0 ≤ s < β . \eta(r,\beta,s) = \frac{2(\beta-s)}{2\beta+2r+1}, \qquad0 \le s < \beta. η ( r , β , s ) = 2 β + 2 r + 1 2 ( β − s ) , 0 ≤ s < β . Assumption 4.5. Let 1 = t N > t N − 1 > ⋯ > t 0 1 = t_N > t_{N-1} > \cdots> t_0 1 = t N > t N − 1 > ⋯ > t 0 Δ t \Delta t Δ t F t \mathcal{F}_t F t Ψ n Δ \Psi_n^\Delta Ψ n Δ ξ n + 1 \xi_{n+1} ξ n + 1 [ t n , t n + 1 ] [t_n,t_{n+1}] [ t n , t n + 1 ] X ^ t n = Ψ n Δ ( X ^ t n + 1 , ξ n + 1 ) \widehat{X}_{t_n} = \Psi_n^\Delta(\widehat{X}_{t_{n+1}},\xi_{n+1}) X t n = Ψ n Δ ( X t n + 1 , ξ n + 1 ) X ^ t N = X 1 θ \widehat{X}_{t_N} = X_1^\theta X t N = X 1 θ x ^ = X ^ t 0 \widehat{x} = \widehat{X}_{t_0} x = X t 0 κ > 0 \kappa> 0 κ > 0 C l o c , C s t a b < ∞ C_{\mathrm{loc}}, C_{\mathrm{stab}} < \infty C loc , C stab < ∞
X t n θ = Ψ n Δ ( X t n + 1 θ , ξ n + 1 ) + R n , (16) X_{t_n}^\theta= \Psi_n^\Delta(X_{t_{n+1}}^\theta,\xi_{n+1}) + R_n, \tag*{(16)} X t n θ = Ψ n Δ ( X t n + 1 θ , ξ n + 1 ) + R n , (16) with local residual
E ∥ R n ∥ H s 2 ≤ C l o c ( Δ t ) 2 κ + 1 , (17) \mathbb{E}\lVert R_n\rVert_{\mathcal{H}^s}^2 \le C_{\mathrm{loc}}(\Delta t)^{2\kappa+1}, \tag*{(17)} E ∥ R n ∥ H s 2 ≤ C loc ( Δ t ) 2 κ + 1 , (17) and, for every pair of F t n + 1 \mathcal{F}_{t_{n+1}} F t n + 1 u , v u,v u , v
E [ ∥ Ψ n Δ ( u , ξ n + 1 ) − Ψ n Δ ( v , ξ n + 1 ) ∥ H s 2 | F t n + 1 ] ≤ ( 1 + C s t a b Δ t ) ∥ u − v ∥ H s 2 . (18) \mathbb{E}\left[\left\lVert\Psi_n^\Delta(u,\xi_{n+1})-\Psi_n^\Delta(v,\xi_{n+1})\right\rVert_{\mathcal{H}^s}^2 \,\middle|\, \mathcal{F}_{t_{n+1}}\right] \le(1+C_{\mathrm{stab}}\Delta t)\lVert u-v\rVert_{\mathcal{H}^s}^2. \tag*{(18)} E [ Ψ n Δ ( u , ξ n + 1 ) − Ψ n Δ ( v , ξ n + 1 ) H s 2 F t n + 1 ] ≤ ( 1 + C stab Δ t ) ∥ u − v ∥ H s 2 . (18) The recovery estimate then compares the exact posterior reverse process, the learned reverse process, and its time discretization, with separate effects from observational noise, learned-score error, and numerical error. The learned-score effect is measured by M r , q , s , Γ ( t ) M_{r,q,s,\Gamma}(t) M r , q , s , Γ ( t ) H − q \mathcal{H}^{-q} H − q H s \mathcal{H}^s H s
Theorem 4.6. Let X t ⋆ X_t^\star X t ⋆ be the exact posterior reverse process and X t θ X_t^\theta X t θ the posterior reverse process under the learned score s θ s_\theta s θ Both processes solve (9) under the assumptions of Proposition 4.1 , start from the same state at t = 1 t = 1 t = 1 and use the same reverse Brownian motion. The reconstruction x ^ \widehat{x} x is the final iterate of the time discretization of X t θ X_t^\theta X t θ in Assumption 4.5 at step size Δ t \Delta t Δ t Assume that A ∈ L ( H , Y ) A \in\mathcal{L}(\mathcal{H},\mathcal{Y}) A ∈ L ( H , Y ) is smoothing of order r > 0 r > 0 r > 0 such that there exist constants 0 < c A ≤ C A < ∞ 0 < c_A \le C_A < \infty 0 < c A ≤ C A < ∞ such that
c A ∥ u ∥ H − r ≤ ∥ A u ∥ Y ≤ C A ∥ u ∥ H − r c_A\lVert u\rVert_{\mathcal{H}^{-r}} \le\lVert Au\rVert_{\mathcal{Y}} \le C_A\lVert u\rVert_{\mathcal{H}^{-r}} c A ∥ u ∥ H − r ≤ ∥ A u ∥ Y ≤ C A ∥ u ∥ H − r for all u ∈ H u \in\mathcal{H} u ∈ H Assume that x † ∈ H β x^\dagger\in\mathcal{H}^\beta x † ∈ H β for some β > 0 \beta> 0 β > 0 and Assumptions 4.4 and 4.5 hold. Suppose s θ s_\theta s θ satisfies (5) , fix a reconstruction index s ≤ β s \le\beta s ≤ β that is recoverable in the sense of Definition 4.2 . Then
E ∥ x ^ − x † ∥ H s 2 ≤ 3 Φ n o i s e ( σ ; r , β , s ) + 3 Φ s c o r e ( δ θ ; r , q , s , Γ ) + 3 Φ d i s c ( Δ t ; r , s , Γ ) , (19) \mathbb{E}\lVert\widehat{x}-x^\dagger\rVert_{\mathcal{H}^s}^2 \le3\Phi_{\mathrm{noise}}(\sigma;r,\beta,s) + 3\Phi_{\mathrm{score}}(\delta_\theta;r,q,s,\Gamma) + 3\Phi_{\mathrm{disc}}(\Delta t;r,s,\Gamma), \tag*{(19)} E ∥ x − x † ∥ H s 2 ≤ 3 Φ noise ( σ ; r , β , s ) + 3 Φ score ( δ θ ; r , q , s , Γ ) + 3 Φ disc ( Δ t ; r , s , Γ ) , (19) where
Φ n o i s e ( σ ; r , β , s ) = C n o i s e σ 2 η ( r , β , s ) , (20) \Phi_{\mathrm{noise}}(\sigma;r,\beta,s) = C_{\mathrm{noise}}\sigma^{2\eta(r,\beta,s)}, \tag*{(20)} Φ noise ( σ ; r , β , s ) = C noise σ 2 η ( r , β , s ) , (20) Φ s c o r e ( δ θ ; r , q , s , Γ ) = ( ∫ t 0 1 g ( t ) 2 M r , q , s , Γ ( t ) d t ) 2 δ θ 2 , (21) \Phi_{\mathrm{score}}(\delta_\theta;r,q,s,\Gamma) = \left(\int_{t_0}^{1} g(t)^2 M_{r,q,s,\Gamma}(t)\,\mathrm{d}t\right)^2\delta_\theta^2, \tag*{(21)} Φ score ( δ θ ; r , q , s , Γ ) = ( ∫ t 0 1 g ( t ) 2 M r , q , s , Γ ( t ) d t ) 2 δ θ 2 , (21) and
Φ d i s c ( Δ t ; r , s , Γ ) = C d i s c ( r , s , Γ ) ( Δ t ) 2 κ , (22) \Phi_{\mathrm{disc}}(\Delta t;r,s,\Gamma) = C_{\mathrm{disc}}(r,s,\Gamma)(\Delta t)^{2\kappa}, \tag*{(22)} Φ disc ( Δ t ; r , s , Γ ) = C disc ( r , s , Γ ) ( Δ t ) 2 κ , (22) where constant C n o i s e C_{\mathrm{noise}} C noise is the noise constant of Assumption 4.4 , and the discretization constant is C d i s c ( r , s , Γ ) = 2 C l o c e 2 C s t a b ( 1 − t 0 ) C_{\mathrm{disc}}(r,s,\Gamma)=2C_{\mathrm{loc}}e^{2C_{\mathrm{stab}}(1-t_0)} C disc ( r , s , Γ ) = 2 C loc e 2 C stab ( 1 − t 0 ) with κ \kappa κ C l o c C_{\mathrm{loc}} C loc and C s t a b C_{\mathrm{stab}} C stab being the constants of Assumption 4.5. The learned-score contribution depends on the weak-norm index q q q the reconstruction index s s s and the correction Γ \Gamma Γ as well as on the score error δ θ \delta_\theta δ θ
The proof of Theorem 4.6 is given in Appendix § 8 . In Equation (21) , the factor ∫ t 0 1 g ( t ) 2 M r , q , s , Γ ( t ) d t \int_{t_0}^{1} g(t)^2 M_{r,q,s,\Gamma}(t)\,\mathrm{d}t ∫ t 0 1 g ( t ) 2 M r , q , s , Γ ( t ) d t H − q \mathcal{H}^{-q} H − q H s \mathcal{H}^s H s
Theorem 4.7. Under the assumptions of Theorem 4.6 and given N ( r , q , s , Γ ) \mathcal{N}(r,q,s,\Gamma) N ( r , q , s , Γ ) and the critical recoverable index
s ⋆ ( r , β , q , Γ ) = sup { s ≤ β ; N ( r , q , s , Γ ) < ∞ } , (23) s_\star(r,\beta,q,\Gamma) = \sup\left\{s \le\beta\,;\, \mathcal{N}(r,q,s,\Gamma) < \infty\right\}, \tag*{(23)} s ⋆ ( r , β , q , Γ ) = sup { s ≤ β ; N ( r , q , s , Γ ) < ∞ } , (23) define
X β ( R ) = { x † ∈ H β ; ∥ x † ∥ H β ≤ R } , \mathcal{X}_\beta(R) = \left\{x^\dagger\in\mathcal{H}^\beta\,;\, \lVert x^\dagger\rVert_{\mathcal{H}^\beta} \le R\right\}, X β ( R ) = { x † ∈ H β ; ∥ x † ∥ H β ≤ R } , for R > 0 R > 0 R > 0 S q ( δ θ ) \mathcal{S}_q(\delta_\theta) S q ( δ θ ) (5) for which Assumption 4.5 holds with C l o c C_{\mathrm{loc}} C loc C s t a b C_{\mathrm{stab}} C stab s θ s_\theta s θ R > 0 R > 0 R > 0 s < s ⋆ ( r , β , q , Γ ) s < s_\star(r,\beta,q,\Gamma) s < s ⋆ ( r , β , q , Γ )
C R = C R ( r , β , q , s , Γ , Σ ϵ ) < ∞ C_R = C_R(r,\beta,q,s,\Gamma,\Sigma_\epsilon) < \infty C R = C R ( r , β , q , s , Γ , Σ ϵ ) < ∞ such that
E ∥ x ^ − x † ∥ H s 2 ≤ C R ( σ 2 η ( r , β , s ) + A ( r , q , s , Γ ) 2 δ θ 2 + ( Δ t ) 2 κ ) (24) \mathbb{E}\lVert\widehat{x} - x^\dagger\rVert_{\mathcal{H}^s}^2 \le C_R\left(\sigma^{2\eta(r,\beta,s)} + \mathcal{A}(r,q,s,\Gamma)^2\delta_\theta^2 + (\Delta t)^{2\kappa}\right) \tag*{(24)} E ∥ x − x † ∥ H s 2 ≤ C R ( σ 2 η ( r , β , s ) + A ( r , q , s , Γ ) 2 δ θ 2 + ( Δ t ) 2 κ ) (24) for every x † ∈ X β ( R ) x^\dagger\in\mathcal{X}_\beta(R) x † ∈ X β ( R ) s θ ∈ S q ( δ θ ) s_\theta\in\mathcal{S}_q(\delta_\theta) s θ ∈ S q ( δ θ )
Appendix C shows the proof of Theorem 4.7 . For indices s < s ⋆ ( r , β , q , Γ ) s < s_\star(r,\beta,q,\Gamma) s < s ⋆ ( r , β , q , Γ ) Theorem 4.7 gives a uniform recovery bound in H s \mathcal{H}^s H s H − q \mathcal{H}^{-q} H − q H s \mathcal{H}^s H s
Theorem 4.8. Assume the hypotheses of Theorem 4.6 , and assume that x ↦ b ⋆ ( x , t ) x \mapsto b^\star(x,t) x ↦ b ⋆ ( x , t ) t ∈ [ t 0 , 1 ] t \in[t_0,1] t ∈ [ t 0 , 1 ] U T , t Γ U^\Gamma_{T,t} U T , t Γ R > 0 R > 0 R > 0
X β ( R ) = { x † ∈ H β ; ∥ x † ∥ H β ≤ R } \mathcal{X}_\beta(R) = \left\{x^\dagger\in\mathcal{H}^\beta;\ \lVert x^\dagger\rVert_{\mathcal{H}^\beta} \le R\right\} X β ( R ) = { x † ∈ H β ; ∥ x † ∥ H β ≤ R } For δ > 0 \delta> 0 δ > 0
S q ( δ ) = { s θ ; sup t ∈ [ t 0 , 1 ] E ∥ s θ ( X t θ , t ) − ∇ log p t ( X t θ ) ∥ H − q 2 ≤ δ 2 } . \mathcal{S}_q(\delta) = \left\{s_\theta;\ \sup_{t\in[t_0,1]} \mathbb{E}\left\lVert s_\theta(X_t^\theta,t)-\nabla\log p_t(X_t^\theta)\right\rVert_{\mathcal{H}^{-q}}^2 \le\delta^2\right\}. S q ( δ ) = { s θ ; t ∈ [ t 0 , 1 ] sup E s θ ( X t θ , t ) − ∇ log p t ( X t θ ) H − q 2 ≤ δ 2 } . Fix s > s ⋆ ( r , β , q , Γ ) s > s_\star(r,\beta,q,\Gamma) s > s ⋆ ( r , β , q , Γ ) η ( r , β , s ) > 0 \eta(r,\beta,s) > 0 η ( r , β , s ) > 0 H − q \mathcal{H}^{-q} H − q H s \mathcal{H}^s H s
sup { ∥ ∫ t 0 1 U 0 , t Γ g ( t ) 2 e ( t ) d t ∥ H s ; e ∈ C ( [ t 0 , 1 ] ; H s ) , sup t ∈ [ t 0 , 1 ] ∥ e ( t ) ∥ H − q ≤ 1 } = ∞ . (25) \sup\left\{\left\lVert\int_{t_0}^{1}U^\Gamma_{0,t}g(t)^2e(t)\,\mathrm{d}t\right\rVert_{\mathcal{H}^s};\ e \in C([t_0,1];\mathcal{H}^s),\ \sup_{t\in[t_0,1]}\lVert e(t)\rVert_{\mathcal{H}^{-q}}\le1\right\} = \infty. \tag*{(25)} sup { ∫ t 0 1 U 0 , t Γ g ( t ) 2 e ( t ) d t H s ; e ∈ C ([ t 0 , 1 ] ; H s ) , t ∈ [ t 0 , 1 ] sup ∥ e ( t ) ∥ H − q ≤ 1 } = ∞. (25) Then there exist R > 0 R > 0 R > 0 c R > 0 c_R > 0 c R > 0 δ n ↓ 0 \delta_n \downarrow0 δ n ↓ 0 σ n ↓ 0 \sigma_n \downarrow0 σ n ↓ 0 Δ t n ↓ 0 \Delta t_n \downarrow0 Δ t n ↓ 0 x n † ∈ X β ( R ) x_n^\dagger\in\mathcal{X}_\beta(R) x n † ∈ X β ( R ) s θ , n ∈ S q ( δ n ) s_{\theta,n} \in\mathcal{S}_q(\delta_n) s θ , n ∈ S q ( δ n ) x ^ n \widehat{x}_n x n
lim inf n → ∞ E ∥ x ^ n − x n † ∥ H s 2 ≥ c R . (26) \liminf_{n\to\infty}\mathbb{E}\lVert\widehat{x}_n-x_n^\dagger\rVert_{\mathcal{H}^s}^2 \ge c_R. \tag*{(26)} n → ∞ lim inf E ∥ x n − x n † ∥ H s 2 ≥ c R . (26) Consequently, for s > s ⋆ ( r , β , q , Γ ) s > s_\star(r,\beta,q,\Gamma) s > s ⋆ ( r , β , q , Γ ) C R < ∞ C_R < \infty C R < ∞
E ∥ x ^ − x † ∥ H s 2 ≤ C R ( σ 2 η ( r , β , s ) + δ θ 2 + ( Δ t ) 2 κ ) (27) \mathbb{E}\lVert\widehat{x}-x^\dagger\rVert_{\mathcal{H}^s}^2 \le C_R\left(\sigma^{2\eta(r,\beta,s)}+\delta_\theta^2+(\Delta t)^{2\kappa}\right) \tag*{(27)} E ∥ x − x † ∥ H s 2 ≤ C R ( σ 2 η ( r , β , s ) + δ θ 2 + ( Δ t ) 2 κ ) (27) for some noise level σ \sigma σ δ θ \delta_\theta δ θ Δ t \Delta t Δ t x † ∈ X β ( R ) x^\dagger\in\mathcal{X}_\beta(R) x † ∈ X β ( R ) s θ ∈ S q ( δ θ ) s_\theta\in\mathcal{S}_q(\delta_\theta) s θ ∈ S q ( δ θ )
The proof of Theorem 4.8 is given in Appendix D. The proof uses condition (25) , which is stronger than A ( r , q , s , Γ ) = ∞ \mathcal{A}(r,q,s,\Gamma)=\infty A ( r , q , s , Γ ) = ∞ Γ t \Gamma_t Γ t
Proposition 4.9. Let Γ t i s o = I \Gamma_t^{\mathrm{iso}}=I Γ t iso = I Γ t o p = ( A ∗ A + λ t L ρ ) − 1 \Gamma_t^{\mathrm{op}}=(A^\ast A+\lambda_tL^\rho)^{-1} Γ t op = ( A ∗ A + λ t L ρ ) − 1 I \mathcal{I} I Definition 4.2 satisfy
M r , q , s , Γ o p ( t ) ≤ M r , q , s , Γ i s o ( t ) , s ∈ I , t ∈ [ t 0 , 1 ] . (28) M_{r,q,s,\Gamma^{\mathrm{op}}}(t)\le M_{r,q,s,\Gamma^{\mathrm{iso}}}(t),\qquad s\in\mathcal{I},\quad t\in[t_0,1]. \tag*{(28)} M r , q , s , Γ op ( t ) ≤ M r , q , s , Γ iso ( t ) , s ∈ I , t ∈ [ t 0 , 1 ] . (28) Then, for every s ∈ I s\in\mathcal{I} s ∈ I
A ( r , q , s , Γ o p ) ≤ A ( r , q , s , Γ i s o ) , (29) \mathcal{A}(r,q,s,\Gamma^{\mathrm{op}})\le\mathcal{A}(r,q,s,\Gamma^{\mathrm{iso}}), \tag*{(29)} A ( r , q , s , Γ op ) ≤ A ( r , q , s , Γ iso ) , (29) and consequently
Φ s c o r e ( δ θ ; r , q , s , Γ o p ) ≤ Φ s c o r e ( δ θ ; r , q , s , Γ i s o ) . (30) \Phi_{\mathrm{score}}(\delta_\theta;r,q,s,\Gamma^{\mathrm{op}})\le\Phi_{\mathrm{score}}(\delta_\theta;r,q,s,\Gamma^{\mathrm{iso}}). \tag*{(30)} Φ score ( δ θ ; r , q , s , Γ op ) ≤ Φ score ( δ θ ; r , q , s , Γ iso ) . (30) Fix s ∈ I s\in\mathcal{I} s ∈ I A ( r , q , s , Γ i s o ) < ∞ \mathcal{A}(r,q,s,\Gamma^{\mathrm{iso}})<\infty A ( r , q , s , Γ iso ) < ∞ (28) holds strictly on a set of positive g ( t ) 2 d t g(t)^2\,\mathrm{d}t g ( t ) 2 d t (29) and (30) are strict for every δ θ > 0 \delta_\theta>0 δ θ > 0
The constants of Assumption 4.4 depend on Γ \Gamma Γ b ⋆ b^\star b ⋆ Γ \Gamma Γ (19) , and the noise contribution requires a separate comparison of these constants.
We prove Proposition 4.9 in Appendix E. The recovery bound of Theorem 4.6 , the obstruction of Theorem 4.8, and the comparison of Proposition 4.9 describe how the score-propagation factor varies with q q q s s s Γ \Gamma Γ (28) holds. The learned-score contribution in the recoverable range then decreases as well.
Experiments
We design three experiments to test our theoretical claims. We first use a diagonal Gaussian inverse problem with a closed-form posterior and examine whether the predicted transition across Hilbert-scale norms is visible in finite-dimensional simulations. We then use non-Gaussian one-dimensional signals and a learned score surrogate to test the same transition under an implicit prior representation. We finally compare isotropic and geometry-aware posterior guidance after matching the observation residual, so differences in reconstruction quality do not come from unequal data fit. For a spectral vector u = ∑ j = 1 d u j e j u = \sum_{j=1}^{d} u_j e_j u = ∑ j = 1 d u j e j
∥ u ∥ H d s 2 = ∑ j = 1 d j 2 s ∣ u j ∣ 2 . \lVert u \rVert_{H_d^s}^{2} = \sum_{j=1}^{d} j^{2s} \lvert u_j \rvert^{2}. ∥ u ∥ H d s 2 = j = 1 ∑ d j 2 s ∣ u j ∣ 2 . Unless stated otherwise, reported values are empirical means over independent test draws.1
Experiment 1: Known-Posterior Recoverability Transition
We use the diagonal Gaussian inverse problem
x j ∼ N ( 0 , τ j 2 ) , τ j 2 ∝ j − ( 2 β + 1 ) , ( A x ) j = a j x j , a j = j − r , y j = a j x j † + σ ξ j , ξ j ∼ N ( 0 , 1 ) . x_j \sim N(0,\tau_j^2), \qquad\tau_j^2 \propto j^{-(2\beta+1)}, \qquad(Ax)_j = a_jx_j, \qquad a_j = j^{-r}, \qquad y_j = a_jx_j^\dagger+ \sigma\xi_j, \qquad\xi_j \sim N(0,1). x j ∼ N ( 0 , τ j 2 ) , τ j 2 ∝ j − ( 2 β + 1 ) , ( A x ) j = a j x j , a j = j − r , y j = a j x j † + σ ξ j , ξ j ∼ N ( 0 , 1 ) . The parameter grid, Monte Carlo budget, learned-surrogate sample size, and weak perturbation level are listed in Table 1 . We evaluate the reconstruction error on the dense regularity grid
Table 1. Implementation details for Experiment 1.
s ∈ { 0 , 0.05 , 0.10 , … , 2.50 } . s \in\{0, 0.05, 0.10, \ldots, 2.50\}. s ∈ { 0 , 0.05 , 0.10 , … , 2.50 } . The diagonal Gaussian structure gives the closed-form posterior
x j ∣ y j ∼ N ( m j , v j ) , v j = ( τ j − 2 + σ − 2 a j 2 ) − 1 , m j = v j σ − 2 a j y j . x_j \mid y_j \sim N(m_j,v_j), \qquad v_j = \left(\tau_j^{-2} + \sigma^{-2}a_j^2\right)^{-1}, \qquad m_j = v_j\sigma^{-2}a_jy_j. x j ∣ y j ∼ N ( m j , v j ) , v j = ( τ j − 2 + σ − 2 a j 2 ) − 1 , m j = v j σ − 2 a j y j . We compare four reconstructions, x ^ p o s t \widehat{x}_{\mathrm{post}} x post x ^ o r a c l e \widehat{x}_{\mathrm{oracle}} x oracle x ^ θ \widehat{x}_{\theta} x θ x ^ θ , w e a k \widehat{x}_{\theta,\mathrm{weak}} x θ , weak x ^ p o s t \widehat{x}_{\mathrm{post}} x post x ^ o r a c l e \widehat{x}_{\mathrm{oracle}} x oracle x ^ θ \widehat{x}_{\theta} x θ x ^ θ , w e a k \widehat{x}_{\theta,\mathrm{weak}} x θ , weak q = 1 q = 1 q = 1 δ = 4 × 10 − 2 \delta= 4 \times10^{-2} δ = 4 × 1 0 − 2 max ( a j , 5 × 10 − 3 ) − 1 \max(a_j, 5 \times10^{-3})^{-1} max ( a j , 5 × 1 0 − 3 ) − 1
E ( s ; r , β , σ ) = E ∥ x ^ − x † ∥ H d s 2 , \mathcal{E}(s;r,\beta,\sigma) = \mathbb{E}\lVert\widehat{x} - x^\dagger\rVert_{H_d^s}^{2}, E ( s ; r , β , σ ) = E ∥ x − x † ∥ H d s 2 , as well as the normalized curve E ( s ) / E ( 0 ) \mathcal{E}(s)/\mathcal{E}(0) E ( s ) / E ( 0 )
s ^ c r i t = sup { s : E ( s ; r , β , σ ) E ( 0 ; r , β , σ ) ≤ C t o l } , C t o l ∈ { 5 , 10 } . \widehat{s}_{\mathrm{crit}} = \sup\left\{s : \frac{\mathcal{E}(s;r,\beta,\sigma)}{\mathcal{E}(0;r,\beta,\sigma)} \le C_{\mathrm{tol}}\right\}, \qquad C_{\mathrm{tol}} \in\{5,10\}. s crit = sup { s : E ( 0 ; r , β , σ ) E ( s ; r , β , σ ) ≤ C tol } , C tol ∈ { 5 , 10 } . Figure 1 shows the normalized error curves at d = 1024 d = 1024 d = 1024 σ = 10 − 2 \sigma= 10^{-2} σ = 1 0 − 2 r r r s s s Figure 2 summarizes the same behavior through s ^ c r i t \widehat{s}_{\mathrm{crit}} s crit β \beta β
Figure 1. Dense transition atlas. Each panel tracks normalized error across target regularity using a denser s s s
Figure 2. Empirical recoverability heatmaps under two tolerance levels. The panels summarize how the critical regularity varies with ( r , β ) (r,\beta) ( r , β )
Table 2 simulates s ^ c r i t \hat{s}_{\mathrm{crit}} s ^ crit d = 1024 d = 1024 d = 1024 σ = 10 − 2 \sigma= 10^{-2} σ = 1 0 − 2 C t o l = 5 C_{\mathrm{tol}} = 5 C tol = 5 s ^ c r i t \hat{s}_{\mathrm{crit}} s ^ crit β = 1 \beta= 1 β = 1 β = 2 \beta= 2 β = 2 β = 3 \beta= 3 β = 3 H d s H_d^s H d s
Table 2. Empirical recoverability index s ^ c r i t \hat{s}_{\mathrm{crit}} s ^ crit d = 1024 d = 1024 d = 1024 σ = 10 − 2 \sigma= 10^{-2} σ = 1 0 − 2 C t o l = 5 C_{\mathrm{tol}} = 5 C tol = 5
Table 3 gives the error scale at s = 0 , 1 , 2 s = 0, 1, 2 s = 0 , 1 , 2 d = 1024 d = 1024 d = 1024 β = 2 \beta= 2 β = 2 σ = 10 − 2 \sigma= 10^{-2} σ = 1 0 − 2
Table 3. Mean reconstruction error at three target regularities for the representative case d = 1024 d=1024 d = 1024 β = 2 \beta=2 β = 2 σ = 10 − 2 \sigma=10^{-2} σ = 1 0 − 2
E ( 0 ) = 1.242 × 10 − 3 , E ( 1 ) = 2.657 × 10 − 2 , E ( 2 ) = 5.203 \mathcal{E}(0) = 1.242 \times10^{-3}, \qquad\mathcal{E}(1) = 2.657 \times10^{-2}, \qquad\mathcal{E}(2) = 5.203 E ( 0 ) = 1.242 × 1 0 − 3 , E ( 1 ) = 2.657 × 1 0 − 2 , E ( 2 ) = 5.203 when r = 0.5 r = 0.5 r = 0.5
E ( 0 ) = 1.056 × 10 − 2 , E ( 1 ) = 9.586 × 10 − 2 , E ( 2 ) = 5.958 \mathcal{E}(0) = 1.056 \times10^{-2}, \qquad\mathcal{E}(1) = 9.586 \times10^{-2}, \qquad\mathcal{E}(2) = 5.958 E ( 0 ) = 1.056 × 1 0 − 2 , E ( 1 ) = 9.586 × 1 0 − 2 , E ( 2 ) = 5.958 when r = 3.0 r = 3.0 r = 3.0 H d 0 H_d^0 H d 0 H d 2 H_d^2 H d 2 Figure 3 and Table 4 demonstrate the effect of the weak score perturbation. For the severe case r = 3 r = 3 r = 3 β = 2 \beta= 2 β = 2
Figure 3. Weak-score stress plot. Values above one indicate amplification of the strong-norm error relative to the exact posterior mean.
Table 4. Strong-norm stress ratio for the weakly perturbed score under the more ill-posed case r = 3 r = 3 r = 3 β = 2 \beta= 2 β = 2
E θ , w e a k ( 2 ) E p o s t ( 2 ) \frac{\mathcal{E}_{\theta,\mathrm{weak}}(2)}{\mathcal{E}_{\mathrm{post}}(2)} E post ( 2 ) E θ , weak ( 2 ) stays between approximately 0.998 and 1.000 across the reported ( d , σ ) (d,\sigma) ( d , σ ) s s s
Experiment 2: Learned-Score Invisible-Mode Stress Test
We generate non-Gaussian one-dimensional signals on a grid with d = 256 d = 256 d = 256
A 1 = mild blur , A 2 = severe blur , A 3 = severe blur with missing frequencies , A_1 = \text{mild blur}, \qquad A_2 = \text{severe blur}, \qquad A_3 = \text{severe blur with missing frequencies}, A 1 = mild blur , A 2 = severe blur , A 3 = severe blur with missing frequencies , with spectral multipliers
a j ( 1 ) = 1 1 + j 1.1 , a j ( 2 ) = 1 1 + j 2.2 , a_j^{(1)} = \frac{1}{1+j^{1.1}}, \qquad a_j^{(2)} = \frac{1}{1+j^{2.2}}, a j ( 1 ) = 1 + j 1.1 1 , a j ( 2 ) = 1 + j 2.2 1 , where A 3 A_3 A 3 α = 0.04 \alpha= 0.04 α = 0.04 α \alpha α x x x x 3 x^3 x 3 x 5 x^5 x 5 Table 5 shows the selected Tikhonov and Sobolev parameters, the guidance strengths of DPS and GeoPD, and the implementation details. We compare the Tikhonov, Sobolev, DPS, and GeoPD reconstructions x ^ T i k \widehat{x}_{\mathrm{Tik}} x Tik x ^ S o b \widehat{x}_{\mathrm{Sob}} x Sob x ^ D P S \widehat{x}_{\mathrm{DPS}} x DPS x ^ G e o P D \widehat{x}_{\mathrm{GeoPD}} x GeoPD H d 0 H_d^0 H d 0 H d 1 H_d^1 H d 1 H d 2 H_d^2 H d 2
Table 5. Implementation details for Experiment 2.
Figure 4 shows a representative reconstruction under A 3 A_3 A 3 Table 6 shows the corresponding H d 0 H_d^0 H d 0 H d 1 H_d^1 H d 1 H d 2 H_d^2 H d 2
Figure 4. Top: representative reconstructions for the missing-frequency operator. Middle: spectral energy of the truth, reconstructions, and reconstruction error. Bottom: high-frequency band error across forward operators.
Table 6. Reconstruction metrics for non-Gaussian one-dimensional signals. The geometry-aware sampler is most distinct from isotropic guidance in stronger norms and in the high-frequency band.
We also evaluate validation score mismatch in weak norms. At the representative diffusion level α ≈ 9.65 × 10 − 2 \alpha\approx9.65 \times10^{-2} α ≈ 9.65 × 1 0 − 2
δ ^ 0 2 ( α ) = 5.767 × 10 3 , δ ^ 0.5 2 ( α ) = 3.860 × 10 2 , δ ^ 1 2 ( α ) = 6.946 × 10 1 , δ ^ 2 2 ( α ) = 9.216. \widehat{\delta}_0^2(\alpha) = 5.767 \times10^3, \qquad\widehat{\delta}_{0.5}^2(\alpha) = 3.860 \times10^2, \qquad\widehat{\delta}_1^2(\alpha) = 6.946 \times10^1, \qquad\widehat{\delta}_2^2(\alpha) = 9.216. δ 0 2 ( α ) = 5.767 × 1 0 3 , δ 0.5 2 ( α ) = 3.860 × 1 0 2 , δ 1 2 ( α ) = 6.946 × 1 0 1 , δ 2 2 ( α ) = 9.216. At the smallest noise level α = 0.03 \alpha= 0.03 α = 0.03 9.480 × 10 4 9.480 \times10^4 9.480 × 1 0 4 4.388 × 10 3 4.388 \times10^3 4.388 × 1 0 3 7.188 × 10 2 7.188 \times10^2 7.188 × 1 0 2 9.317 × 10 1 9.317 \times10^1 9.317 × 1 0 1
The Sobolev regularizer achieves the lowest reconstruction error for each of the three operators. For A 1 A_1 A 1 9.019 × 10 7 9.019 \times10^7 9.019 × 1 0 7 2.497 × 10 8 2.497 \times10^8 2.497 × 1 0 8 2.395 × 10 8 2.395 \times10^8 2.395 × 1 0 8 A 2 A_2 A 2 A 3 A_3 A 3 H d 2 H_d^2 H d 2 H d 2 H_d^2 H d 2 1.556 × 10 − 1 1.556 \times10^{-1} 1.556 × 1 0 − 1 1.215 × 10 − 1 1.215 \times10^{-1} 1.215 × 1 0 − 1
Experiment 3: Matched-Residual Comparison of Isotropic and Geometry-Aware Guidance
We compare isotropic and geometry-aware posterior guidance after matching the observation residual. Both samplers use the same score model, reverse-time integrator, and terminal distribution; they differ only in the likelihood correction. The isotropic sampler uses a correction proportional to A ∗ ( y − A x ) A^*(y - Ax) A ∗ ( y − A x )
Γ t = ( A ∗ A + λ t L ρ ) − 1 . \Gamma_t = (A^*A + \lambda_t L^\rho)^{-1}. Γ t = ( A ∗ A + λ t L ρ ) − 1 . In diagonal spectral coordinates,
( Γ t z ) j = 1 a j 2 + λ t j 2 ρ z j , ρ = 2. (\Gamma_t z)_j = \frac{1}{a_j^2 + \lambda_t j^{2\rho}} z_j, \qquad\rho= 2. ( Γ t z ) j = a j 2 + λ t j 2 ρ 1 z j , ρ = 2. We evaluate this comparison in two domains. The spectral domain uses
d = 512 , β = 2 , σ = 10 − 2 , r ∈ { 0.5 , 1 , 2 , 3 } . d = 512, \qquad\beta= 2, \qquad\sigma= 10^{-2}, \qquad r \in\{0.5, 1, 2, 3\}. d = 512 , β = 2 , σ = 1 0 − 2 , r ∈ { 0.5 , 1 , 2 , 3 } . The learned-signal domain reuses the score surrogate from Experiment 2 and focuses on the severe operators A 2 A_2 A 2 A 3 A_3 A 3
γ ∈ { 0.25 , 0.40 , 0.55 , 0.70 , 0.85 , 1.00 , 1.20 , 1.45 , 1.70 , 2.00 , 2.30 } . \gamma\in\{0.25, 0.40, 0.55, 0.70, 0.85, 1.00, 1.20, 1.45, 1.70, 2.00, 2.30\}. γ ∈ { 0.25 , 0.40 , 0.55 , 0.70 , 0.85 , 1.00 , 1.20 , 1.45 , 1.70 , 2.00 , 2.30 } . For each scenario, we sweep this grid separately for isotropic and geometry-aware guidance, record the mean residual, and choose the pair ( γ i s o , γ g e o ) (\gamma_{\mathrm{iso}}, \gamma_{\mathrm{geo}}) ( γ iso , γ geo ) H d 0 H_d^0 H d 0 H d 1 H_d^1 H d 1 H d 2 H_d^2 H d 2
Table 7. Implementation details for Experiment 3.
Table 8 summarizes the selected guidance strengths and residual gaps. In the spectral domain, the matched pairs are
Table 8. Guidance strengths selected for the matched-residual comparison
( γ i s o , γ g e o ) = ( 1.70 , 2.30 ) , ( 1.70 , 2.30 ) , ( 1.00 , 2.00 ) , ( 0.40 , 0.70 ) (\gamma_{\mathrm{iso}}, \gamma_{\mathrm{geo}}) = (1.70, 2.30),\ (1.70, 2.30),\ (1.00, 2.00),\ (0.40, 0.70) ( γ iso , γ geo ) = ( 1.70 , 2.30 ) , ( 1.70 , 2.30 ) , ( 1.00 , 2.00 ) , ( 0.40 , 0.70 ) for r = 0.5 , 1.0 , 2.0 , 3.0 r = 0.5, 1.0, 2.0, 3.0 r = 0.5 , 1.0 , 2.0 , 3.0 3.396 × 10 − 4 3.396 \times10^{-4} 3.396 × 1 0 − 4 3.969 × 10 − 3 3.969 \times10^{-3} 3.969 × 1 0 − 3 ( 0.85 , 0.85 ) (0.85, 0.85) ( 0.85 , 0.85 ) A 2 A_2 A 2 ( 1.00 , 0.40 ) (1.00, 0.40) ( 1.00 , 0.40 ) A 3 A_3 A 3 Figure 5 shows the residual-error tradeoff curves before matching, which verifies that the comparison is made at comparable data fit.
Figure 5. Residual-error tradeoff curves for representative severe cases in Experiment 3. Each curve traces the effect of guidance strength before the matched-residual point is selected.
Table 9 gives the matched-residual reconstruction errors. In the spectral domain, geometry-aware guidance gives small gains in most cases. For the severe case r = 3.0 r = 3.0 r = 3.0
Table 9. Experiment 3 matched-residual comparison of isotropic and geometry-aware posterior guidance. Residuals are tuned to be comparable within each scenario before comparing reconstruction stability.
∥ A x ^ i s o − y ∥ 2 ≈ 7.859 × 10 − 1 , ∥ A x ^ g e o − y ∥ 2 ≈ 7.827 × 10 − 1 . \lVert A\widehat{x}_{\mathrm{iso}} - y\rVert_2 \approx7.859 \times10^{-1}, \qquad\lVert A\widehat{x}_{\mathrm{geo}} - y\rVert_2 \approx7.827 \times10^{-1}. ∥ A x iso − y ∥ 2 ≈ 7.859 × 1 0 − 1 , ∥ A x geo − y ∥ 2 ≈ 7.827 × 1 0 − 1 . and the corresponding errors are
H d 0 : 1.510 vs. 1.446 , H d 1 : 4.685 × 10 4 vs. 4.637 × 10 4 , H_d^0 : 1.510\ \text{vs.}\ 1.446, \qquad H_d^1 : 4.685 \times10^4\ \text{vs.}\ 4.637 \times10^4, H d 0 : 1.510 vs. 1.446 , H d 1 : 4.685 × 1 0 4 vs. 4.637 × 1 0 4 , H d 2 : 7.489 × 10 9 vs. 7.392 × 10 9 . H_d^2 : 7.489 \times10^9\ \text{vs.}\ 7.392 \times10^9. H d 2 : 7.489 × 1 0 9 vs. 7.392 × 1 0 9 . Table 10 and Figure 6 summarize these ratios. For r = 3.0 r = 3.0 r = 3.0 H d 0 H_d^0 H d 0 H d 1 H_d^1 H d 1 H d 2 H_d^2 H d 2 r = 2.0 r = 2.0 r = 2.0 H d 0 H_d^0 H d 0
Figure 6. Matched-residual gain heatmaps for Experiment 3. Values above one favor geometry-aware guidance.
Table 10. Relative gain ratio ∥ x ^ i s o − x ⋆ ∥ H s 2 / ∥ x ^ g e o − x ⋆ ∥ H s 2 \lVert\hat{x}_{\mathrm{iso}}-x^\star\rVert_{H^s}^2/\lVert\hat{x}_{\mathrm{geo}}-x^\star\rVert_{H^s}^2 ∥ x ^ iso − x ⋆ ∥ H s 2 / ∥ x ^ geo − x ⋆ ∥ H s 2
The learned-signal domain gives a more mixed comparison. For A 2 A_2 A 2
H d 0 : 2.585 × 10 2 vs. 2.671 × 10 2 , H_d^0 : 2.585 \times10^2\ \text{vs.}\ 2.671 \times10^2, H d 0 : 2.585 × 1 0 2 vs. 2.671 × 1 0 2 , H d 1 : 1.030 × 10 5 vs. 1.071 × 10 5 , H_d^1 : 1.030 \times10^5\ \text{vs.}\ 1.071 \times10^5, H d 1 : 1.030 × 1 0 5 vs. 1.071 × 1 0 5 , H d 2 : 3.550 × 10 8 vs. 3.613 × 10 8 . H_d^2 : 3.550 \times10^8\ \text{vs.}\ 3.613 \times10^8. H d 2 : 3.550 × 1 0 8 vs. 3.613 × 1 0 8 . The same pattern holds for A 3 A_3 A 3 H d 2 H_d^2 H d 2 Figure 7 gives additional spectral detail: in the severe spectral case, the geometry-aware correction reduces the high-mode tail of both coefficient-wise error and posterior sample variability, which agrees with the intended role of operator-adapted likelihood injection.
Figure 7. Mode-wise reconstruction error and posterior variability for representative severe cases in Experiment 3.
Conclusion
We studied posterior diffusion for ill-posed linear inverse problems on Hilbert scales and showed that recovery depends on operator smoothing, truth regularity, weak-norm score error, the reconstruction norm, and the posterior correction. The recovery estimate separates observational noise, learned-score approximation, and reverse-time discretization, with the score contribution measured by a propagation functional from H − q H^{-q} H − q H s H^s H s
Proof of Proposition 4.1
Proof. We prove this proposition for X t ⋆ X_t^\star X t ⋆ X t θ X_t^\theta X t θ
S 2 ( H s ) = { X adapted and continuous on [ t 0 , 1 ] ; E sup t ∈ [ t 0 , 1 ] ∥ X t ∥ H s 2 < ∞ } , \mathbb{S}^{2}(\mathcal{H}^{s})=\left\{X\text{ adapted and continuous on }[t_{0},1]\ ;\ \mathbb{E}\sup_{t\in[t_{0},1]}\lVert X_{t}\rVert_{\mathcal{H}^{s}}^{2}<\infty\right\}, S 2 ( H s ) = { X adapted and continuous on [ t 0 , 1 ] ; E t ∈ [ t 0 , 1 ] sup ∥ X t ∥ H s 2 < ∞ } , equipped with the norm
∥ X ∥ S 2 ( H s ) 2 = E sup t ∈ [ t 0 , 1 ] ∥ X t ∥ H s 2 . \lVert X\rVert_{\mathbb{S}^{2}(\mathcal{H}^{s})}^{2}=\mathbb{E}\sup_{t\in[t_{0},1]}\lVert X_{t}\rVert_{\mathcal{H}^{s}}^{2}. ∥ X ∥ S 2 ( H s ) 2 = E t ∈ [ t 0 , 1 ] sup ∥ X t ∥ H s 2 . Define the map T ⋆ \mathcal{T}^{\star} T ⋆ S 2 ( H s ) \mathbb{S}^{2}(\mathcal{H}^{s}) S 2 ( H s )
( T ⋆ X ) t = ξ − ∫ t 1 b ⋆ ( X u , u ) d u − ∫ t 1 g ( u ) d W ‾ u . (\mathcal{T}^{\star}X)_{t}=\xi-\int_{t}^{1}b^{\star}(X_{u},u)\,\mathrm{d}u-\int_{t}^{1}g(u)\,\mathrm{d}\overline{W}_{u}. ( T ⋆ X ) t = ξ − ∫ t 1 b ⋆ ( X u , u ) d u − ∫ t 1 g ( u ) d W u . A fixed point X = T ♯ X X=\mathcal{T}^{\sharp}X X = T ♯ X (9) , and any adapted continuous process satisfying that identity is a fixed point. To this end, we first show that T ⋆ \mathcal{T}^{\star} T ⋆ S 2 ( H s ) \mathbb{S}^{2}(\mathcal{H}^{s}) S 2 ( H s )
∥ a + b + c ∥ H s 2 ≤ 3 ∥ a ∥ H s 2 + 3 ∥ b ∥ H s 2 + 3 ∥ c ∥ H s 2 , (31) \lVert a+b+c\rVert_{\mathcal{H}^{s}}^{2}\leq3\lVert a\rVert_{\mathcal{H}^{s}}^{2}+3\lVert b\rVert_{\mathcal{H}^{s}}^{2}+3\lVert c\rVert_{\mathcal{H}^{s}}^{2}, \tag*{(31)} ∥ a + b + c ∥ H s 2 ≤ 3 ∥ a ∥ H s 2 + 3 ∥ b ∥ H s 2 + 3 ∥ c ∥ H s 2 , (31) we obtain
sup t ∈ [ t 0 , 1 ] ∥ ( T ⋆ X ) t ∥ H s 2 ≤ 3 ∥ ξ ∥ H s 2 + 3 sup t ∈ [ t 0 , 1 ] ∥ ∫ t 1 b ⋆ ( X u , u ) d u ∥ H s 2 + 3 sup t ∈ [ t 0 , 1 ] ∥ ∫ t 1 g ( u ) d W ‾ u ∥ H s 2 . (32) \sup_{t\in[t_{0},1]}\lVert(\mathcal{T}^{\star}X)_{t}\rVert_{\mathcal{H}^{s}}^{2}\leq3\lVert\xi\rVert_{\mathcal{H}^{s}}^{2}+3\sup_{t\in[t_{0},1]}\left\lVert\int_{t}^{1}b^{\star}(X_{u},u)\,\mathrm{d}u\right\rVert_{\mathcal{H}^{s}}^{2}+3\sup_{t\in[t_{0},1]}\left\lVert\int_{t}^{1}g(u)\,\mathrm{d}\overline{W}_{u}\right\rVert_{\mathcal{H}^{s}}^{2}. \tag*{(32)} t ∈ [ t 0 , 1 ] sup ∥( T ⋆ X ) t ∥ H s 2 ≤ 3 ∥ ξ ∥ H s 2 + 3 t ∈ [ t 0 , 1 ] sup ∫ t 1 b ⋆ ( X u , u ) d u H s 2 + 3 t ∈ [ t 0 , 1 ] sup ∫ t 1 g ( u ) d W u H s 2 . (32) For each fixed t ∈ [ t 0 , 1 ] t\in[t_{0},1] t ∈ [ t 0 , 1 ]
∥ ∫ t 1 b ⋆ ( X u , u ) d u ∥ H s 2 ≤ ( 1 − t ) ∫ t 1 ∥ b ⋆ ( X u , u ) ∥ H s 2 d u . \left\lVert\int_{t}^{1}b^{\star}(X_{u},u)\,\mathrm{d}u\right\rVert_{\mathcal{H}^{s}}^{2}\leq(1-t)\int_{t}^{1}\lVert b^{\star}(X_{u},u)\rVert_{\mathcal{H}^{s}}^{2}\,\mathrm{d}u. ∫ t 1 b ⋆ ( X u , u ) d u H s 2 ≤ ( 1 − t ) ∫ t 1 ∥ b ⋆ ( X u , u ) ∥ H s 2 d u . Since 1 − t ≤ 1 − t 0 1-t\leq1-t_{0} 1 − t ≤ 1 − t 0 [ t , 1 ] ⊂ [ t 0 , 1 ] [t,1]\subset[t_{0},1] [ t , 1 ] ⊂ [ t 0 , 1 ]
sup t ∈ [ t 0 , 1 ] ∥ ∫ t 1 b ⋆ ( X u , u ) d u ∥ H s 2 ≤ ( 1 − t 0 ) ∫ t 0 1 ∥ b ⋆ ( X u , u ) ∥ H s 2 d u . (33) \sup_{t\in[t_{0},1]}\left\lVert\int_{t}^{1}b^{\star}(X_{u},u)\,\mathrm{d}u\right\rVert_{\mathcal{H}^{s}}^{2}\leq(1-t_{0})\int_{t_{0}}^{1}\lVert b^{\star}(X_{u},u)\rVert_{\mathcal{H}^{s}}^{2}\,\mathrm{d}u. \tag*{(33)} t ∈ [ t 0 , 1 ] sup ∫ t 1 b ⋆ ( X u , u ) d u H s 2 ≤ ( 1 − t 0 ) ∫ t 0 1 ∥ b ⋆ ( X u , u ) ∥ H s 2 d u . (33) From the linear growth bound (7) ,
∥ b ⋆ ( X u , u ) ∥ H s 2 ≤ K s 2 ( 1 + ∥ X u ∥ H s ) 2 ≤ 2 K s 2 ( 1 + ∥ X u ∥ H s 2 ) , (34) \lVert b^{\star}(X_{u},u)\rVert_{\mathcal{H}^{s}}^{2}\leq K_{s}^{2}(1+\lVert X_{u}\rVert_{\mathcal{H}^{s}})^{2}\leq2K_{s}^{2}(1+\lVert X_{u}\rVert_{\mathcal{H}^{s}}^{2}), \tag*{(34)} ∥ b ⋆ ( X u , u ) ∥ H s 2 ≤ K s 2 ( 1 + ∥ X u ∥ H s ) 2 ≤ 2 K s 2 ( 1 + ∥ X u ∥ H s 2 ) , (34) where the last inequality follows from ( a + b ) 2 ≤ 2 a 2 + 2 b 2 (a+b)^{2}\leq2a^{2}+2b^{2} ( a + b ) 2 ≤ 2 a 2 + 2 b 2 (34) with drift estimate (33) , we have
E sup t ∈ [ t 0 , 1 ] ∥ ∫ t 1 b ⋆ ( X u , u ) d u ∥ H s 2 ≤ 2 ( 1 − t 0 ) K s 2 ∫ t 0 1 E ( 1 + ∥ X u ∥ H s 2 ) d u . \mathbb{E}\sup_{t\in[t_{0},1]}\left\lVert\int_{t}^{1}b^{\star}(X_{u},u)\,\mathrm{d}u\right\rVert_{\mathcal{H}^{s}}^{2}\leq2(1-t_{0})K_{s}^{2}\int_{t_{0}}^{1}\mathbb{E}(1+\lVert X_{u}\rVert_{\mathcal{H}^{s}}^{2})\,\mathrm{d}u. E t ∈ [ t 0 , 1 ] sup ∫ t 1 b ⋆ ( X u , u ) d u H s 2 ≤ 2 ( 1 − t 0 ) K s 2 ∫ t 0 1 E ( 1 + ∥ X u ∥ H s 2 ) d u . Since
E ∥ X u ∥ H s 2 ≤ E sup v ∈ [ t 0 , 1 ] ∥ X v ∥ H s 2 = ∥ X ∥ S 2 ( H s ) 2 , \mathbb{E}\lVert X_{u}\rVert_{\mathcal{H}^{s}}^{2}\leq\mathbb{E}\sup_{v\in[t_{0},1]}\lVert X_{v}\rVert_{\mathcal{H}^{s}}^{2}=\lVert X\rVert_{\mathbb{S}^{2}(\mathcal{H}^{s})}^{2}, E ∥ X u ∥ H s 2 ≤ E v ∈ [ t 0 , 1 ] sup ∥ X v ∥ H s 2 = ∥ X ∥ S 2 ( H s ) 2 , we obtain
∫ t 0 1 E ( 1 + ∥ X u ∥ H s 2 ) d u ≤ ( 1 − t 0 ) ( 1 + ∥ X ∥ S 2 ( H s ) 2 ) . \int_{t_{0}}^{1}\mathbb{E}(1+\lVert X_{u}\rVert_{\mathcal{H}^{s}}^{2})\,\mathrm{d}u\leq(1-t_{0})\left(1+\lVert X\rVert_{\mathbb{S}^{2}(\mathcal{H}^{s})}^{2}\right). ∫ t 0 1 E ( 1 + ∥ X u ∥ H s 2 ) d u ≤ ( 1 − t 0 ) ( 1 + ∥ X ∥ S 2 ( H s ) 2 ) . Hence,
E sup t ∈ [ t 0 , 1 ] ∥ ∫ t 1 b ⋆ ( X u , u ) d u ∥ H s 2 ≤ 2 ( 1 − t 0 ) 2 K s 2 ( 1 + ∥ X ∥ S 2 ( H s ) 2 ) . \mathbb{E}\sup_{t\in[t_0,1]}\left\|\int_t^1 b^\star(X_u,u)\,du\right\|_{\mathcal{H}^s}^2 \le2(1-t_0)^2K_s^2\left(1+\|X\|_{\mathcal{S}^2(\mathcal{H}^s)}^2\right). E t ∈ [ t 0 , 1 ] sup ∫ t 1 b ⋆ ( X u , u ) d u H s 2 ≤ 2 ( 1 − t 0 ) 2 K s 2 ( 1 + ∥ X ∥ S 2 ( H s ) 2 ) . For the stochastic term, Doob’s L 2 L^2 L 2
E sup t ∈ [ t 0 , 1 ] ∥ ∫ t 1 g ( u ) d W ‾ u ∥ H s 2 ≤ 4 ∫ t 0 1 ∣ g ( u ) ∣ 2 d u . \mathbb{E}\sup_{t\in[t_0,1]}\left\|\int_t^1 g(u)\,d\overline{W}_u\right\|_{\mathcal{H}^s}^2 \le4\int_{t_0}^1 |g(u)|^2\,du. E t ∈ [ t 0 , 1 ] sup ∫ t 1 g ( u ) d W u H s 2 ≤ 4 ∫ t 0 1 ∣ g ( u ) ∣ 2 d u . The drift and stochastic estimates, applied to (32) , give
∥ T ⋆ X ∥ S 2 ( H s ) 2 ≤ C ( 1 + E ∥ ξ ∥ H s 2 + ∥ X ∥ S 2 ( H s ) 2 ) . \|T^\star X\|_{\mathcal{S}^2(\mathcal{H}^s)}^2 \le C\left(1+\mathbb{E}\|\xi\|_{\mathcal{H}^s}^2+\|X\|_{\mathcal{S}^2(\mathcal{H}^s)}^2\right). ∥ T ⋆ X ∥ S 2 ( H s ) 2 ≤ C ( 1 + E ∥ ξ ∥ H s 2 + ∥ X ∥ S 2 ( H s ) 2 ) . Thus T ⋆ X ∈ S 2 ( H s ) T^\star X\in\mathcal{S}^2(\mathcal{H}^s) T ⋆ X ∈ S 2 ( H s ) X ∈ S 2 ( H s ) X\in\mathcal{S}^2(\mathcal{H}^s) X ∈ S 2 ( H s )
We prove existence and uniqueness by contraction on a short interval and iteration. Let X , Y ∈ S 2 ( H s ) X,Y\in\mathcal{S}^2(\mathcal{H}^s) X , Y ∈ S 2 ( H s )
( T ⋆ X ) t − ( T ⋆ Y ) t = − ∫ t 1 ( b ⋆ ( X u , u ) − b ⋆ ( Y u , u ) ) d u , (T^\star X)_t-(T^\star Y)_t=-\int_t^1\left(b^\star(X_u,u)-b^\star(Y_u,u)\right)\,du, ( T ⋆ X ) t − ( T ⋆ Y ) t = − ∫ t 1 ( b ⋆ ( X u , u ) − b ⋆ ( Y u , u ) ) d u , hence
sup t ∈ [ t 0 , 1 ] ∥ ( T ⋆ X ) t − ( T ⋆ Y ) t ∥ H s 2 ≤ ( 1 − t 0 ) ∫ t 0 1 ∥ b ⋆ ( X u , u ) − b ⋆ ( Y u , u ) ∥ H s 2 d u . \sup_{t\in[t_0,1]}\|(T^\star X)_t-(T^\star Y)_t\|_{\mathcal{H}^s}^2\le(1-t_0)\int_{t_0}^1\|b^\star(X_u,u)-b^\star(Y_u,u)\|_{\mathcal{H}^s}^2\,du. t ∈ [ t 0 , 1 ] sup ∥ ( T ⋆ X ) t − ( T ⋆ Y ) t ∥ H s 2 ≤ ( 1 − t 0 ) ∫ t 0 1 ∥ b ⋆ ( X u , u ) − b ⋆ ( Y u , u ) ∥ H s 2 d u . Using Lipschitz condition (6) , ∥ b ⋆ ( X u , u ) − b ⋆ ( Y u , u ) ∥ H s \|b^\star(X_u,u)-b^\star(Y_u,u)\|_{\mathcal{H}^s} ∥ b ⋆ ( X u , u ) − b ⋆ ( Y u , u ) ∥ H s
∥ T ⋆ X − T ⋆ Y ∥ S 2 ( H s ) 2 ≤ ( 1 − t 0 ) 2 L s 2 ∥ X − Y ∥ S 2 ( H s ) 2 . \|T^\star X-T^\star Y\|_{\mathcal{S}^2(\mathcal{H}^s)}^2\le(1-t_0)^2L_s^2\|X-Y\|_{\mathcal{S}^2(\mathcal{H}^s)}^2. ∥ T ⋆ X − T ⋆ Y ∥ S 2 ( H s ) 2 ≤ ( 1 − t 0 ) 2 L s 2 ∥ X − Y ∥ S 2 ( H s ) 2 . If the interval is short enough that ( 1 − t 0 ) 2 L s 2 < 1 (1-t_0)^2L_s^2<1 ( 1 − t 0 ) 2 L s 2 < 1 T ⋆ T^\star T ⋆ [ t 0 , 1 ] [t_0,1] [ t 0 , 1 ] [ t 0 , 1 ] [t_0,1] [ t 0 , 1 ] 1 1 1
Let X t ⋆ = ξ − ∫ t 1 b ⋆ ( X u ⋆ , u ) d u − ∫ t 1 g ( u ) d W ‾ u X_t^\star=\xi-\int_t^1b^\star(X_u^\star,u)\,du-\int_t^1g(u)\,d\overline{W}_u X t ⋆ = ξ − ∫ t 1 b ⋆ ( X u ⋆ , u ) d u − ∫ t 1 g ( u ) d W u M ( a ) = E sup t ∈ [ a , 1 ] ∥ X t ⋆ ∥ H s 2 M(a)=\mathbb{E}\sup_{t\in[a,1]}\|X_t^\star\|_{\mathcal{H}^s}^2 M ( a ) = E sup t ∈ [ a , 1 ] ∥ X t ⋆ ∥ H s 2 a ∈ [ t 0 , 1 ] a\in[t_0,1] a ∈ [ t 0 , 1 ] (7) , Doob’s inequality, and Itô isometry gives us
M ( a ) ≤ C ( 1 + E ∥ ξ ∥ H s 2 + ∫ a 1 ∣ g ( u ) ∣ 2 d u + ∫ a 1 M ( u ) d u ) . M(a)\le C\left(1+\mathbb{E}\|\xi\|_{\mathcal{H}^s}^2+\int_a^1|g(u)|^2\,du+\int_a^1M(u)\,du\right). M ( a ) ≤ C ( 1 + E ∥ ξ ∥ H s 2 + ∫ a 1 ∣ g ( u ) ∣ 2 d u + ∫ a 1 M ( u ) d u ) . Gronwall’s lemma in backward time gives
M ( t 0 ) = E sup t ∈ [ t 0 , 1 ] ∥ X t ⋆ ∥ H s 2 ≤ C s ( 1 + E ∥ ξ ∥ H s 2 ) , M(t_0)=\mathbb{E}\sup_{t\in[t_0,1]}\|X_t^\star\|_{\mathcal{H}^s}^2\le C_s\left(1+\mathbb{E}\|\xi\|_{\mathcal{H}^s}^2\right), M ( t 0 ) = E t ∈ [ t 0 , 1 ] sup ∥ X t ⋆ ∥ H s 2 ≤ C s ( 1 + E ∥ ξ ∥ H s 2 ) , which proves (10) .
Under (11) , the operator family
B t Γ = D x b ⋆ ( X t ⋆ , t ) B_t^\Gamma=D_xb^\star(X_t^\star,t) B t Γ = D x b ⋆ ( X t ⋆ , t ) is uniformly bounded on H s \mathcal{H}^s H s
d d t Z t = B t Γ Z t \frac{d}{dt}Z_t=B_t^\Gamma Z_t d t d Z t = B t Γ Z t therefore has a unique evolution family U τ , t Γ \mathcal{U}_{\tau,t}^\Gamma U τ , t Γ H s \mathcal{H}^s H s Z t = U τ , t Γ z Z_t=\mathcal{U}_{\tau,t}^\Gamma z Z t = U τ , t Γ z
d d t ∥ Z t ∥ H s ≤ ∥ B t Γ ∥ L ( H s ) ∥ Z t ∥ H s ≤ L s ′ ∥ Z t ∥ H s , \frac{d}{dt}\|Z_t\|_{\mathcal{H}^s}\le\|B_t^\Gamma\|_{\mathcal{L}(\mathcal{H}^s)}\|Z_t\|_{\mathcal{H}^s}\le L_s'\|Z_t\|_{\mathcal{H}^s}, d t d ∥ Z t ∥ H s ≤ ∥ B t Γ ∥ L ( H s ) ∥ Z t ∥ H s ≤ L s ′ ∥ Z t ∥ H s , and Gronwall’s inequality gives
∥ Z t ∥ H s ≤ e L s ′ ∣ t − τ ∣ ∥ z ∥ H s . \|Z_t\|_{\mathcal{H}^s}\le e^{L_s'|t-\tau|}\|z\|_{\mathcal{H}^s}. ∥ Z t ∥ H s ≤ e L s ′ ∣ t − τ ∣ ∥ z ∥ H s . Since z ∈ H s z\in\mathcal{H}^s z ∈ H s (13) .
Proof of Theorem 4.6
Lemma B.1. Given Assumption 4.4, let m y m_y m y C y C_y C y X 0 ⋆ X_0^\star X 0 ⋆ y y y
E [ ∥ m y − x † ∥ H s 2 ] ≤ C b i a s σ 2 η ( r , β , s ) , E [ Tr ( L s / 2 C y L s / 2 ) ] ≤ C v a r σ 2 η ( r , β , s ) . (35) \mathbb{E}\left[\lVert m_y-x^\dagger\rVert_{\mathcal{H}_s}^2\right]\le C_{\mathrm{bias}}\sigma^{2\eta(r,\beta,s)},\qquad\mathbb{E}\left[\operatorname{Tr}\left(L^{s/2}C_yL^{s/2}\right)\right]\le C_{\mathrm{var}}\sigma^{2\eta(r,\beta,s)}. \tag*{(35)} E [ ∥ m y − x † ∥ H s 2 ] ≤ C bias σ 2 η ( r , β , s ) , E [ Tr ( L s /2 C y L s /2 ) ] ≤ C var σ 2 η ( r , β , s ) . (35) Then
E [ ∥ X 0 ⋆ − x † ∥ H s 2 ] ≤ Φ n o i s e ( σ ; r , β , s ) , (36) \mathbb{E}\left[\lVert X_0^\star-x^\dagger\rVert_{\mathcal{H}_s}^2\right]\le\Phi_{\mathrm{noise}}(\sigma;r,\beta,s), \tag*{(36)} E [ ∥ X 0 ⋆ − x † ∥ H s 2 ] ≤ Φ noise ( σ ; r , β , s ) , (36) where
Φ n o i s e ( σ ; r , β , s ) = C n o i s e σ 2 η ( r , β , s ) , C n o i s e = C b i a s + C v a r . \Phi_{\mathrm{noise}}(\sigma;r,\beta,s)=C_{\mathrm{noise}}\sigma^{2\eta(r,\beta,s)},\qquad C_{\mathrm{noise}}=C_{\mathrm{bias}}+C_{\mathrm{var}}. Φ noise ( σ ; r , β , s ) = C noise σ 2 η ( r , β , s ) , C noise = C bias + C var . Proof. By Assumption 4.4,
E [ X 0 ⋆ ∣ y ] = m y , Cov ( X 0 ⋆ ∣ y ) = C y . \mathbb{E}[X_0^\star\mid y]=m_y,\qquad\operatorname{Cov}(X_0^\star\mid y)=C_y. E [ X 0 ⋆ ∣ y ] = m y , Cov ( X 0 ⋆ ∣ y ) = C y . Define Z = X 0 ⋆ − m y Z=X_0^\star-m_y Z = X 0 ⋆ − m y Z ∣ y Z\mid y Z ∣ y C y C_y C y
E [ ∥ X 0 ⋆ − x † ∥ H s 2 ∣ y ] = E [ ∥ Z + ( m y − x † ) ∥ H s 2 ∣ y ] \mathbb{E}\left[\lVert X_0^\star-x^\dagger\rVert_{\mathcal{H}_s}^2\mid y\right]=\mathbb{E}\left[\lVert Z+(m_y-x^\dagger)\rVert_{\mathcal{H}_s}^2\mid y\right] E [ ∥ X 0 ⋆ − x † ∥ H s 2 ∣ y ] = E [ ∥ Z + ( m y − x † ) ∥ H s 2 ∣ y ] = ( a ) E [ ∥ Z ∥ H s 2 ∣ y ] + ∥ m y − x † ∥ H s 2 + 2 E [ ⟨ Z , m y − x † ⟩ H s ∣ y ] = E [ ∥ Z ∥ H s 2 ∣ y ] + ∥ m y − x † ∥ H s 2 . (37) \begin{aligned}
\overset{(a)}{=}\mathbb{E}\left[\lVert Z\rVert_{\mathcal{H}_s}^2\mid y\right]+\lVert m_y-x^\dagger\rVert_{\mathcal{H}_s}^2+2\mathbb{E}\left[\langle Z,m_y-x^\dagger\rangle_{\mathcal{H}_s}\mid y\right] \\
&=\mathbb{E}\left[\lVert Z\rVert_{\mathcal{H}_s}^2\mid y\right]+\lVert m_y-x^\dagger\rVert_{\mathcal{H}_s}^2. \tag*{(37)}
\end{aligned} = ( a ) E [ ∥ Z ∥ H s 2 ∣ y ] + ∥ m y − x † ∥ H s 2 + 2 E [ ⟨ Z , m y − x † ⟩ H s ∣ y ] = E [ ∥ Z ∥ H s 2 ∣ y ] + ∥ m y − x † ∥ H s 2 . (37) where (a) expands the squared Hilbert norm, and the cross product vanishes since
E [ ⟨ Z , m y − x † ⟩ H s ∣ y ] = ⟨ E [ Z ∣ y ] , m y − x † ⟩ H s = 0. \mathbb{E}\left[\langle Z,m_y-x^\dagger\rangle_{\mathcal{H}_s}\mid y\right]=\langle\mathbb{E}[Z\mid y],m_y-x^\dagger\rangle_{\mathcal{H}_s}=0. E [ ⟨ Z , m y − x † ⟩ H s ∣ y ] = ⟨ E [ Z ∣ y ] , m y − x † ⟩ H s = 0. By definition of the Hilbert-scale norm, ∥ Z ∥ H s 2 = ∥ L s / 2 Z ∥ H 2 \lVert Z\rVert_{\mathcal{H}_s}^2=\lVert L^{s/2}Z\rVert_{\mathcal{H}}^2 ∥ Z ∥ H s 2 = ∥ L s /2 Z ∥ H 2 { e j } j ≥ 1 \{e_j\}_{j\ge1} { e j } j ≥ 1 H \mathcal{H} H L s / 2 C y L s / 2 L^{s/2}C_yL^{s/2} L s /2 C y L s /2
E [ ∥ Z ∥ H s 2 ∣ y ] = E [ ∥ L s / 2 Z ∥ H 2 ∣ y ] = E [ ∑ j = 1 ∞ ∣ ⟨ L s / 2 Z , e j ⟩ H ∣ 2 ∣ y ] = ∑ j = 1 ∞ E [ ∣ ⟨ L s / 2 Z , e j ⟩ H ∣ 2 ∣ y ] = ∑ j = 1 ∞ E [ ⟨ Z , L s / 2 e j ⟩ H ⟨ L s / 2 e j , Z ⟩ H ∣ y ] = ∑ j = 1 ∞ ⟨ C y L s / 2 e j , L s / 2 e j ⟩ H = ∑ j = 1 ∞ ⟨ L s / 2 C y L s / 2 e j , e j ⟩ H = Tr ( L s / 2 C y L s / 2 ) . (38) \begin{aligned}
\mathbb{E}\left[\lVert Z\rVert_{\mathcal{H}_s}^2\mid y\right]=\mathbb{E}\left[\lVert L^{s/2}Z\rVert_{\mathcal{H}}^2\mid y\right] \\
&=\mathbb{E}\left[\sum_{j=1}^{\infty}\left|\langle L^{s/2}Z,e_j\rangle_{\mathcal{H}}\right|^2\mid y\right] \\
&=\sum_{j=1}^{\infty}\mathbb{E}\left[\left|\langle L^{s/2}Z,e_j\rangle_{\mathcal{H}}\right|^2\mid y\right] \\
&=\sum_{j=1}^{\infty}\mathbb{E}\left[\langle Z,L^{s/2}e_j\rangle_{\mathcal{H}}\langle L^{s/2}e_j,Z\rangle_{\mathcal{H}}\mid y\right] \\
&=\sum_{j=1}^{\infty}\langle C_yL^{s/2}e_j,L^{s/2}e_j\rangle_{\mathcal{H}} \\
&=\sum_{j=1}^{\infty}\langle L^{s/2}C_yL^{s/2}e_j,e_j\rangle_{\mathcal{H}} \\
&=\operatorname{Tr}(L^{s/2}C_yL^{s/2}). \tag*{(38)}
\end{aligned} E [ ∥ Z ∥ H s 2 ∣ y ] = E [ ∥ L s /2 Z ∥ H 2 ∣ y ] = E [ j = 1 ∑ ∞ ⟨ L s /2 Z , e j ⟩ H 2 ∣ y ] = j = 1 ∑ ∞ E [ ⟨ L s /2 Z , e j ⟩ H 2 ∣ y ] = j = 1 ∑ ∞ E [ ⟨ Z , L s /2 e j ⟩ H ⟨ L s /2 e j , Z ⟩ H ∣ y ] = j = 1 ∑ ∞ ⟨ C y L s /2 e j , L s /2 e j ⟩ H = j = 1 ∑ ∞ ⟨ L s /2 C y L s /2 e j , e j ⟩ H = Tr ( L s /2 C y L s /2 ) . (38) Using the law of iterated expectation, we have
E ∥ X 0 ⋆ − x † ∥ H s 2 = E [ E [ ∥ X 0 ⋆ − x † ∥ H s 2 ∣ y ] ] = E [ Tr ( L s / 2 C y L s / 2 ) ] + E [ ∥ m y − x † ∥ H s 2 ] ≤ ( C b i a s + C v a r ) σ 2 η ( r , β , s ) . (39) \begin{aligned}
\mathbb{E}\lVert X_0^\star-x^\dagger\rVert_{\mathcal{H}_s}^2=\mathbb{E}\left[\mathbb{E}\left[\lVert X_0^\star-x^\dagger\rVert_{\mathcal{H}_s}^2\mid y\right]\right] \\
&=\mathbb{E}\left[\operatorname{Tr}(L^{s/2}C_yL^{s/2})\right]+\mathbb{E}\left[\lVert m_y-x^\dagger\rVert_{\mathcal{H}_s}^2\right] \\
&\le(C_{\mathrm{bias}}+C_{\mathrm{var}})\sigma^{2\eta(r,\beta,s)}. \tag*{(39)}
\end{aligned} E ∥ X 0 ⋆ − x † ∥ H s 2 = E [ E [ ∥ X 0 ⋆ − x † ∥ H s 2 ∣ y ] ] = E [ Tr ( L s /2 C y L s /2 ) ] + E [ ∥ m y − x † ∥ H s 2 ] ≤ ( C bias + C var ) σ 2 η ( r , β , s ) . (39) Lemma B.2. Let s ⋆ ( x , t ) = ∇ log p t ( x ) s^{\star}(x,t)=\nabla\log p_t(x) s ⋆ ( x , t ) = ∇ log p t ( x ) X t ⋆ X_t^{\star} X t ⋆ X t θ X_t^{\theta} X t θ
d X t ⋆ = ( f ( X t ⋆ , t ) − g ( t ) 2 s ⋆ ( X t ⋆ , t ) − g ( t ) 2 Γ t ∇ x log p ( y ∣ X t ⋆ ) ) d t + g ( t ) d W ‾ t , (40) dX_t^{\star}=\left(f(X_t^{\star},t)-g(t)^2s^{\star}(X_t^{\star},t)-g(t)^2\Gamma_t\nabla_x\log p(y\mid X_t^{\star})\right)dt+g(t)d\overline{W}_t, \tag*{(40)} d X t ⋆ = ( f ( X t ⋆ , t ) − g ( t ) 2 s ⋆ ( X t ⋆ , t ) − g ( t ) 2 Γ t ∇ x log p ( y ∣ X t ⋆ ) ) d t + g ( t ) d W t , (40) and
d X t θ = ( f ( X t θ , t ) − g ( t ) 2 s θ ( X t θ , t ) − g ( t ) 2 Γ t ∇ x log p ( y ∣ X t θ ) ) d t + g ( t ) d W ‾ t , (41) dX_t^{\theta}=\left(f(X_t^{\theta},t)-g(t)^2s_{\theta}(X_t^{\theta},t)-g(t)^2\Gamma_t\nabla_x\log p(y\mid X_t^{\theta})\right)dt+g(t)d\overline{W}_t, \tag*{(41)} d X t θ = ( f ( X t θ , t ) − g ( t ) 2 s θ ( X t θ , t ) − g ( t ) 2 Γ t ∇ x log p ( y ∣ X t θ ) ) d t + g ( t ) d W t , (41) with the same terminal state and the same reverse Brownian motion. Assume also that
sup t ∈ [ t 0 , 1 ] E ∥ s θ ( X t θ , t ) − s ⋆ ( X t θ , t ) ∥ H − q 2 ≤ δ θ 2 (42) \sup_{t\in[t_0,1]}\mathbb{E}\left\|s_{\theta}(X_t^{\theta},t)-s^{\star}(X_t^{\theta},t)\right\|_{H^{-q}}^2\leq\delta_{\theta}^2 \tag*{(42)} t ∈ [ t 0 , 1 ] sup E s θ ( X t θ , t ) − s ⋆ ( X t θ , t ) H − q 2 ≤ δ θ 2 (42) for some q ≥ 0 q\geq0 q ≥ 0 U t , τ Γ U_{t,\tau}^{\Gamma} U t , τ Γ B ‾ t Γ = ∫ 0 1 D x b ⋆ ( X t ⋆ + λ ( X t θ − X t ⋆ ) , t ) d λ \overline{B}_t^{\Gamma}=\int_0^1D_xb^{\star}(X_t^{\star}+\lambda(X_t^{\theta}-X_t^{\star}),t)d\lambda B t Γ = ∫ 0 1 D x b ⋆ ( X t ⋆ + λ ( X t θ − X t ⋆ ) , t ) d λ
∥ U 0 , t Γ ∥ L ( H − q , H s ) ≤ M r , q , s , Γ ( t ) , ∫ t 0 1 g ( t ) 2 M r , q , s , Γ ( t ) d t < ∞ . (43) \left\|U_{0,t}^{\Gamma}\right\|_{\mathcal{L}(H^{-q},H^s)}\leq M_{r,q,s,\Gamma}(t),\qquad\int_{t_0}^1g(t)^2M_{r,q,s,\Gamma}(t)\,dt<\infty. \tag*{(43)} U 0 , t Γ L ( H − q , H s ) ≤ M r , q , s , Γ ( t ) , ∫ t 0 1 g ( t ) 2 M r , q , s , Γ ( t ) d t < ∞. (43) Then
E ∥ X 0 θ − X 0 ⋆ ∥ H s 2 ≤ Φ s c o r e ( δ θ ; r , q , s , Γ ) , (44) \mathbb{E}\left\|X_0^{\theta}-X_0^{\star}\right\|_{H^s}^2\leq\Phi_{\mathrm{score}}(\delta_{\theta};r,q,s,\Gamma), \tag*{(44)} E X 0 θ − X 0 ⋆ H s 2 ≤ Φ score ( δ θ ; r , q , s , Γ ) , (44) where
Φ s c o r e ( δ θ ; r , q , s , Γ ) = ( ∫ t 0 1 g ( t ) 2 M r , q , s , Γ ( t ) d t ) 2 δ θ 2 . \Phi_{\mathrm{score}}(\delta_{\theta};r,q,s,\Gamma)=\left(\int_{t_0}^1g(t)^2M_{r,q,s,\Gamma}(t)\,dt\right)^2\delta_{\theta}^2. Φ score ( δ θ ; r , q , s , Γ ) = ( ∫ t 0 1 g ( t ) 2 M r , q , s , Γ ( t ) d t ) 2 δ θ 2 . Proof. Define Z t = X t θ − X t ⋆ Z_t=X_t^{\theta}-X_t^{\star} Z t = X t θ − X t ⋆ e t = s θ ( X t θ , t ) − s ⋆ ( X t θ , t ) e_t=s_{\theta}(X_t^{\theta},t)-s^{\star}(X_t^{\theta},t) e t = s θ ( X t θ , t ) − s ⋆ ( X t θ , t ) (40) from (41) gives
d Z t = ( f ( X t θ , t ) − f ( X t ⋆ , t ) − g ( t ) 2 ( s ⋆ ( X t θ , t ) − s ⋆ ( X t ⋆ , t ) ) − g ( t ) 2 Γ t ( ∇ x log p ( y ∣ X t θ ) − ∇ x log p ( y ∣ X t ⋆ ) ) − g ( t ) 2 e t ) d t . dZ_t=\left(f(X_t^{\theta},t)-f(X_t^{\star},t)-g(t)^2(s^{\star}(X_t^{\theta},t)-s^{\star}(X_t^{\star},t))-g(t)^2\Gamma_t\left(\nabla_x\log p(y\mid X_t^{\theta})-\nabla_x\log p(y\mid X_t^{\star})\right)-g(t)^2e_t\right)dt. d Z t = ( f ( X t θ , t ) − f ( X t ⋆ , t ) − g ( t ) 2 ( s ⋆ ( X t θ , t ) − s ⋆ ( X t ⋆ , t )) − g ( t ) 2 Γ t ( ∇ x log p ( y ∣ X t θ ) − ∇ x log p ( y ∣ X t ⋆ ) ) − g ( t ) 2 e t ) d t . Applying the mean-value expansion to the first three terms along the segment joining X t ⋆ X_t^{\star} X t ⋆ X t θ X_t^{\theta} X t θ B ‾ t Γ \overline{B}_t^{\Gamma} B t Γ
d Z t = ( B ‾ t Γ Z t − g ( t ) 2 e t ) d t , Z 1 = 0. (45) dZ_t=\left(\overline{B}_t^{\Gamma}Z_t-g(t)^2e_t\right)dt,\qquad Z_1=0. \tag*{(45)} d Z t = ( B t Γ Z t − g ( t ) 2 e t ) d t , Z 1 = 0. (45) Equation (45) is linear in Z t Z_t Z t − g ( t ) 2 e t -g(t)^2e_t − g ( t ) 2 e t U 0 , t Γ U_{0,t}^{\Gamma} U 0 , t Γ
d Y t = B ‾ t Γ Y t d t . dY_t=\overline{B}_t^{\Gamma}Y_t\,dt. d Y t = B t Γ Y t d t . The variation-of-constants formula then gives
Z 0 = ∫ t 0 1 U 0 , t Γ g ( t ) 2 e t d t . (46) Z_0=\int_{t_0}^1U_{0,t}^{\Gamma}g(t)^2e_t\,dt. \tag*{(46)} Z 0 = ∫ t 0 1 U 0 , t Γ g ( t ) 2 e t d t . (46) Consequently,
∥ Z 0 ∥ H s 2 ≤ ∫ t 0 1 g ( t ) 2 ∥ U 0 , t Γ ∥ L ( H − q , H s ) ∥ e t ∥ H − q d t ≤ ∫ t 0 1 g ( t ) 2 M r , q , s , Γ ( t ) ∥ e t ∥ H − q d t . \|Z_0\|_{H^s}^2\leq\int_{t_0}^1g(t)^2\left\|U_{0,t}^{\Gamma}\right\|_{\mathcal{L}(H^{-q},H^s)}\|e_t\|_{H^{-q}}\,dt\leq\int_{t_0}^1g(t)^2M_{r,q,s,\Gamma}(t)\|e_t\|_{H^{-q}}\,dt. ∥ Z 0 ∥ H s 2 ≤ ∫ t 0 1 g ( t ) 2 U 0 , t Γ L ( H − q , H s ) ∥ e t ∥ H − q d t ≤ ∫ t 0 1 g ( t ) 2 M r , q , s , Γ ( t ) ∥ e t ∥ H − q d t . Taking expectations and applying Cauchy–Schwarz gives
E ∥ Z 0 ∥ H s 2 ≤ ( ∫ t 0 1 g ( t ) 2 M r , q , s , Γ ( t ) d t ) 2 sup t ∈ [ t 0 , 1 ] E ∥ e t ∥ H − q 2 . \mathbb{E}\|Z_0\|_{H^s}^2\leq\left(\int_{t_0}^1g(t)^2M_{r,q,s,\Gamma}(t)\,dt\right)^2\sup_{t\in[t_0,1]}\mathbb{E}\|e_t\|_{H^{-q}}^2. E ∥ Z 0 ∥ H s 2 ≤ ( ∫ t 0 1 g ( t ) 2 M r , q , s , Γ ( t ) d t ) 2 t ∈ [ t 0 , 1 ] sup E ∥ e t ∥ H − q 2 . Using (42) , we obtain Equation (44) .
Lemma B.3. Given Assumption 4.5, we have
E ∥ x ^ − X 0 θ ∥ H s 2 ≤ Φ d i s c ( Δ t ; r , s , Γ ) , (47) \mathbb{E}\|\widehat{x}-X_0^{\theta}\|_{H^s}^2\leq\Phi_{\mathrm{disc}}(\Delta t;r,s,\Gamma), \tag*{(47)} E ∥ x − X 0 θ ∥ H s 2 ≤ Φ disc ( Δ t ; r , s , Γ ) , (47) where
C d i s c ( r , s , Γ ) = 2 C l o c e 2 C s t a b ( 1 − t 0 ) C_{\mathrm{disc}}(r,s,\Gamma)=2C_{\mathrm{loc}}e^{2C_{\mathrm{stab}}(1-t_0)} C disc ( r , s , Γ ) = 2 C loc e 2 C stab ( 1 − t 0 ) and
Φ d i s c ( Δ t ; r , s , Γ ) = C d i s c ( r , s , Γ ) ( Δ t ) 2 κ . \Phi_{\mathrm{disc}}(\Delta t;r,s,\Gamma)=C_{\mathrm{disc}}(r,s,\Gamma)(\Delta t)^{2\kappa}. Φ disc ( Δ t ; r , s , Γ ) = C disc ( r , s , Γ ) ( Δ t ) 2 κ . Proof. Define the grid error E n = X ^ t n − X t n θ E_n=\widehat{X}_{t_n}-X^\theta_{t_n} E n = X t n − X t n θ (16) ,
E n = Ψ n Δ ( X ^ t n + 1 , ξ n + 1 ) − Ψ n Δ ( X t n + 1 θ , ξ n + 1 ) − R n . E_n=\Psi_n^{\Delta}(\widehat{X}_{t_{n+1}},\xi_{n+1})-\Psi_n^{\Delta}(X^\theta_{t_{n+1}},\xi_{n+1})-R_n. E n = Ψ n Δ ( X t n + 1 , ξ n + 1 ) − Ψ n Δ ( X t n + 1 θ , ξ n + 1 ) − R n . Therefore,
∥ E n ∥ H s 2 ≤ 2 ∥ Ψ n Δ ( X ^ t n + 1 , ξ n + 1 ) − Ψ n Δ ( X t n + 1 θ , ξ n + 1 ) ∥ H s 2 + 2 ∥ R n ∥ H s 2 . \lVert E_n\rVert_{\mathcal{H}^s}^2\leq2\lVert\Psi_n^{\Delta}(\widehat{X}_{t_{n+1}},\xi_{n+1})-\Psi_n^{\Delta}(X^\theta_{t_{n+1}},\xi_{n+1})\rVert_{\mathcal{H}^s}^2+2\lVert R_n\rVert_{\mathcal{H}^s}^2. ∥ E n ∥ H s 2 ≤ 2 ∥ Ψ n Δ ( X t n + 1 , ξ n + 1 ) − Ψ n Δ ( X t n + 1 θ , ξ n + 1 ) ∥ H s 2 + 2 ∥ R n ∥ H s 2 . We take conditional expectation with respect to F t n + 1 \mathcal{F}_{t_{n+1}} F t n + 1 (18) , then
E ∥ E n ∥ H s 2 ≤ 2 ( 1 + C s t a b Δ t ) E ∥ E n + 1 ∥ H s 2 + 2 C l o c ( Δ t ) 2 κ + 1 . \mathbb{E}\lVert E_n\rVert_{\mathcal{H}^s}^2\leq2(1+C_{\mathrm{stab}}\Delta t)\mathbb{E}\lVert E_{n+1}\rVert_{\mathcal{H}^s}^2+2C_{\mathrm{loc}}(\Delta t)^{2\kappa+1}. E ∥ E n ∥ H s 2 ≤ 2 ( 1 + C stab Δ t ) E ∥ E n + 1 ∥ H s 2 + 2 C loc ( Δ t ) 2 κ + 1 . Since E N = 0 E_N=0 E N = 0
E ∥ E 0 ∥ H s 2 ≤ 2 C l o c e 2 C s t a b ( 1 − t 0 ) ( Δ t ) 2 κ . \mathbb{E}\lVert E_0\rVert_{\mathcal{H}^s}^2\leq2C_{\mathrm{loc}}e^{2C_{\mathrm{stab}}(1-t_0)}(\Delta t)^{2\kappa}. E ∥ E 0 ∥ H s 2 ≤ 2 C loc e 2 C stab ( 1 − t 0 ) ( Δ t ) 2 κ . Proof. To prove Theorem 4.6, we split the reconstruction error and use inequality (31)
E ∥ x ^ − x † ∥ H s 2 ≤ 3 E ∥ X 0 ⋆ − x † ∥ H s 2 + 3 E ∥ X 0 θ − X 0 ⋆ ∥ H s 2 + 3 E ∥ x ^ − X 0 θ ∥ H s 2 . (48) \mathbb{E}\lVert\widehat{x}-x^\dagger\rVert_{\mathcal{H}^s}^2\leq3\mathbb{E}\lVert X_0^\star-x^\dagger\rVert_{\mathcal{H}^s}^2+3\mathbb{E}\lVert X_0^\theta-X_0^\star\rVert_{\mathcal{H}^s}^2+3\mathbb{E}\lVert\widehat{x}-X_0^\theta\rVert_{\mathcal{H}^s}^2. \tag*{(48)} E ∥ x − x † ∥ H s 2 ≤ 3 E ∥ X 0 ⋆ − x † ∥ H s 2 + 3 E ∥ X 0 θ − X 0 ⋆ ∥ H s 2 + 3 E ∥ x − X 0 θ ∥ H s 2 . (48) Using Lemma B.1, Lemma B.2, and Lemma B.3, and applying them to (48) proves Theorem 4.6.
Proof of Theorem 4.7
Proof. Fix R > 0 R>0 R > 0 X β ( R ) = { x † ∈ H β : ∥ x † ∥ H β ≤ R } \mathcal{X}_\beta(R)=\{x^\dagger\in\mathcal{H}^\beta:\lVert x^\dagger\rVert_{\mathcal{H}^\beta}\leq R\} X β ( R ) = { x † ∈ H β : ∥ x † ∥ H β ≤ R } S q ( δ θ ) \mathcal{S}_q(\delta_\theta) S q ( δ θ ) s < s ⋆ ( r , β , q , Γ ) s<s_\star(r,\beta,q,\Gamma) s < s ⋆ ( r , β , q , Γ ) s s s A ( r , q , s , Γ ) < ∞ \mathcal{A}(r,q,s,\Gamma)<\infty A ( r , q , s , Γ ) < ∞ Φ s c o r e ( δ θ ; r , q , s , Γ ) = A ( r , q , s , Γ ) 2 δ θ 2 \Phi_{\mathrm{score}}(\delta_\theta;r,q,s,\Gamma)=\mathcal{A}(r,q,s,\Gamma)^2\delta_\theta^2 Φ score ( δ θ ; r , q , s , Γ ) = A ( r , q , s , Γ ) 2 δ θ 2
Using Theorem 4.6, for every x † ∈ X β ( R ) x^\dagger\in\mathcal{X}_\beta(R) x † ∈ X β ( R ) s θ ∈ S q ( δ θ ) s_\theta\in\mathcal{S}_q(\delta_\theta) s θ ∈ S q ( δ θ )
E ∥ x ^ − x † ∥ H s 2 ≤ 3 C n o i s e σ 2 η ( r , β , s ) + 3 A ( r , q , s , Γ ) 2 δ θ 2 + 3 C d i s c ( r , s , Γ ) ( Δ t ) 2 κ . \mathbb{E}\lVert\widehat{x}-x^\dagger\rVert_{\mathcal{H}^s}^2\leq3C_{\mathrm{noise}}\sigma^{2\eta(r,\beta,s)}+3\mathcal{A}(r,q,s,\Gamma)^2\delta_\theta^2+3C_{\mathrm{disc}}(r,s,\Gamma)(\Delta t)^{2\kappa}. E ∥ x − x † ∥ H s 2 ≤ 3 C noise σ 2 η ( r , β , s ) + 3 A ( r , q , s , Γ ) 2 δ θ 2 + 3 C disc ( r , s , Γ ) ( Δ t ) 2 κ . By Assumption 4.4 and the uniformity of C l o c C_{\mathrm{loc}} C loc C s t a b C_{\mathrm{stab}} C stab S q ( δ θ ) \mathcal{S}_q(\delta_\theta) S q ( δ θ ) C n o i s e C_{\mathrm{noise}} C noise C d i s c ( r , s , Γ ) C_{\mathrm{disc}}(r,s,\Gamma) C disc ( r , s , Γ ) X β ( R ) × S q ( δ θ ) \mathcal{X}_\beta(R)\times\mathcal{S}_q(\delta_\theta) X β ( R ) × S q ( δ θ )
C R ≥ 3 max { C n o i s e , 1 , C d i s c ( r , s , Γ ) } , C_R\geq3\max\{C_{\mathrm{noise}},1,C_{\mathrm{disc}}(r,s,\Gamma)\}, C R ≥ 3 max { C noise , 1 , C disc ( r , s , Γ )} , with C R C_R C R ( R , r , β , q , s , Γ , Σ ϵ ) (R,r,\beta,q,s,\Gamma,\Sigma_\epsilon) ( R , r , β , q , s , Γ , Σ ϵ )
E ∥ x ^ − x † ∥ H s 2 ≤ C R ( σ 2 η ( r , β , s ) + A ( r , q , s , Γ ) 2 δ θ 2 + ( Δ t ) 2 κ ) , \mathbb{E}\lVert\widehat{x}-x^\dagger\rVert_{\mathcal{H}^s}^2\leq C_R\left(\sigma^{2\eta(r,\beta,s)}+\mathcal{A}(r,q,s,\Gamma)^2\delta_\theta^2+(\Delta t)^{2\kappa}\right), E ∥ x − x † ∥ H s 2 ≤ C R ( σ 2 η ( r , β , s ) + A ( r , q , s , Γ ) 2 δ θ 2 + ( Δ t ) 2 κ ) , which proves the uniform recovery estimate over X β ( R ) × S q ( δ θ ) \mathcal{X}_\beta(R)\times\mathcal{S}_q(\delta_\theta) X β ( R ) × S q ( δ θ )
Proof of Theorem 4.8
Proof. Fix R > 0 R>0 R > 0 x n † = 0 ∈ X β ( R ) x_n^\dagger=0\in\mathcal{X}_\beta(R) x n † = 0 ∈ X β ( R ) c R = 1 c_R=1 c R = 1 n n n e n ∈ C ( [ t 0 , 1 ] ; H s ) e_n\in C([t_0,1];\mathcal{H}^s) e n ∈ C ([ t 0 , 1 ] ; H s ) sup t ∈ [ t 0 , 1 ] ∥ e n ( t ) ∥ H − q ≤ δ n \sup_{t\in[t_0,1]}\lVert e_n(t)\rVert_{\mathcal{H}^{-q}}\leq\delta_n sup t ∈ [ t 0 , 1 ] ∥ e n ( t ) ∥ H − q ≤ δ n ∥ ∫ t 0 1 U 0 , t g ( t ) 2 e n ( t ) d t ∥ H s ≥ 3 \left\lVert\int_{t_0}^1\mathcal{U}_{0,t}g(t)^2e_n(t)\,\mathrm{d}t\right\rVert_{\mathcal{H}^s}\geq3 ∫ t 0 1 U 0 , t g ( t ) 2 e n ( t ) d t H s ≥ 3 s θ , n ( x , t ) = s ⋆ ( x , t ) + e n ( t ) s_{\theta,n}(x,t)=s^\star(x,t)+e_n(t) s θ , n ( x , t ) = s ⋆ ( x , t ) + e n ( t ) s θ , n ( x , t ) − ∇ x log p t ( x ) = e n ( t ) s_{\theta,n}(x,t)-\nabla_x\log p_t(x)=e_n(t) s θ , n ( x , t ) − ∇ x log p t ( x ) = e n ( t ) x x x s θ , n ∈ S q ( δ n ) s_{\theta,n}\in\mathcal{S}_q(\delta_n) s θ , n ∈ S q ( δ n ) b θ , n = b ⋆ − g 2 e n b^{\theta,n}=b^\star-g^2e_n b θ , n = b ⋆ − g 2 e n e n e_n e n H s \mathcal{H}^s H s X t ⋆ , n X_t^{\star,n} X t ⋆ , n X t θ , n X_t^{\theta,n} X t θ , n Z t n = X t θ , n − X t ⋆ , n Z_t^n=X_t^{\theta,n}-X_t^{\star,n} Z t n = X t θ , n − X t ⋆ , n b ⋆ b^\star b ⋆ x x x Z n Z^n Z n d Z t n = ( B t Γ Z t n − g ( t ) 2 e n ( t ) ) d t \mathrm{d}Z_t^n=(B_t^\Gamma Z_t^n-g(t)^2e_n(t))\,\mathrm{d}t d Z t n = ( B t Γ Z t n − g ( t ) 2 e n ( t )) d t Z 1 n = 0 Z_1^n=0 Z 1 n = 0 B t Γ B_t^\Gamma B t Γ Z 0 n = ∫ t 0 1 U 0 , t g ( t ) 2 e n ( t ) d t Z_0^n=\int_{t_0}^1\mathcal{U}_{0,t}g(t)^2e_n(t)\,\mathrm{d}t Z 0 n = ∫ t 0 1 U 0 , t g ( t ) 2 e n ( t ) d t ∥ Z 0 n ∥ H s ≥ 3 \lVert Z_0^n\rVert_{\mathcal{H}^s}\geq3 ∥ Z 0 n ∥ H s ≥ 3 x n † = 0 x_n^\dagger=0 x n † = 0 E ∥ X 0 ⋆ , n − x n † ∥ H s 2 ≤ C n o i s e σ n 2 η ( r , β , s ) → 0 \mathbb{E}\lVert X_0^{\star,n}-x_n^\dagger\rVert_{\mathcal{H}^s}^2\leq C_{\mathrm{noise}}\sigma^{2\eta(r,\beta,s)}_n\to0 E ∥ X 0 ⋆ , n − x n † ∥ H s 2 ≤ C noise σ n 2 η ( r , β , s ) → 0 η ( r , β , s ) > 0 \eta(r,\beta,s)>0 η ( r , β , s ) > 0 s θ , n s_{\theta,n} s θ , n Δ t n ↓ 0 \Delta t_n\downarrow0 Δ t n ↓ 0 E ∥ x ^ n − X 0 θ , n ∥ H s 2 ≤ 1 / n \mathbb{E}\lVert\widehat{x}_n-X_0^{\theta,n}\rVert_{\mathcal{H}^s}^2\leq1/n E ∥ x n − X 0 θ , n ∥ H s 2 ≤ 1/ n L 2 ( Ω ; H s ) L^2(\Omega;\mathcal{H}^s) L 2 ( Ω ; H s )
( E ∥ x ^ n − x n † ∥ H s 2 ) 1 / 2 ≥ ∥ Z 0 n ∥ H s − ( C n o i s e σ n 2 η ( r , β , s ) ) 1 / 2 − n − 1 / 2 , (49) \left(\mathbb{E}\lVert\widehat{x}_n-x_n^\dagger\rVert_{\mathcal{H}^s}^2\right)^{1/2}\geq\lVert Z_0^n\rVert_{\mathcal{H}^s}-\left(C_{\mathrm{noise}}\sigma^{2\eta(r,\beta,s)}_n\right)^{1/2}-n^{-1/2}, \tag*{(49)} ( E ∥ x n − x n † ∥ H s 2 ) 1/2 ≥ ∥ Z 0 n ∥ H s − ( C noise σ n 2 η ( r , β , s ) ) 1/2 − n − 1/2 , (49) hence
lim inf n → ∞ E ∥ x ^ n − x n † ∥ H s 2 ≥ 9 ≥ c R , (50) \liminf_{n\to\infty}\mathbb{E}\lVert\hat{x}_n-x_n^\dagger\rVert_{\mathcal{H}^s}^2 \ge9 \ge c_R, \tag*{(50)} n → ∞ lim inf E ∥ x ^ n − x n † ∥ H s 2 ≥ 9 ≥ c R , (50) which proves (26) .
If a uniform estimate
E ∥ x ^ − x † ∥ H s 2 ≤ C R ( σ 2 η ( r , β , s ) + δ θ 2 + ( Δ t ) 2 κ ) \mathbb{E}\lVert\hat{x}-x^\dagger\rVert_{\mathcal{H}^s}^2 \le C_R\left(\sigma^{2\eta(r,\beta,s)}+\delta_\theta^2+(\Delta t)^{2\kappa}\right) E ∥ x ^ − x † ∥ H s 2 ≤ C R ( σ 2 η ( r , β , s ) + δ θ 2 + ( Δ t ) 2 κ ) holds for all x † ∈ X β ( R ) x^\dagger\in X_\beta(R) x † ∈ X β ( R ) s θ ∈ S q ( δ θ ) s_\theta\in S_q(\delta_\theta) s θ ∈ S q ( δ θ )
E ∥ x ^ n − x n † ∥ H s 2 ≤ C R ( σ n 2 η ( r , β , s ) + δ n 2 + ( Δ t n ) 2 κ ) ⟶ 0. \mathbb{E}\lVert\hat{x}_n-x_n^\dagger\rVert_{\mathcal{H}^s}^2 \le C_R\left(\sigma_n^{2\eta(r,\beta,s)}+\delta_n^2+(\Delta t_n)^{2\kappa}\right)\longrightarrow0. E ∥ x ^ n − x n † ∥ H s 2 ≤ C R ( σ n 2 η ( r , β , s ) + δ n 2 + ( Δ t n ) 2 κ ) ⟶ 0. This contradicts the constructed bound (49) and therefore there is no such uniform estimate can hold once s > s ⋆ ( r , β , q , Γ ) s>s_\star(r,\beta,q,\Gamma) s > s ⋆ ( r , β , q , Γ )
Proof of Proposition 4.9
Proof. Fix s ∈ I s\in\mathcal{I} s ∈ I (28) by the nonnegative weight g ( t ) 2 g(t)^2 g ( t ) 2 [ t 0 , 1 ] [t_0,1] [ t 0 , 1 ] (15) , the score-propagation functional, we have
A ( r , q , s , Γ o p ) = ∫ t 0 1 g ( t ) 2 M r , q , s , Γ o p ( t ) d t ≤ ∫ t 0 1 g ( t ) 2 M r , q , s , Γ i s o ( t ) d t = A ( r , q , s , Γ i s o ) . \mathcal{A}(r,q,s,\Gamma^{\mathrm{op}})=\int_{t_0}^{1}g(t)^2M_{r,q,s,\Gamma^{\mathrm{op}}}(t)\,\mathrm{d}t\le\int_{t_0}^{1}g(t)^2M_{r,q,s,\Gamma^{\mathrm{iso}}}(t)\,\mathrm{d}t=\mathcal{A}(r,q,s,\Gamma^{\mathrm{iso}}). A ( r , q , s , Γ op ) = ∫ t 0 1 g ( t ) 2 M r , q , s , Γ op ( t ) d t ≤ ∫ t 0 1 g ( t ) 2 M r , q , s , Γ iso ( t ) d t = A ( r , q , s , Γ iso ) . The integrands are nonnegative, and the two integrals are therefore well defined in [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ] (29) thus holds without any finiteness assumption on A ( r , q , s , Γ i s o ) \mathcal{A}(r,q,s,\Gamma^{\mathrm{iso}}) A ( r , q , s , Γ iso ) (21) , the score Φ s c o r e ( δ θ ; r , q , s , Γ ) = A ( r , q , s , Γ ) 2 δ θ 2 \Phi_{\mathrm{score}}(\delta_\theta;r,q,s,\Gamma)=\mathcal{A}(r,q,s,\Gamma)^2\delta_\theta^2 Φ score ( δ θ ; r , q , s , Γ ) = A ( r , q , s , Γ ) 2 δ θ 2 A \mathcal{A} A [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) (30) is a direct consequence of (29) .
The strict inequalities of Proposition 4.9 require A ( r , q , s , Γ i s o ) < ∞ \mathcal{A}(r,q,s,\Gamma^{\mathrm{iso}})<\infty A ( r , q , s , Γ iso ) < ∞ s ∈ I s\in\mathcal{I} s ∈ I E ⊂ [ t 0 , 1 ] E\subset[t_0,1] E ⊂ [ t 0 , 1 ] g ( t ) 2 d t g(t)^2\,\mathrm{d}t g ( t ) 2 d t Γ o p \Gamma^{\mathrm{op}} Γ op Γ i s o \Gamma^{\mathrm{iso}} Γ iso A ( r , q , s , Γ o p ) ≤ A ( r , q , s , Γ i s o ) < ∞ \mathcal{A}(r,q,s,\Gamma^{\mathrm{op}})\le\mathcal{A}(r,q,s,\Gamma^{\mathrm{iso}})<\infty A ( r , q , s , Γ op ) ≤ A ( r , q , s , Γ iso ) < ∞ g ( t ) 2 d t g(t)^2\,\mathrm{d}t g ( t ) 2 d t
A ( r , q , s , Γ i s o ) − A ( r , q , s , Γ o p ) = ∫ t 0 1 g ( t ) 2 ( M r , q , s , Γ i s o ( t ) − M r , q , s , Γ o p ( t ) ) d t ≥ ∫ E g ( t ) 2 ( M r , q , s , Γ i s o ( t ) − M r , q , s , Γ o p ( t ) ) d t > 0. \begin{aligned}
\mathcal{A}(r,q,s,\Gamma^{\mathrm{iso}})-\mathcal{A}(r,q,s,\Gamma^{\mathrm{op}})
=\int_{t_0}^{1}g(t)^2\left(M_{r,q,s,\Gamma^{\mathrm{iso}}}(t)-M_{r,q,s,\Gamma^{\mathrm{op}}}(t)\right)\,\mathrm{d}t \\
&\ge\int_E g(t)^2\left(M_{r,q,s,\Gamma^{\mathrm{iso}}}(t)-M_{r,q,s,\Gamma^{\mathrm{op}}}(t)\right)\,\mathrm{d}t>0.
\end{aligned} A ( r , q , s , Γ iso ) − A ( r , q , s , Γ op ) = ∫ t 0 1 g ( t ) 2 ( M r , q , s , Γ iso ( t ) − M r , q , s , Γ op ( t ) ) d t ≥ ∫ E g ( t ) 2 ( M r , q , s , Γ iso ( t ) − M r , q , s , Γ op ( t ) ) d t > 0. The nonnegativity of the integrand on [ t 0 , 1 ] [t_0,1] [ t 0 , 1 ] E E E 0 ≤ A ( r , q , s , Γ o p ) < A ( r , q , s , Γ i s o ) < ∞ 0\le\mathcal{A}(r,q,s,\Gamma^{\mathrm{op}})<\mathcal{A}(r,q,s,\Gamma^{\mathrm{iso}})<\infty 0 ≤ A ( r , q , s , Γ op ) < A ( r , q , s , Γ iso ) < ∞ δ θ > 0 \delta_\theta>0 δ θ > 0 u ↦ u 2 δ θ 2 u\mapsto u^2\delta_\theta^2 u ↦ u 2 δ θ 2 [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) (30) . □
