ELPAC: Endpoint-Anchored Latent Progression with Stage-Varying Multimodal Coordination

Introduction

Aging-related neurodegeneration is observed indirectly: pathology accumulates for years before symptoms emerge, and two patients indistinguishable on a cognitive test may differ markedly in pathology, structural change, and future risk [1, 2]. For each subject, three latent quantities are of central scientific interest: the progression subtype that best describes them, including preserved aging; their continuous stage along that subtype trajectory; and the stage-specific structure of how measurements across modalities covary during progression. For example, consider two subjects with the same Mini-Mental State Examination (MMSE) score of 26, one 30% along the early-onset AD (EOAD) amyloid+tau trajectory (stage $S_i \approx0.3$), the other far along healthy aging (stage $S_i \approx0.9$), which is advanced healthy aging rather than advanced disease.

None of these quantities is directly observed. Aging studies now routinely collect brain imaging, plasma assays, and cognitive measurements [3, 4], yet most participants contribute only a single, partially observed cross-sectional snapshot, without repeated measurements or follow-up diagnoses that would reveal where they lie along progression. This creates two coupled inference problems. First, supervision is asymmetric: endpoint groups are well characterized, while early and intermediate subjects, the part of the cohort whose stage and risk are most uncertain, carry the noisiest labels. A model that grows trajectories forward from this uncertain origin therefore rests on the cohort’s weakest signal, whereas one anchored at the endpoints rests on its strongest. Second, dependence is non-stationary: the coupling between measurements changes along progression, so marginal correlations mix subjects from different latent stages and subtypes and can wash out biologically meaningful coordination. The target is therefore not subtype, stage, or covariance separately, but a joint posterior over subtype assignment, continuous stage, and stage-varying multimodal dependence, with uncertainty propagated across all of them.

Prior work has advanced each ingredient of this problem, but mostly in isolation. Disease progression models [5, 6, 7, 8, 9, 10, 11, 12, 13, 14] recover event orderings or subtype-specific mean trajectories, targeting the expected trajectory, not the multivariate dependence around it. Condition-varying covariance and dynamic factor models [15, 16, 17, 18] learn covariance as a function of an observed condition; in our setting that condition is latent. Multimodal biomedical architectures [19, 20, 21, 22, 23] fuse heterogeneous data and absorb missingness, but read it through a predictive lens rather than a progression-resolved one. Identifiable latent variable learning [24, 25, 26, 27, 28, 29] shows that auxiliary structure resolves rotational and permutation ambiguities in factor models, but typically requires the auxiliary variable to be observed, not inferred. None learns the latent disease axis and the dependence around it as mutually identifying objects.

To address the gap, we introduce ELPAC (Endpoint-anchored Latent Progression And Coordination), a Bayesian framework with two interlocking constructions that recover the latent progression and its multimodal coordination from cross-sectional snapshots (Figure 1). We integrate subtype trajectories backward from clinically reliable endpoint groups towards a shared healthy reference along per-modality velocity components that encode monotone progression, modality-specific direction, and optional sequential ordering priors. These biological constraints supply the inductive bias absent from cross-sectional snapshots, while endpoint anchoring calibrates the stage axis from the cohort’s best supervised subjects. Around that axis, residual multimodal variation decomposes into coordination axes, each a sparse coalition of regions and modalities whose covariation strengthens or attenuates with stage; the loadings expose which programs are operative in each subtype, the activations expose when they emerge. The two constructions are fitted jointly: stage-resolved variation supplies the auxiliary structure for the coordination decomposition, and the resolved coordination feeds back through the multivariate likelihood to sharpen the posterior over subtype and stage. We use a scalable amortized variational scheme paired with structured low-rank updates over the global parameters, enabling joint posterior inference at cohort scale with single-pass deployment on new subjects.

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These design choices yield three contributions:

• We incorporate asymmetric endpoint supervision as an inductive bias for subject-level continuous stage inference, with subtype trajectories anchored at clinically reliable groups and integrated backward through biologically structured velocity components.

• We establish conditional identifiability of the stage-varying multimodal coordination via the inferred stage axis as auxiliary variation, yielding interpretable sparse coordination axes across and within multimodal measurements.

• We validate ELPAC on synthetic data and on a harmonized ADNI + NACC/SCAN cohort ($N = 7{,}501$; NACC March 2026 data freeze), demonstrating accurate latent-structure recovery, well-calibrated prediction intervals, and biologically interpretable coordination programs that differ by subtype.

Method

Problem formulation

We observe a cross-sectional cohort $\mathcal{D} = \{(Y_i, X_i, Y_i^{\mathrm{cog}})\}_{i=1}^{n}$, where $Y_i \in\mathbb{R}^{R}$ is a partially observed multimodal biomarker vector with each coordinate $r \in\{1,\ldots,R\}$ assigned to a modality $m(r) \in\mathcal{M}$, $X_i \in\mathbb{R}^{P}$ collects demographic and genetic covariates, and $Y_i^{\mathrm{cog}} \in\mathbb{R}$ is an auxiliary cognitive outcome that may be missing. Each subject carries a latent subtype $c_i \in\{1, \ldots, C\}$, with subtypes defined by clinically distinct endpoint phenotypes, and a latent progression stage $S_i \in(0, 1)$. The cohort comprises three strata: endpoint anchors $\mathcal{A}_c$ with clinically confirmed subtype $c$ (near $s \to1$); a healthy reference set $\mathcal{H}$ near $s \approx0$; and a general unsupervised population $\mathcal{U}$ with unknown $(c_i, S_i)$. The objects of inference are the subject-level latents $(c_i, S_i)$, the subtype-specific mean trajectories $\theta_c(s) \in\mathbb{R}^R$, and the stage-varying within-stage covariance $\Sigma_c(s) \in\mathbb{R}^{R \times R}$. Boundary supervision from $\mathcal{A}_c$ and $\mathcal{H}$ grounds the latent stage, and the resulting stage axis serves as the auxiliary variation under which $\Sigma_c(s)$ is identifiable. Table 1 summarizes the notation.

Symbol	Meaning	Symbol	Meaning
$c_i$; $S_i \in(0, 1)$	subtype; progression stage	$C$; $R$; $m(r) \in\mathcal{M}$	no. subtypes, biomarkers; modality of $r$
$Y_i$; $O_i$	biomarker vector; observed entries	$X_i$; $Y_i^{\mathrm{cog}}$; $\chi_c(s)$	covariates; cognition; its stage trend
$\mathcal{A}_c$; $\mathcal{H}$; $\mathcal{U}$	anchors; healthy reference; unlabeled	$s_A$; $s_H$	anchor; healthy-reference stage location
$\theta_c(s)$; $\Sigma_c(s)$	mean trajectory; within-stage covariance	$\mu_c$; $\Theta_0$	subtype endpoint; shared healthy origin
$\Theta_{c,j}$; $V_{c,j}$; $J$	shell state; shell velocity; no. shells	$\operatorname{sgn}_m$; $\tilde{t}_{c,m}$	modality direction; modality onset center
$\delta_{c,r}$; $t_{c,r}$; $\omega_{c,r}$	inclusion; onset timing; onset sharpness	$F_{c,r}$; $B_q$; $Q$	signed velocity magnitude; spline basis, size
$E_i$; $z_i$; $\eta_i$	residual; factor scores; idiosyncratic noise	$A_c$; $\nu_c(s)$; $D_c$	loadings; activations; noise covariance
$K$; $\psi_p$; $P$	no. factors; activation B-spline basis, size	$\Psi$; $q_{\phi,\psi}$	global parameters; variational family

Observation model

Let $M_{ir} \in\{0,1\}$ indicate whether biomarker $r$ is observed for subject $i$, with $O_i = \{r : M_{ir} = 1\}$, and let $O_i^{\mathrm{cog}} \in\{0,1\}$ indicate whether the auxiliary cognitive outcome $Y_i^{\mathrm{cog}}$ is observed. The observed biomarker likelihood is

where $\theta_c(s) \in\mathbb{R}^R$ and $\Sigma_c(s) \in\mathbb{R}^{R \times R}$ denote the subtype-$c$ mean trajectory and within-stage covariance. The subscript $O_i$ selects the observed entries and corresponding submatrix, so unobserved biomarker coordinates are integrated out under the multivariate Gaussian model. When $O_i^{\mathrm{cog}}=1$, the cognitive outcome enters as

where $\gamma_0 \sim\mathcal{N}(0,1)$, $\gamma_X \sim\mathcal{N}(0,I_p)$, $\log\sigma_{\mathrm{cog}} \sim\mathcal{N}(0,1)$, and $\chi_c(s)=\boldsymbol{b}^{I}(s)^\top\xi_c$ is a monotone I-spline with non-negative coefficients $\xi_{c,j}\sim\operatorname{Half\text{-}Normal}(1)$ that encodes the monotone progressionconditioned trend of $Y_i^{\operatorname{cog}}$. The corresponding factor is omitted from the joint likelihood when $O_i^{\operatorname{cog}}=0$.

Construction I: endpoint-anchored progression module

The subtype mean trajectory $\theta_c(s)$ is defined as a reverse-time integral on $[0,1]$ between two latent boundary states. Partition the interval by shells $0=\tau_0<\cdots<\tau_J=1$ with width $\Delta=1/J$, and let $\theta_c(s)=\mathcal{I}(\{\boldsymbol{\Theta}_{c,j},V_{c,j}\}_{j=0}^{J};s)$ denote the cubic Hermite interpolant of shell states $\boldsymbol{\Theta}_{c,j}\in\mathbb{R}^{R}$ and shell-endpoint velocities $V_{c,j}\in\mathbb{R}^{R}$ (explicit form in equation (25)) satisfying

This right-endpoint update interprets $V_{c,j}$ as the arrival velocity at shell boundary $\tau_j$, yielding a triangular reverse-time construction in which $\boldsymbol{\Theta}_{c,j}$ depends on $(\boldsymbol{\Theta}_{c,J},V_{c,j+1:J})$ alone. We assign weakly informative Gaussian hyperpriors to the subtype endpoint $\mu_c$ and the shared healthy-reference origin $\boldsymbol{\Theta}_0$,

The recursion equation (3) fixes $\boldsymbol{\Theta}_{c,0}$ deterministically as the integrated origin $\boldsymbol{\Theta}_{c,0}=\boldsymbol{\Theta}_{c,J}-\Delta\sum_{j=1}^{J}V_{c,j}$, and we attach a Gaussian coupling potential that draws each subtype’s integrated origin toward the shared healthy-reference origin,

where $\sigma_0$ controls the strength of the coupling: all subtypes depart from a common healthy state, while each subtype’s integrated origin may deviate from it. equation (5) is a coupling factor on the joint over $(\boldsymbol{\Theta}_0,\{\boldsymbol{\Theta}_{c,J},V_{c,1:J}\})$, not a conditional density on the deterministic $\boldsymbol{\Theta}_{c,0}(\Psi)$; combined with the Gaussian boundary priors of equation (4) it yields a normalized joint prior since $\psi_{\mathrm{soft}}\leq1$ pointwise. The posteriors of $\mu_c$ and $\boldsymbol{\Theta}_0$ are informed by the anchor and healthy-reference cohorts $\mathcal{A}_c$ and $\mathcal{H}$ through the observation likelihood equation (1) evaluated at the boundary stages introduced below. The recursion equation (3) together with equation (5) then determines $\theta_c(s)$ from the velocity field $V_{c,1:J}$.

The velocity admits a biologically structured decomposition,

in which $\delta_{c,r}\in\{0,1\}$ encodes biomarker inclusion in subtype $c$, $(t_{c,r},\omega_{c,r})$ govern onset timing and sharpness, $\{B_q\}$ is a $Q$-dimensional B-spline basis with coefficients $\{\beta_{c,r,q}\}$, and the modality sign $\operatorname{sgn}_{m(r)}\in\{+1,-1\}$ is a fixed modality-level constant encoding the direction of evolution (e.g., monotone accumulation versus monotone atrophy). The associated priors are

The modality-monotone direction encoded by $\operatorname{sgn}_{m(r)}$ is an inductive bias common to event-based progression models [5, 6]. Domain knowledge enters through the modality-level onset centers $\widetilde{t}_{c,m}$: when biomarker ordering or late-onset behavior is supported by prior evidence, the diffuse $\mathcal{N}(0.5,1)$ on $\widetilde{t}_{c,m}$ is replaced by an informative location or ordering prior.

Subtype assignment is modeled as a covariate-modulated categorical, and the subtype-conditional stage as logit-normal. For unsupervised subjects $i\in\mathcal{U}$,

with $\alpha\sim\mathcal{N}(0,I_C)$, $B\sim\mathcal{N}(0,I_{C\times p})$, $b_c\sim\mathcal{N}(0,I_p)$, and $\sigma_c\sim\operatorname{Half\text{-}Cauchy}(0,1)$. The diffuse prior on $b_c$ is replaced by an informative location prior on the relevant covariate when subtype-specific covariate effects (e.g., age-of-onset) are supported by external evidence. For anchored subjects $i\in\mathcal{A}_c$, the subtype is fixed at $c$ and the stage is concentrated near the endpoint,

and for healthy-reference subjects $i \in\mathcal{H}$, the subtype is unconstrained and the stage is concentrated near the origin,

Here $s_A, s_H \in(0,1)$ are hyperparameters localizing anchors near the endpoint and healthy subjects near the origin.

Construction II: stage-varying coordination module

With $(c,s)$ latent, the iterated law of total covariance separates raw pairwise correlation into a within-stage coordination term and a between-$(c,s)$ mean effect,

The second term is captured by the mean trajectory $\theta_c(s)$ of Construction I. We model the first by endowing the residual $E_i \equiv Y_i-\theta_{c_i}(S_i)$ with a stage-varying latent factor representation,

where $z_i\in\mathbb{R}^K$ are latent factor scores and $\eta_i$ is idiosyncratic noise. Each column $A_{c,\cdot,k}$ defines a coordination axis, encoding which biomarkers and modalities co-vary, and the activation $\nu_{c,k}(s)$ governs when along progression that axis strengthens or vanishes; a column may concentrate within a single modality or span multiple modalities, capturing within-modality and across-modality coordination respectively. The induced within-stage covariance has the rank-$K$-plus-diagonal form

The loadings matrix $A_c\in\mathbb{R}^{R\times K}$ is constrained lower triangular with positive diagonal, fixing the rotational ambiguity inherent to factor models [27, 30]. A multiplicative-gamma cascade [28, 29] adaptively controls the effective number of factors: $\kappa_{c,1}\sim\operatorname{Gamma}(2,1)$, $\kappa_{c,h\geq2}\sim\operatorname{Gamma}(a_2,1)$ with $a_2>1$, $\Lambda_{c,k}=\prod_{h\leq k}\kappa_{c,h}$, and, conditional on $\Lambda_{c,k}$, $A_{c,r,k}\sim\mathcal{N}(0,\Lambda_{c,k}^{-1})$ for $r>k$, $A_{c,k,k}\sim\operatorname{Half\text{-}Normal}(\Lambda_{c,k}^{-1/2})$. Since $\mathbb{E}[\Lambda_{c,k}]$ grows with $k$, later columns are increasingly shrunk toward zero. The activation $\nu_{c,k}(s)=\exp\left(\sum_p w_{c,k,p}\psi_p(s)\right)$ is expanded on a $P$-dimensional B-spline basis $\{\psi_p\}$. A constant shift in $\log\nu_{c,k}$ may be absorbed into $A_{c,\cdot,k}$, so we resolve the resulting scale ambiguity by supporting $w_{c,k}$ on the centered subspace $\mathcal{P}_0=\{w\in\mathbb{R}^P:1^{\mathsf{T}}w=0\}$, via the projection $\Pi_0=I_P-\frac{1}{P}11^{\mathsf{T}}$, with the singular Gaussian prior

where $K_B$ denotes the second-difference smoothness penalty matrix, so coordination strengthens or fades gradually along progression rather than jumping between adjacent stages, and $\epsilon_B$ is a small ridge; $\tau_k$ scales how strongly factor $k$'s activation varies with stage. The prior covariance has rank $P-1$ and is supported on $\mathcal{P}_0$, so the centering identity $1^{\mathsf{T}}w_{c,k}=0$ holds almost surely. $D_c$ is diagonal, so all within-stage cross-biomarker dependencies pass through the coordination axes; the idiosyncratic noise variances follow a log-Gaussian prior, $\log d_{c,r}\sim\mathcal{N}(m_D,\sigma_D^2)$, with $D_c=\operatorname{diag}(d_{c,r}+d_{\min})$ and a small positive floor $d_{\min}$ that keeps every noise variance bounded away from zero. Downstream summaries, including stage-dependent pairwise correlations $\rho_c(r,r';s)$ and Fisher modality informativeness $W_{c,m}^{\mathrm{FI}}(s)$, are deterministic functionals of $(A_c,\nu_c(s),D_c)$.

Conditional identifiability

The latent stage recovered through Construction I supplies the auxiliary variation under which the Construction II decomposition $(A_c,\nu_c(\cdot),D_c)$ is identifiable, and the resolved $\Sigma_c(s)$ feeds back through the likelihood equation (1) to sharpen the $(c_i,S_i)$ posterior; the two blocks are fitted jointly.

regularity conditions of section C.2), suppose the within-stage covariance is identified at two stages $s_1 \ne s_2$ whose activation ratios are pairwise distinct,

Then the loadings $\mathbf{A}$, the activation function $\nu(\cdot)$, and the idiosyncratic noise $\mathbf{D}$ are uniquely determined.

Proof sketch. (i) At each fixed $s$, the Anderson–Rubin row-deletion condition with lower-triangular $\mathbf{A}$ and $R \ge2K+1$ identifies $\mathbf{D}$ and $\mathbf{A}\operatorname{diag}(\nu(s))\mathbf{A}^{\mathsf{T}}$ [27, 30]. (ii) Setting $\mathbf{\Pi}_{\bullet}:=\mathbf{\Sigma}(s_{\bullet})-\mathbf{D}$, the generalized eigenproblem $\mathbf{\Pi}_2v=\lambda\mathbf{\Pi}_1v$ has eigenvalues $\nu_k(s_2)/\nu_k(s_1)$, distinct by equation (16); primal columns recover as $\widetilde{\mathbf{a}}_k=\mathbf{\Pi}_1v_k/\lVert\mathbf{\Pi}_1v_k\rVert_2$, and the lower-triangular positive-diagonal normalization pins scale and sign. (iii) The centering identity $\mathbf{1}^{\mathsf{T}}\mathbf{w}_k=0$ (with partition-of-unity $\{\psi_p\}$) pins $\nu(s_1),\nu(s_2)$ pointwise. (iv) Spline-design coverage extends $\nu(\cdot)$ to all of $[0,1]$; boundary anchoring equations (10) and (11) together with the trajectory-feature regularity rule out non-identity boundary-fixing warps. Full proof, warp invariance, and full-model identification in section C.

Inference

Let $\mathbf{\Psi}$ denote the global parameters of equations (1) to (11) and $\mathcal{Z}=\{(c_i,S_i)\}_{i=1}^{n}$ the subject-level latents. We approximate $p(\mathbf{\Psi},\mathcal{Z}\mid\mathcal{D})$ by a structured variational family that pairs a low-rank multivariate Gaussian on $\mathbf{\Psi}$ with $C$ subtype-conditional Gaussians on $\operatorname{logit}(S_i)$ per subject,

where $\mathbf{W}_{\phi}\in\mathbb{R}^{|\Psi|\times r_g}$ is the low-rank factor of the global Gaussian (rank $r_g$), and $(\boldsymbol{\mu}_{\psi}^{(c)},\boldsymbol{\tau}_{\psi}^{(c)})$ are produced by an amortized recognition network with weights $\psi$ shared across subjects [31, 32]. Marginalizing $c_i$ exactly inside the evidence lower bound gives the collapsed bound

with per-candidate-subtype data factor $\ell_i^{(c)}=p(\mathbf{Y}_i,\mathbf{o}_i\mid c,S_i^{(c)},\mathbf{\Psi})p(\mathbf{Y}_i^{\mathrm{cog}}\mid c,S_i^{(c)},\mathbf{X}_i,\mathbf{\Psi})^{\mathbf{o}_i^{\mathrm{cog}}}$. The expectations in equation (18) are estimated by single-sample Monte Carlo with reparameterized noise,

with $\boldsymbol{\epsilon}^{(W)}\sim\mathcal{N}(\mathbf{0},\mathbf{I}_{r_g})$, $\boldsymbol{\epsilon}^{(d)}\sim\mathcal{N}(\mathbf{0},\mathbf{I}_{|\Psi|})$, and $\epsilon_i^{(c)}\sim\mathcal{N}(0,1)$, and $(\phi,\psi)$ are optimized with Adam [33]. Each $\ell_i^{(c)}$ is evaluated under the rank-$K$-plus-diagonal covariance $\mathbf{\Sigma}_{c,\mathbf{o}_i}(S_i^{(c)})$ via the Sylvester–Woodbury identity [34] at cost $O(RK^2+K^3)$. Algorithm 1 provides the pseudocode.

The subtype posterior is

and the subtype-conditional stage posterior is $\widehat{p}(S_i\mid c_i=c,\mathbf{Y}_i,\mathbf{X}_i,\mathcal{D})=q_{\psi}(S_i^{(c)}\mid\mathbf{Y}_i,\mathbf{X}_i)$. Marginalizing the subtype yields a $C$-component Gaussian mixture on $\operatorname{logit}(S_i)$,

Posterior expectations and credible intervals are obtained from equations (20) and (21) by Monte Carlo over $\mathbf{\Psi}\sim q_{\phi}^{\mathrm{glob}}$, with the inner stage integral of $\ell_i^{(c)}$ against the stage prior $p(s\mid c,\mathbf{X}_i,\mathbf{\Psi})$ computed by quadrature on a logit-stage grid. For a held-out subject the encoder is evaluated in a single forward pass.

Experiments

We evaluate ELPAC along two axes that test the two constructions of section 2. Synthetic experiments¹ measure recoverability of the anchored stage and the stage-varying factor decomposition against ground truth. The harmonized ADNI+NACC/SCAN evaluation measures held-out generative fit, subtype and stage recovery, predictive-interval (PI) coverage of biomarker observations, and the biological structure of the recovered coordination programs in a regime where most subjects are partially observed and unlabeled. We compare ELPAC against representative baselines per relevant family: SuStaIn [6] and DEBM [7] represent subtype-and-stage and event-based progression models, Static-FA [27] separates factor analysis from downstream subtype or stage prediction, MoPoE-VAE [22] and HEALNet [23] represent multimodal neural fusion, while XGBoost [35] and label propagation [36] directly use the available endpoint labels and serve as supervised reference.

Metrics. We report held-out NLL/cell for generative fit; reconstruction RMSE and masked NLL (one modality held out) for imputation; BAcc and Macro-F1 for subtype recovery; Spearman $\rho$ of recovered stage against ground-truth latent (synthetic) or MMSE (real); AUC against case–control labels for diagnostic discrimination; $\lvert\mathrm{Cog}\ r\rvert$ for cognition alignment; and PI coverage at the 90% nominal level for calibration. Metrics scale by 100; best bolded, second underlined.

Synthetic cohorts: joint latent-structure recovery and construction ablation

Setup. We generate synthetic data from the model in section 2 using $C=3$ subtypes, $K=3$ latent coordination factors, $J=8$ trajectory shells, and $R=20$ biomarkers split across four modalities. Each replicate has 70% endpoint anchors per subtype, 10% healthy-reference controls, and 60% unlabeled subjects. Observation masks are drawn independently per modality to match the sparse multimodal observation pattern in the real cohort. For each replicate, endpoint anchors are sampled with $\operatorname{logit}(S_i)\sim\mathcal{N}(\operatorname{logit}(s_A),\epsilon_S^2)$ for each subtype, healthy-reference controls with $\operatorname{logit}(S_i)\sim\mathcal{N}(\operatorname{logit}(s_H),\epsilon_S^2)$, and the remaining unlabeled subjects with $(c_i,S_i)$ drawn from equation (9). We generate 100 independent replicates at $N=1{,}000$, $2{,}000$, and $5{,}000$ subjects, partition each into a stratified 80/20 train–test split, and evaluate predictive performance on held-out test subjects.

Latent-structure recovery. ELPAC jointly recovers subtype assignment, latent stage, mean trajectories, and stage-varying coordination, leading on every metric tied to the latent stage axis (Table 2).

Figure 2 provides a closer look: the subtype posterior cleanly separates the unlabeled population, the recovered mean trajectories and onset locations are close to the truth, and the stage-varying cross-modal correlations track the truth pattern across stages.

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Ablation and sensitivity. We evaluate the roles of endpoint anchoring and stage-varying coordination (Table 3). Removing endpoint anchors damages subtype recovery (BAcc $-46.5$ pp), cognition alignment ($-21.8$ pp), and held-out likelihood ($+2.8$ nats), while leaving stage Spearman $\rho$ within 4 pp of the full model: boundary supervision carries the subtype signal and orients the latent stage axis. Replacing the factor covariance by a diagonal covariance damages held-out likelihood ($+4.7$ nats), masked reconstruction ($+0.38$ nats), and subtype recovery (BAcc $-26.7$ pp): residual covariance carries both density-modeling and subtype-discriminative information. The full model also leads stage-binned posterior predictive correlation (PPC) over the static-factor variant, evidence that the stage-varying activations carry information beyond a single stationary covariance. Sensitivity analyses in section D.6 show the robustness across the factor-module hyperparameters and sample-sizes. Sweeping each of the hyperparameters one at a time leaves held-out NLL within the $\pm1.96$ SE band (Figure 6, 50 paired cohorts/cell); scaling $N$ from 1,000 to 5,000 leaves all four recovery metrics within replicate-cohort SE of the $N=1,000$ values in Table 2 (Figure 7).

Method	NLL/cell $\downarrow$	BAcc $\uparrow$	Macro-F1 $\uparrow$	Stage-score $\rho\uparrow$	AUC$_{\{1,2\}/3} \uparrow$	Masked NLL $\downarrow$	$\lvert$Cog r$\rvert\uparrow$
SuStaIn	–	$44.0\pm0.8$	$43.3\pm0.8$	$45.4\pm0.6$	$54.9\pm1.3$	–	$47.7\pm0.6$
DEBM	–	–	–	$38.4\pm0.8$	$71.5\pm0.6$	–	$13.4\pm0.6$
Static-FA	–	$60.4\pm0.8$	$59.2\pm0.9$	$27.5\pm1.2$	$52.1\pm1.2$	–	$42.1\pm1.3$
MoPoE-VAE	$1.47\pm0.01$	$47.8\pm0.8$	$43.8\pm0.9$	$39.7\pm0.9$	$38.1\pm0.7$	$1.46\pm0.01$	$38.5\pm0.9$
XGBoost	–	$87.5\pm0.5$	$86.4\pm0.5$	$95.6\pm0.0$	$75.9\pm2.3$	–	$68.2\pm0.8$
Label-prop	–	$87.6\pm0.5$	$86.7\pm0.5$	$43.9\pm0.7$	$77.3\pm0.3$	–	$42.3\pm0.9$
HEALNet	–	$89.8\pm0.5$	$89.9\pm0.5$	$64.9\pm0.5$	$48.1\pm0.6$	–	$87.8\pm0.3$
ELPAC (ours)	$1.07\pm0.00$	$95.4\pm0.2$	$96.0\pm0.1$	$96.2\pm0.0$	$81.0\pm2.3$	$0.91\pm0.00$	$97.9\pm0.1$

Model	Held-out NLL $\downarrow$	BAcc $\uparrow$	Stage-score $\rho\uparrow$	Masked NLL $\downarrow$	Stage-binned PPC $\uparrow$	$\lvert\mathrm{Cog}\ r\rvert\uparrow$
ELPAC, no-anchor	$22.07\pm0.05$	$48.9\pm0.6$	$92.4\pm0.1$	$1.109\pm0.003$	$81.4\pm0.5$	$76.1\pm1.1$
Diagonal covariance	$23.97\pm0.05$	$68.7\pm0.5$	$95.7\pm0.0$	$1.289\pm0.002$	–	$96.1\pm0.1$
Static-factor covariance	$19.38\pm0.05$	$\mathbf{95.5\pm0.2}$	$\mathbf{97.4\pm0.0}$	$0.912\pm0.002$	$78.7\pm0.5$	$\mathbf{98.7\pm0.1}$
ELPAC (full)	$\mathbf{19.27\pm0.05}$	$95.4\pm0.2$	$96.2\pm0.0$	$\mathbf{0.910\pm0.002}$	$\mathbf{82.8\pm0.4}$	$97.9\pm0.1$

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Harmonized AD cohort: stage-resolved coordination and clinical alignment

Cohort and protocol. We evaluate ELPAC on a harmonized cross-sectional Alzheimer’s disease (AD) cohort of $N=7,501$ subjects pooled from ADNI [37] and NACC [38] with SCAN-standardized imaging [39] (NACC March 2026 data freeze). Subjects are retained whenever any of the four modalities (amyloid PET, tau PET, structural MRI, plasma biomarkers) is observed, with the per-modality availability ranging from 13.3% to 73.0%. The cohort comprises 641 endpoint anchors, 219 healthy-reference startpoints, and 6,641 unlabeled subjects. The endpoint anchors split into early-onset AD (EOAD, $n=68$), late-onset AD (LOAD, $n=377$), and healthy aging (HA, $n=196$). Details of dataset characteristics are summarized in Table 9. Performance is evaluated over 20 random 80/20 train–test splits, each stratified by anchor stratum (per-subtype anchors, healthy reference, unlabeled).

Generative fit, subtype and stage recovery, and calibrated prediction intervals. ELPAC leads generative fit, reconstruction, stage correlation with MMSE, and Macro-F1 in Table 4, validating the subtype and stage-axis recovery. HEALNet, paired with a supervised classification head trained on the endpoint anchors, leads diagnostic AUCs by $\le2.3$ pp over ELPAC; ELPAC additionally returns a coherent posterior over subtype, stage, and trajectories, with per-subtype, per-modality PI coverage tracking the 90% and 95% nominal targets (Figure 3, column 5).

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Ablation. The AD ablation reproduces the synthetic per-axis attribution at smaller deltas (Table 5). Endpoint anchoring lifts subtype recovery (BAcc $+4.3$ pp) and both diagnostic AUCs ($+1.7$ and $+1.9$ pp). The diagonal-covariance variant widens the held-out and masked NLL gaps by $+1.8$ nats each relative to the full model, and the static-factor variant drops BAcc by $-9.8$ pp (58.7 vs. 68.5 full), supporting stage-varying activations as the source of subtype-discriminative residual structure. The full model leads held-out NLL, subtype recovery, both diagnostic AUCs, and masked reconstruction.

Model	Held-out NLL ↓	BAcc ↑	AUC AD/(CN+MCI) ↑	AUC AD/MCI ↑	Masked NLL ↓	$|\operatorname{Cog}\ r|$ ↑
ELPAC, no-anchor	11.19±0.03	64.2±1.5	84.7±0.3	77.0±0.6	7.38±0.05	59.2±0.5
Diagonal covariance	12.83±0.05	66.5±1.5	84.9±0.3	76.3±0.5	8.98±0.04	57.0±0.5
Static-factor covariance	11.85±0.04	58.7±1.6	84.0±0.3	74.0±0.6	7.25±0.05	50.2±0.6
ELPAC (full)	11.01±0.03	68.5±1.4	86.4±0.2	78.9±0.5	7.22±0.05	59.0±0.5

Recovered subtype trajectories and multimodal coordination programs. The mean biomarker trajectories track subtype-specific dominant pathologies (Figure 3): amyloid and tau in EOAD, amyloid in LOAD with structural change following and modest tau elevation, and a structural-only signal in HA that remains below the pathology threshold on amyloid and tau. The coordination programs differ qualitatively across subtypes (Figure 4). EOAD’s leading axes are cross-modal A/T/N coalitions, including one axis that ties medial-temporal tau to lateral-temporal atrophy. LOAD’s leading axis is amyloid-dominant within-modality, with cross-modal A/N and T/N structure on later axes. HA’s leading axes are atrophy-dominant within-modality. The stage-varying variance share confirms the EOAD concurrent-multimodal versus LOAD sequential-amyloid-then-tau pattern (Figure 11).

Vector (PDF)

Vector (PDF)

Robustness to anchor corruption. Across the four targeted corruptions, ELPAC retains stage ordering and subtype separation: we corrupted anchors—subtype mislabeling, the true stage redrawn at random, an inverted group-mean direction, and a localized non-monotone bump—and refit at each level; the mislabeling arms are mirrored on the real cohort (section D.7). Anchor mislabeling degrades subtype and stage recovery smoothly as the labels get noisier (at 20% uniform mislabeling, held-out synthetic subtype accuracy and stage $\rho$ fall by 3.4 and 5.7 pp and real-cohort BAcc by 7.0 pp); the stage redraw affects staging while leaving subtype recovery unchanged; and the two mean-shape perturbations have the smallest stage-ordering effects. On the real cohort, held-out likelihood stays well inside the 0.84-nat full-vs-static-factor margin of Table 5.

Cross-cohort robustness and longitudinal consistency. Leave-one-cohort-out folds preserve the stage axis and subtype mapping (section E.1), indicating cross-cohort generalization. On subjects with longitudinal follow-up visits, ELPAC’s cross-sectional stage estimates advance on average with subsequent visits, and the same model attains the highest cognition recovery and three-year MCI-to-

AD conversion AUC among the baselines (section E.2). These support the anchored stage axis as a transferable severity coordinate.

Discussion

ELPAC resolves two coupled problems in latent-progression modeling: locating each subject on a disease axis whose endpoints are reliably labeled but whose origin is not, and capturing how multimodal biomarker coalitions reorganize along that axis. Endpoint anchoring inverts the supervision direction for the first; the stage-varying factor decomposition handles the second, with the inferred stage axis supplying the auxiliary variation that conditionally identifies its loadings and activations.

Limitations. The stage axis is pseudo-temporal, recovered from cross-sectional snapshots, so the coordination factors describe co-variation along it rather than mechanism. The dependency layer in equation (13) models second-order multimodal coordination; higher-order joint dependencies are a future extension via copula or normalizing-flow residuals. Theorem 1 identifies the dynamic factor block under the conditions of section C.2; the full-model proposition (section C.4) further invokes a Gaussian mixing-family hypothesis, with broader mixing families a separate theoretical direction. The amortized variational scheme produces cohort-scale joint inference and single-pass test-time deployment at the per-evaluation cost stated in § 2.6; richer guide families (e.g., normalizing-flow posteriors) remain an inference-side extension.

Acknowledgments and Disclosure of Funding

Funding. This work was partially funded by awards R01AG081413, R01EB034720, and R01AG068191 from the NIH. The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIH.

Competing interests. The authors declare no competing interests.

Data. Data collection and sharing for the Alzheimer’s Disease Neuroimaging Initiative (ADNI) is funded by the National Institute on Aging (National Institutes of Health Grant U19 AG024904). The grantee organization is the Northern California Institute for Research and Education. In the past, ADNI has also received funding from the National Institute of Biomedical Imaging and Bioengineering, the Canadian Institutes of Health Research, and private sector contributions through the Foundation for the National Institutes of Health (FNIH) including generous contributions from the following: AbbVie, Alzheimer’s Association; Alzheimer’s Drug Discovery Foundation; Araclon Biotech; BioClinica, Inc.; Biogen; Bristol-Myers Squibb Company; CereSpir, Inc.; Cogstate; Eisai Inc.; Elan Pharmaceuticals, Inc.; Eli Lilly and Company; EuroImmun; F. Hoffmann-La Roche Ltd and its affiliated company Genentech, Inc.; Fujirebio; GE Healthcare; IXICO Ltd.; Janssen Alzheimer Immunotherapy Research & Development, LLC.; Johnson & Johnson Pharmaceutical Research &Development LLC.; Lumosity; Lundbeck; Merck & Co., Inc.; Meso Scale Diagnostics, LLC.; NeuroRx Research; Neurotrack Technologies; Novartis Pharmaceuticals Corporation; Pfizer Inc.; Piramal Imaging; Servier; Takeda Pharmaceutical Company; and Transition Therapeutics.

The NACC database is funded by NIA/NIH Grant U24 AG072122. NACC data are contributed by the NIA-funded ADRCs: P30 AG062429 (PI James Brewer, MD, PhD), P30 AG066468 (PI Oscar Lopez, MD), P30 AG062421 (PI Teresa Gomez-Isla, MD), P30 AG066509 (PI Thomas Grabowski, MD), P30 AG066514 (PI Mary Sano, PhD), P30 AG066530 (PI Helena Chui, MD, Arthur Toga, PhD), P30 AG066507 (PI Marilyn Albert, PhD), P30 AG066444 (PI David Holtzman, MD), P30 AG066518 (PIs Lisa Silbert, MD, Kevin Duff, PhD), P30 AG066512 (PI Thomas Wisniewski, MD), P30 AG066462 (PI Scott Small, MD), P30 AG072979 (PI David Wolk, MD), P30 AG072972 (PIs Charles DeCarli, MD, Rachel Whitmer, PhD), P30 AG072976 (PI Andrew Saykin, PsyD), P30 AG072975 (PI Julie Schneider, MD, MS), P30 AG072978 (PI Ann McKee, MD), P30 AG072977 (PI Robert Vassar, PhD), P30 AG066519 (PI Joshua Grill, PhD), P30 AG062677 (PIs Brad Boeve, MD, Ronald Petersen, MD, PhD), P30 AG079280 (PI Jessica Langbaum, PhD), P30 AG062422 (PI Gil Rabinovici, MD), P30 AG066511 (PI Allan Levey, MD, PhD), P30 AG072946 (PI Linda Van Eldik, PhD), P30 AG062715 (PI Sanjay Asthana, MD, FRCP), P30 AG072973 (PI Russell Swerdlow, MD), P30 AG066506 (PIs Glenn Smith, PhD, ABPP, David Lowenstein, PhD, Ranjan Duara, MD), P30 AG066508 (PIs Stephen Strittmatter, MD, PhD, Christopher Van Dyck, MD), P30 AG066515 (PI Victor Henderson, MD, MS), P30 AG072947 (PI Suzanne Craft, PhD), P30 AG072931 (PI Henry Paulson, MD, PhD), P30 AG066546 (PIs Sudha Seshadri, MD, Gladys Maestre, MD, PhD), P30 AG086401 (PI Erik Roberson, MD, PhD), P30 AG086404 (PI Gary Rosenberg, MD), P30 AG086403 (PI Angela Jefferson, PhD), P30 AG072958 (PIs Heather Whitson, MD, Gwenn Garden, MD, PhD), P30 AG072959 (PI Jagan Pillai, MD, PhD), P30 AG092752 (Ihab Hajjar, MD, MS).

SCAN is a multi-institutional project that was funded as a U24 grant (AG067418) by the National Institute on Aging in May 2020. Data collected by SCAN and shared by NACC are contributed by the NIA-funded ADRCs as follows: Arizona Alzheimer’s Center - P30 AG072980 (PI: Eric Reiman, MD); R01 AG069453 (PI: Eric Reiman (contact), MD); P30 AG019610 (PI: Eric Reiman, MD); and the State of Arizona which provided additional funding supporting our center; Boston University - P30 AG013846 (PI Neil Kowall MD); Cleveland ADRC - P30 AG062428 (James Leverenz, MD); Cleveland Clinic, Las Vegas – P20 AG068058; Columbia - P50 AG008702 (PI Scott Small MD); Duke/UNC ADRC – P30 AG072958; Emory University - P30AG066511 (PI Levey Allan, MD, PhD); Indiana University - R01 AG19771 (PI Andrew Saykin, PsyD); P30 AG10133 (PI Andrew Saykin, PsyD); P30 AG072976 (PI Andrew Saykin, PsyD); R01 AG061788 (PI Shannon Risacher, PhD); R01 AG053993 (PI Yu-Chien Wu, MD, PhD); U01 AG057195 (PI Liana Apostolova, MD); U19 AG063911 (PI Bradley Boeve, MD); and the Indiana University Department of Radiology and Imaging Sciences; Johns Hopkins - P30 AG066507 (PI Marilyn Albert, Ph.D.); Mayo Clinic - P50 AG016574 (PI Ronald Petersen MD PhD); Mount Sinai - P30 AG066514 (PI Mary Sano, PhD); R01 AG054110 (PI Trey Hedden, PhD); R01 AG053509 (PI Trey Hedden, PhD); New York University - P30AG066512-01S2 (PI Thomas Wisniewski, MD); R01AG056031 (PI Ricardo Osorio, MD); R01AG056531 (PIs Ricardo Osorio, MD; Girardin Jean-Louis, PhD); Northwestern University - P30 AG013854 (PI Robert Vassar PhD); R01 AG045571 (PI Emily Rogalski, PhD); R56 AG045571, (PI Emily Rogalski, PhD); R01 AG067781, (PI Emily Rogalski, PhD); U19 AG073153, (PI Emily Rogalski, PhD); R01 DC008552, (M.-Marsel Mesulam, MD); R01 AG077444, (PIs M.-Marsel Mesulam, MD, Emily Rogalski, PhD); R01 NS075075 (PI Emily Rogalski, PhD); R01 AG056258 (PI Emily Rogalski, PhD); Oregon Health and Science University - P30 AG066518 (PIs Lisa Silbert, MD, Kevin Duff, PhD); R56 AG074321 (PI Jeffrey Kaye, MD); Rush University - P30 AG010161 (PI David Bennett MD); Stanford – P30AG066515; P50 AG047366 (PI Victor Henderson MD MS); University of Alabama, Birmingham – P20; University of California, Davis - P30 AG012019 (PI Charles DeCarli, MD); P30 AG072972 (PI Charles DeCarli, MD); University of California, Irvine - P50 AG016573 (PI Frank LaFerla PhD); University of California, San Diego - P30AG062429 (PI James Brewer, MD, PhD); University of California, San Francisco - P30 AG062422 (Rabinovici, Gil D., MD); University of Kansas - P30 AG035982 (Russell Swerdlow, MD); University of Kentucky - P30 AG028283-15S1 (PIs Linda Van Eldik, PhD and Brian Gold, PhD); University of Michigan ADRC - P30AG053760 (PI Henry Paulson, MD, PhD) P30AG072931 (PI Henry Paulson, MD, PhD) Cure Alzheimer’s Fund 200775 - (PI Henry Paulson, MD, PhD) U19 NS120384 (PI Charles DeCarli, MD, University of Michigan Site PI Henry Paulson, MD, PhD) R01 AG068338 (MPI Bruno Giordani, PhD, Carol Persad, PhD, Yi Murphey, PhD) S100D026738-01 (PI Douglas Noll, PhD) R01 AG058724 (PI Benjamin Hampstead, PhD) R35 AG072622 (PI Benjamin Hampstead, PhD) W81XWH2110743 (PI Benjamin Hampstead, PhD) R01 AG073235 (PI Nancy Chiaravalloti, University of Michigan Site PI Benjamin Hampstead, PhD) 1I01RX001534 (PI Benjamin Hampstead, PhD) IRX001381 (PI Benjamin Hampstead, PhD); University of New Mexico - P20 AG068077 (Gary Rosenberg, MD); University of Pennsylvania - State of PA project 2019NF4100087335 (PI David Wolk, MD); Rooney Family Research Fund (PI David Wolk, MD); R01 AG055005 (PI David Wolk, MD); University of Pittsburgh - P50 AG005133 (PI Oscar Lopez MD); University of Southern California - P50 AG005142 (PI Helena Chui MD); University of Washington - P50 AG005136 (PI Thomas Grabowski MD); University of Wisconsin - P50 AG033514 (PI Sanjay Asthana MD FRCP); Vanderbilt University – P20 AG068082; Wake Forest - P30AG072947 (PI Suzanne Craft, PhD); Washington University, St. Louis - P01 AG03991 (PI John Morris MD); P01 AG026276 (PI John Morris MD); P20 MH071616 (PI Dan Marcus); P30 AG066444 (PI John Morris MD); P30 NS098577 (PI Dan Marcus); R01 AG021910 (PI Randy Buckner); R01 AG043434 (PI Catherine Roe); R01 EB009352 (PI Dan Marcus); UL1 TR000448 (PI Brad Evanoff); U24 RR021382 (PI Bruce Rosen); Avid Radiopharmaceuticals / Eli Lilly; Yale - P50 AG047270 (PI Stephen Strittmatter MD PhD); R01AG052560 (MPI: Christopher van Dyck, MD; Richard Carson, PhD); R01AG062276 (PI: Christopher van Dyck, MD); 1Florida - P30AG066506-03 (PI Glenn Smith, PhD); P50 AG047266 (PI Todd Golde MD PhD)

Appendix roadmap

The appendix supports the main paper with the full model specification, priors and hyperparameters, the exact ELBO and variational parameterization, training pseudocode, identifiability proofs, parameter-block recovery against truth, sensitivity analyses, leave-one-cohort-out and longitudinal validation, and the closed-form derivations and Sylvester–Woodbury reductions for the derived covariance-path summaries.

Section	Contents
App. B	Full model specification and inference: joint likelihood, variational family, ELBO and reparameterized gradients, training algorithm, test-time inference, and closed-form derived summaries (Sylvester–Woodbury reductions, pairwise correlation, Fisher modality weights)
App. C	Formal identifiability definitions, full proof of Theorem 1, warp invariance, the Theorem-1 checklist at the fitted real-cohort posterior, and full-model identification proposition
App. D	Experimental protocol: cohort summary, informative priors, competing methods and runtime, metric definitions, sensitivity analysis (hyperparameters and cohort size), robustness to anchor corruption, and test-time refinement
App. E	Real-data validation and biology: leave-one-cohort-out cohort-shift validation, longitudinal validation (stage velocity, cognition recovery, MCI-to-AD conversion), subtype biology summaries (stage discriminability, endpoint pathology, cross-modal correlations), per-factor multimodal mixture, and stage-varying Fisher modality weights

Full model specification and inference

Joint likelihood

Let $\boldsymbol{\Psi}$ collect the global model parameters of §2 and $\mathcal{Z} = \{(c_i, S_i)\}_{i=1}^{n}$ the subject-level latents. The full joint over observables, latents, and parameters factorizes as

The subtype–stage priors $p(c_i \mid X_i, \boldsymbol{\Psi})$ and $p(S_i \mid c_i, X_i, \boldsymbol{\Psi})$ are stratum-specific: equation (9) for $i \in\mathcal{U}$; equation (10) for $i \in\mathcal{A}_{c}$, with $p(c_i=c \mid X_i, \boldsymbol{\Psi})=1$ encoding the observed endpoint subtype as a Dirac prior; and equation (11) for $i \in\mathcal{H}$, with $c_i$ unconstrained. The biomarker factor $\log p(Y_i,O_i \mid c_i, S_i, \boldsymbol{\Psi})$ is conditionally independent of $X_i$ given $(c_i, S_i, \boldsymbol{\Psi})$, while the cognitive factor depends on $X_i$ linearly through equation (2).

The parameter prior $p(\boldsymbol{\Psi})$ factorizes across the boundary, velocity, coordination, subtype–stage, and cognitive blocks of the generative process,

with per-component priors specified in §2, $\psi_{\mathrm{soft}}$ the Gaussian coupling potential of equation (5), and $V_c$ the parametric velocity-field components $\{\delta_{c,r}, t_{c,r}, \omega_{c,r}, \beta_{c,r}\}_{r=1}^{R}$ of equations (7) and (8), which determine the discretized shell-endpoint velocities $V_{c,j}$ for $j=0,\ldots,J$ via equation (6); in particular $V_{c,0}$ enters the cubic Hermite equation (25) (left endpoint of the leftmost shell) and is governed by the same prior as $V_{c,1:J}$.

Per-subject biomarker likelihood, fully composed. We write $\theta_c(s;\boldsymbol{\Psi})$, $\boldsymbol{\Sigma}_c(s;\boldsymbol{\Psi})$ when emphasizing dependence on the global parameters $\boldsymbol{\Psi}$ and recover the bare $\theta_c(s)$, $\boldsymbol{\Sigma}_c(s)$ of §2 when $\boldsymbol{\Psi}$-dependence is implicit. The mean trajectory and within-stage covariance entering the per-subject Gaussian likelihood equation (1) are deterministic functionals of disjoint blocks of $\boldsymbol{\Psi}$. For the mean, the closed-form solution to the backward recursion equation (3) gives

with each shell-endpoint velocity $\mathbf{V}_{c,j}$ assembled coordinatewise from equation (6) ($\mathbf{V}_{c,0}$ in particular is equation (6) evaluated at $\tau_0=0$, supplying the slope at the leftmost shell boundary). The continuous-time trajectory is the cubic Hermite interpolant of the shell states with shell-endpoint slopes,

with the standard cubic Hermite basis $h_{00}(u)=2u^3-3u^2+1$, $h_{10}(u)=u^3-2u^2+u$, $h_{01}(u)=-2u^3+3u^2$, $h_{11}(u)=u^3-u^2$. For the covariance, $\boldsymbol{\Sigma}_c(s;\boldsymbol{\Psi})$ has the rank-$K$-plus-diagonal form equation (14) with activation spline equation (15) and idiosyncratic floor $D_c=\operatorname{diag}(d_{c,r}+d_{\min})$.

Substituting these compositions into equation (1) gives the explicit log-likelihood evaluated on the observed coordinates $\mathcal{O}_i$:

with $\lVert\mathbf{x}\rVert_M^2=\mathbf{x}^{\mathsf{T}}M\mathbf{x}$, where $\boldsymbol{\theta}_{c_i,\mathcal{O}_i}(S_i;\boldsymbol{\Psi})\in\mathbb{R}^{|\mathcal{O}_i|}$ is the row-restriction of equation (25) to $\mathcal{O}_i$, and $\boldsymbol{\Sigma}_{c_i,\mathcal{O}_i}(S_i;\boldsymbol{\Psi})\in\mathbb{R}^{|\mathcal{O}_i|\times|\mathcal{O}_i|}$ is the principal sub-block

with $\mathbf{A}_{c,\mathcal{O}_i,:}\in\mathbb{R}^{|\mathcal{O}_i|\times K}$ the row-restriction of $\mathbf{A}_c$ and $\mathbf{D}_{c,\mathcal{O}_i,\mathcal{O}_i}$ the corresponding diagonal sub-block. The restriction preserves the low-rank-plus-diagonal structure with rank $\min(K,|\mathcal{O}_i|)$, so the Sylvester–Woodbury reduction in equation (29) applies verbatim with $\mathbf{A}_{c,\mathcal{O}_i,:}$ in place of $\mathbf{A}_c$ and $\mathbf{D}_{c,\mathcal{O}_i,\mathcal{O}_i}$ in place of $\mathbf{D}_c$, keeping the per-stage cost of the determinant and quadratic form at $O(|\mathcal{O}_i|K^2+K^3)$ regardless of the missingness pattern. The lower-triangular constraint on $\mathbf{A}_c$ is not preserved under the row restriction to $\mathcal{O}_i$ but enters only through the global identifiability of $\mathbf{A}_c$ (Theorem 1); equation (26) depends on $\mathbf{A}_c$ only through $\mathbf{A}_{c,\mathcal{O}_i,:}\operatorname{diag}(\boldsymbol{\nu}_{c}(S_i))\mathbf{A}_{c,\mathcal{O}_i,:}^{\mathsf{T}}$ and is well-defined on any sub-block.

Inference targets the posterior $p(\boldsymbol{\Psi},Z\mid\mathcal{D},\mathbf{X})$, approximated by the structured variational family of equation (17) optimized against the collapsed ELBO of equation (18).

Inference

Reparameterized gradient. Writing $\mathbf{w}=(\boldsymbol{\Psi},\{S_i^{(c)}\})$ for the continuous latents and $\boldsymbol{\theta}=(\boldsymbol{\phi},\boldsymbol{\psi})$ for the variational parameters, equation (19) gives $\mathbf{w}=g(\boldsymbol{\epsilon};\boldsymbol{\theta})$ with $\boldsymbol{\epsilon}\sim\mathcal{N}(\mathbf{0},I)$ independent of $\boldsymbol{\theta}$. The reparameterized gradient of equation (18) is

estimated by single-sample Monte Carlo per minibatch step. The Bernoulli gates $\delta_{c,r}$ inside $\boldsymbol{\Psi}$ are reparameterized through a Concrete relaxation (below).

Recognition network. The encoder producing $\{(\boldsymbol{\mu}_{\psi}^{(c)},\boldsymbol{\tau}_{\psi}^{(c)})\}_{c=1}^{C}$ in equation (17) is a feed-forward MLP with two hidden layers of width 128, LayerNorm, SiLU activations, and dropout 0.1, taking $(\mathbf{Y}_{i,\mathcal{O}_i},\mathbf{X}_i)$ as input. Output heads are $C$ linear projections to $(\boldsymbol{\mu}_{\psi}^{(c)},\log\boldsymbol{\tau}_{\psi}^{(c)})$, with $\boldsymbol{\tau}_{\psi}^{(c)}=\operatorname{softplus}(\log\boldsymbol{\tau}_{\psi}^{(c)})+10^{-3}$. The global Gaussian $q_{\boldsymbol{\phi}}^{\mathrm{glob}}$ is realized through AutoLowRankMultivariateNormal [40] with rank $r_g=10$.

Reparameterization of the singular activation prior. The activation-spline prior equation (15) is supported on the centered subspace $P_0=\{w\in\mathbb{R}^{P}: \mathbf{1}^{\top}w=0\}$. We sample $w_{c,k}^{\mathrm{std}}\sim\mathcal{N}(0,I_P)$ and reparameterize $w_{c,k}=\tau_k\Pi_0Lw_{c,k}^{\mathrm{std}}$, with $L=\operatorname{chol}((K_B+\epsilon_BI_P)^{-1})$ precomputed once and the projection $\Pi_0=I_P-\frac{1}{P}\mathbf{1}\mathbf{1}^{\top}$ enforcing $\mathbf{1}^{\top}w_{c,k}=0$ exactly.

Concrete relaxation of the region-inclusion gates. The Bernoulli gates $\delta_{c,r}\sim\operatorname{Bern}(\pi_{c,m(r)})$ of equation (7) are realized at training time through a Concrete (Gumbel-Softmax) relaxation [41, 42] with temperature $\lambda(t)=1.0-0.9\,t/T\in[0.1,1.0]$, annealed linearly across the optimization horizon $T$. At evaluation, $\delta_{c,r}$ is rounded to its hard Bernoulli value.

Optimization. The variational parameters $(\phi,\psi)$ are optimized with Adam under global-norm gradient clipping at threshold 5 and minibatch size $\lvert B\rvert=256$, in two phases. Phase 1 holds the activation block at $\Sigma_c=D_c$ and uses $\mathrm{lr}_1=3\times10^{-3}$ for $T_1=5\times10^4$ steps; phase 2 releases the activation block and uses $\mathrm{lr}_2=10^{-3}$ for $T_2=5\times10^4$ steps, initialized from the phase-1 parameters. The Concrete temperature schedule $\lambda(t)$ runs across the combined horizon $T=T_1+T_2$.

Algorithm 1 Stochastic variational inference for ELPAC.

Require: Cohort $\{(Y_i,O_i,X_i,c_i^\star)\}_{i=1}^{n}$; learning rate $\mathrm{lr}$; horizon $T$

1. Initialize global variational parameters $(\mu_\phi,W_\phi,d_\phi)$ and encoder weights $\psi$

2. for $t=1,\ldots,T$ do

3. &nbsp;&nbsp;&nbsp;&nbsp;Sample minibatch $B\subseteq\{1,\ldots,n\}$

4. &nbsp;&nbsp;&nbsp;&nbsp;Reparameterized draw $\widetilde{\Psi}\leftarrow\mu_\phi+W_\phi\epsilon^{(W)}+d_\phi\odot\epsilon^{(d)}$, with $\epsilon^{(W)},\epsilon^{(d)}\sim\mathcal{N}(0,I)$ and Concrete temperature $\lambda(t)$ for the gates $\delta_{c,r}$ in $\widetilde{\Psi}$

5. &nbsp;&nbsp;&nbsp;&nbsp;for each $i\in B$ do

6. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Encode $(\mu_\psi^{(c)},\tau_\psi^{(c)})_{c=1}^{C}\leftarrow q_\psi^{\mathrm{enc}}(Y_i,O_i,X_i)$

7. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Reparameterized draw $\operatorname{logit}(\widetilde{S}_i^{(c)})\leftarrow\mu_\psi^{(c)}+\tau_\psi^{(c)}\epsilon_i^{(c)}$, $\epsilon_i^{(c)}\sim\mathcal{N}(0,1)$, for each $c$

8. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Evaluate $\ell_i^{(c)}\leftarrow p(Y_i,O_i\mid c,\widetilde{S}_i^{(c)},\widetilde{\Psi})p(Y_i^{\mathrm{cog}}\mid c,\widetilde{S}_i^{(c)},X_i,\widetilde{\Psi})^{O_i^{\mathrm{cog}}}$ via Sylvester–Woodbury (section B.3)

9. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$\log\hat{p}_i\leftarrow\operatorname{logsumexp}_{c}\left[\log p(c\mid X_i,\widetilde{\Psi})+\log\ell_i^{(c)}\right]$

10. &nbsp;&nbsp;&nbsp;&nbsp;end for

11. &nbsp;&nbsp;&nbsp;&nbsp;$\widehat{\mathcal{L}}_B\leftarrow\frac{n}{\lvert B\rvert}\sum_{i\in B}\left[\log\hat{p}_i-\sum_{c=1}^{C}\operatorname{KL}\left(q_\psi(S_i^{(c)})\mathbin{\|}p(S_i\mid c_i=c,X_i,\widetilde{\Psi})\right)\right]+\log p(\widetilde{\Psi})-\log q_\phi^{\mathrm{glob}}(\widetilde{\Psi})$

12. &nbsp;&nbsp;&nbsp;&nbsp;$g\leftarrow\nabla_{(\phi,\psi)}\widehat{\mathcal{L}}_B$; clip $\lVert g\rVert_2\leq5$; update $(\phi,\psi)$ with Adam$(\mathrm{lr})$

13. end for

14. return $(\widehat{\phi},\widehat{\psi})$; evaluate posterior summaries equations (20) and (21) by Monte Carlo over $\Psi\sim q_{\widehat{\phi}}^{\mathrm{glob}}$

Test-time inference. The posterior summaries equations (20) and (21) are evaluated by Monte Carlo over $M=50$ samples $\Psi^{(m)}\sim q_\phi^{\mathrm{glob}}$, with the inner stage integral $\int_0^1p(s\mid c,X_i,\Psi^{(m)})\ell_i^{(c)}(\Psi^{(m)},s)\,\mathrm{d}s$ computed by Gauss–Legendre quadrature on a $G=19$-point logit-stage grid. For a held-out subject, this requires a single encoder forward pass and $M\cdot C\cdot G$ Sylvester–Woodbury likelihood evaluations. Per-subject refinement of this pass is evaluated in section D.8.

Closed forms and derived summaries

This subsection collects the closed-form expressions used for the per-step likelihood evaluation in algorithm 1 and the post-hoc derived summaries reported in section E. For each subtype $c$, $A_c\in\mathbb{R}^{R\times K}$ is the loading matrix, $D_c=\operatorname{diag}(d_{c,1},\ldots,d_{c,R})\succ0$ the idiosyncratic noise, and $\nu_c(s)\in\mathbb{R}_{>0}^{K}$ the activation vector. Write $M_c=A_c^{\top}D_c^{-1}A_c\in\mathbb{R}^{K\times K}$ (computed once per subtype) and $G_c(s)=(\operatorname{diag}(\nu_c(s))^{-1}+M_c)^{-1}\in\mathbb{R}^{K\times K}$ (computed per (subtype, stage) pair).

Determinant and inverse (Sylvester / Woodbury). The Sylvester and Woodbury identities reduce the per-stage cost from $O(R^3)$ to $O(RK^2 + K^3)$:

Stage-dependent pairwise correlation $\rho_c(r,r';s)$. For $r \ne r'$ the cross-modal entry of $\Sigma_c(s)$ and the implied correlation are

with $\Sigma_c(s)_{r,r} = a_{c,r}^{\mathsf T}\operatorname{diag}(\nu_c(s))a_{c,r} + d_{c,r}$.

Fisher modality weights $w_{c,m}^{\mathrm{FI}}(s)$ (marginal, blockwise-exact). Let $\theta'_c(s)=\mathrm{d}\theta_c/\mathrm{d}s \in\mathbb{R}^R$, computed via spline-interpolated finite differences from the inferred mean trajectory, and

with $\nu'_{c,k}(s)=\nu_{c,k}(s)\sum_p w_{c,k,p}\psi'_p(s)$. The marginal Fisher information from modality $m$ alone is

with $\Sigma_{c,m}(s)=\Sigma_c(s)_{R_m,R_m}$ the marginal covariance restricted to the $R_m \times R_m$ block. Both $\Sigma_{c,m}^{-1}$ and $\Sigma'_{c,m}$ admit the same Sylvester–Woodbury reduction as the full-rank counterparts of equation (29), applied with $A_{c,R_m}$ in place of $A_c$ and $D_{c,R_m,R_m}$ in place of $D_c$, keeping the per-stage cost at $O(RK^2 + K^3)$. The per-modality weight is the share of total marginal Fisher information,

the marginal-modality decomposition reported by Figure 12.

Vector (PDF)

Identifiability details

Formal definitions

Definition 1 (Observational equivalence). Two parameter settings $\Psi$, $\Psi'$ are observationally equivalent if they induce the same marginal distribution of the observables $(Y_i, X_i, Y_i^{\mathrm{cog}})$ for every admissible cohort configuration.

Definition 2 (Identifiability up to subtype permutation). The model is identifiable up to subtype permutation if observational equivalence implies equality of $\Psi$ and $\Psi'$ up to a permutation of subtype labels and the corresponding relabeling of all subtype-indexed parameters.

Proof of Theorem 1

The subtype index $c$ is suppressed; we write $\Sigma(s)=A\operatorname{diag}(\nu(s))A^{\mathsf T}+D$ as in equation (14). We adopt the conditions of Theorem 1; in addition, $\nu_k(s_1)>0$ for every $k$ at the reference stage $s_1$, and the B-spline basis $\{\psi_p\}_{p=1}^P$ used for $\log\nu_k(\cdot)$ forms a partition of unity, $\sum_p\psi_p(s)=1$ for all $s\in[0,1]$.

Generic regularity (holds by construction in our model+priors). (i) Spline-design coverage. The observed stages span the activation basis: there exist $S\ge P$ distinct posterior stage values yielding a full-rank design matrix $\Phi_S=[\psi_p(s_\ell)]_{\ell,p}$. Holds for any continuous-stage prior with $[0,1]$ support, including the logit-normal of equation (9). (ii) Anderson–Rubin row-deletion. For every subset $\mathcal{I}\subset[R]$ of $2K+1$ rows, the sub-matrix $[A\operatorname{diag}(\sqrt{\nu(s)})]_{\mathcal{I}}$ has rank $K$. Holds Lebesgue-a.e. on $A$, and with probability one under the multiplicative-gamma cascade prior of section 2.4. (iii) Distinguishable trajectory features. $\theta_c(s)$ admits at least one identifiable interior feature (extremum, inflection, or modality-discordant onset) preserved only by the identity warp on $[0,1]$. Generic under the structured velocity equation (6) with distinct biomarker onsets $\{t_{c,r}\}$. Step 1 (identification of $A\operatorname{diag}(\nu(s))A^{\top}$ and $D$ at a fixed stage). For any stage $s$, the model imposes the rank-$K$-plus-diagonal decomposition $\Sigma(s) = A\operatorname{diag}(\nu(s))A^{\top} + D$ with $A$ lower triangular with positive diagonal and $D$ diagonal. Under $R \ge2K + 1$, the lower-triangular constraint, and the AR row-deletion regularity (ii) above, the classical Anderson–Rubin theorem [27, 30] identifies the factor model $\Sigma(s) = \widetilde{A}(s)\widetilde{A}(s)^{\top} + D$ up to per-column signs of $\widetilde{A}(s) := A\operatorname{diag}(\nu(s))^{1/2}$, which the positive-diagonal constraint pins to $+1$. In particular $D$ and $A\operatorname{diag}(\nu(s))A^{\top} = \widetilde{A}(s)\widetilde{A}(s)^{\top}$ are uniquely determined by $\Sigma(s)$. Applied at $s_1$ and $s_2$ with $D$ shared, this yields a single $D$ and the rank-$K$ matrices $\Pi_{\bullet} := \Sigma(s_{\bullet}) - D = A\operatorname{diag}(\nu(s_{\bullet}))A^{\top}$.

Step 2 (factor decomposition across two stages). Restricted to the column space of $A$, the matrix $\Pi_1$ is symmetric positive definite under $\nu_k(s_1) > 0$, so the generalized eigenvalue problem $\Pi_2v = \lambda\Pi_1v$ on $\operatorname{col}(A)$ is well-posed and admits $K$ generalized eigenvalues $\lambda_k = \nu_k(s_2)/\nu_k(s_1)$ with eigenvectors $v_k$ dual to the columns of $A$. Under the ratio-distinctness condition equation (16), the eigenvalues are pairwise distinct and the eigenspaces are one-dimensional. The primal columns of $A$ are recovered as $\widetilde{a}_k = \Pi_1v_k/\lVert\Pi_1v_k\rVert_2$, determined up to a per-column positive scale.

Step 3 (resolution of the per-column scale). The lower-triangular structure of $A$ fixes the sign and ordering of the columns: column $k$ has its first non-zero entry in row $k$, with that entry positive. There remains a one-parameter rescaling per column, $A_{\cdot,k} \mapsto\rho_k A_{\cdot,k}$ and $\log\nu_k(\cdot) \mapsto\log\nu_k(\cdot) - 2\log\rho_k$ for $\rho_k > 0$, that leaves $\Sigma(s)$ invariant. Under the spline parameterization $\log\nu_k(s) = \sum_p w_{k,p}\psi_p(s)$ with the partition-of-unity property $\sum_p \psi_p(s) = 1$, the uniform shift $-2\log\rho_k$ in $\log\nu_k$ corresponds to $w_{k,p} \mapsto w_{k,p} - 2\log\rho_k$ for every $p$, so $\sum_p w_{k,p}$ shifts by $-2P\log\rho_k$. The centering identity $\sum_p w_{k,p} = 0$ then forces $\rho_k = 1$. With the per-column scale resolved, $\nu(s_1)$ and $\nu(s_2)$ are identified pointwise; under the spline-design coverage regularity (i) the coefficients $\{w_{k,p}\}$ are identified from $\Sigma(s)$ at $S \ge P$ stages with full-rank design $\Phi_S$, after which $\nu(\cdot)$ is identified throughout $[0,1]$.

Step 4 (warp equivalence). The activation ratios in equation (16) are invariant under any strictly monotone reparameterization $\varphi: [0,1] \to[0,1]$ of the latent stage. Boundary pinning by equations (10) and (11) forces $\varphi(0)=0$ and $\varphi(1)=1$. The trajectory-feature regularity (iii) then forbids any non-identity such $\varphi$ from preserving the identified interior feature set of $\theta_c(s)$, so $\varphi$ is the identity on $[0,1]$ and the latent stage is identified up to the model’s natural symmetries.

Theorem-1 conditions at the fitted real-cohort posterior

The regularity conditions of section C.2 are genericity statements, which certify a prior draw. We therefore decided each condition at the fitted estimate itself: Table 7 reports the status of each, Table 8 the equation (16) margin measured there. Equation (16) asks for distinctness, and distinctness holds strictly in every seed and both folds, at both the nominal and the data-supported rank.

Every condition a fitted point can decide holds at the real-cohort posterior means, exhaustively so on the row-deletion condition, in every seed, and in both leave-one-cohort-out folds where scored. These diagnostics show the fitted estimates are consistent with every checkable algebraic and numerical condition; they do not establish real-data identifiability, nor discharge the theorem’s two-stage covariance-identification premise, which a fit cannot exhibit.

Full-model identification (Proposition 1)

Theorem 1 identifies the dynamic factor block conditional on $\Sigma_c(s)$ being available at two distinct stages. Closing this conditioning at the full-model level introduces a single substantive hypothesis on the family of continuous-stage Gaussian mixtures induced by the generative process.

Proposition 1 (Full-model identifiability). Within the structural model of Theorem 1, assume that the family of continuous-stage Gaussian mixtures with stage-varying mean $\theta_c(s)$ and covariance $\Sigma_c(s)$, induced by the generative process of §2 and indexed by subtype $c$, is identifiable from the marginal distribution $p(\mathbf{Y}\mid c)$. Then the full parameter vector $(A_c,\nu_c(\cdot),D_c,\theta_c(\cdot))_{c=1}^{C}$ and the per-subject latents are generically identifiable up to a permutation of subtype labels.

Proof sketch. Under the mixing-family identifiability hypothesis, the conditional distributions $\{p(\mathbf{Y}\mid c,s)\}_{s\in[0,1]}$ are identified from the marginal $p(\mathbf{Y}\mid c)$ together with the latent stage measure. The boundary supervision of equations (10) and (11), by anchoring $s \to0$ on $\mathcal{H}$ and $s \to1$ on $A_c$, fixes the strictly-monotone reparameterization ambiguity inherent to Theorem 1 (Step 4) and thereby pins the latent stage measure on $[0,1]$. Pointwise identifiability of $p(Y \mid c,s)$ yields $\theta_c(s)$ from first moments and $\Sigma_c(s)$ from second moments at every $s$ in the support; Theorem 1 applied at any two stages with pairwise distinct activation ratios identifies the dynamic factor block $(A_c,\nu_c(\cdot),D_c)$. □

Class	Theorem-1 condition	Status
Structural	$R \ge2K+1$; $A_c$ lower-triangular, strictly positive diagonal; $\nu_{c,k}(s_1)>0$	all hold — $R=22 \ge17$ at nominal $K=8$
	centered coefficients, diagonal-positive $D_c$	hold by parameterization
	non-constant activations	holds in all 20 pooled seeds; every subtype carries stage-monotone columns
Regularity	(i) spline-design coverage	full rank $P=5$ in all 240 designs, worst $\sigma_{\min}/\sigma_{\max}=1.7 \times10^{-2}$
	(ii) Anderson–Rubin row deletion	exhaustive — all $\binom{22}{17}=26{,}334$ 17-row sub-blocks of $A_c\operatorname{diag}(\nu_c(s))^{1/2}$ retain full rank 8
	(iii) distinguishable trajectory features	generic under the structured velocity, which places interior features in $\theta_c(s)$; fitted onsets separate across modalities in every subtype
equation (16)	pairwise-distinct $\nu_{c,k}(s_2)/\nu_{c,k}(s_1)$	strict inequality, no tolerance, at both scopes ($K=8$ and $K_\star$)

Scope	Subtype	min equation (16) gap over $K_\star$	min $\nu_{c,k}(s_1)$, $K=8$	$K_\star$
Pooled	EOAD	0.0012	0.0678	3.66
	LOAD	0.0002	0.0560	3.95
	HA	0.0001	0.1181	6.17
Leave-ADNI	EOAD	0.0054	0.0956	3.43
	LOAD	0.0029	0.0274	3.29
	HA	0.0011	0.1205	6.15
Leave-NACC	EOAD	0.0101	0.2942	2.47
	LOAD	0.0063	0.3329	2.97
	HA	0.0037	0.0662	2.55

Scope of the mixing-family hypothesis. This subsection takes the continuous-stage extension of the finite-mixture results it cites [43, 44] as a hypothesis, and we identify the result that hypothesis corresponds to. Bruni and Koch’s identifiability theorem [45] recovers the mixing measure from the mean and covariance functions on a compact index under two regularity classes. Our construction satisfies the first, $\Lambda_1$, by design: splines on $[0,1]$ give bounded $C^1$ paths and the noise floor keeps $\Sigma_c(s)$ uniformly positive definite. The second, $\Lambda_2$, is the pairwise injectivity they single out as crucial, and we adopt it as a modeling assumption on the admissible class.

The fitted posteriors are consistent with the case $\Lambda_2$ reduces to within a single path: injectivity there needs one stage-monotone activation column per subtype, and Table 7 records that every subtype carries them. That matters most for healthy aging, whose near-flat mean cannot carry injectivity alone. The monotonicity is of $\nu_{c,k}(s)$ itself—the interior peaks of Figure 11 are in the normalized variance share, which peaks interiorly even when every activation is monotone.

Bruni and Koch write out the scalar case and state the multivariate extension without a general-$p$ proof. Their subtype-mixture result identifies labels through disjoint ranges; under ELPAC’s shared healthy origin (equations (5) and (11)) we instead assume the endpoint anchors at $s \to1$ (equation (10)) select the labeled components, carried inward by the backward recursion (equation (3)). Thus Proposition 1 stays explicitly conditional, and no reported result depends on it. Theorem 1 is established without invoking the mixing-family hypothesis.

Experimental protocol

Real-data cohort summary

Data used in the preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu). The ADNI was launched in 2003 as a public-private partnership, led by Principal Investigator Michael W. Weiner, MD. The original goal of ADNI was to test whether serial magnetic resonance imaging (MRI), positron emission tomography (PET), other biological markers, and clinical and neuropsychological assessment can be combined to measure the progression of mild cognitive impairment (MCI) and early Alzheimer’s disease (AD). The current goals include validating biomarkers for clinical trials, improving the generalizability of ADNI data by increasing diversity in the participant cohort, and to provide data concerning the diagnosis and progression of Alzheimer’s disease to the scientific community. For up-to-date information, see adni.loni.usc.edu. ADNI clinical, cognitive, and fluid-biomarker data were taken from the ADNIMERGE R package (version 0.0.1, June 2022), and amyloid and tau PET regional SUVRs from the UC Berkeley processed tables (IDA release September 2025, downloaded December 2025).

Table 9 summarizes the harmonized ADNI/NACC cohort introduced in § 3.2 and used for the ablations of Table 5. The cohort comprises every ADNI or NACC subject in the canonical extraction with at least one of four modalities measured (amyloid PET, tau PET, structural MRI, plasma biomarkers). Subtype anchors are subjects with clinically confirmed endpoint phenotypes (EOAD, LOAD, HA), and the remaining subjects are unlabeled at the subtype level and contribute through the unsupervised likelihood. Demographic denominators reflect non-missing values per field, and modality-availability rates report the percentage of cohort subjects with at least one feature observed in each modality. Per-feature missingness exceeds 80% on some plasma cells, and the observation likelihood equation (1) integrates out the unobserved coordinates analytically. Site and scanner effects are harmonized fold-wise with ComBat [46], and intracranial volume is residualized fold-wise from sMRI volumes. Age, education, female fraction, and APOE-$\epsilon$4 rate agree across sources within 1.7 years and within sample-size-weighted SDs, consistent with the cross-cohort transfer in section E.1.

	ADNI	NACC	Total
Subjects	1,722	5,779	7,501
Subtype anchors
EOAD (early-onset Alzheimer’s disease)	40	28	68
LOAD (late-onset Alzheimer’s disease)	263	114	377
HA (healthy aging)	83	113	196
NC startpoint pool2	49	170	219
Unlabeled (general cohort)	1,287	5,354	6,641
Demographics, mean (SD) or %
Age at scan (years)	72.9 (7.8)	71.2 (8.5)	71.6 (8.4)
Female (%)	49.0	59.6	57.2
Education (years)	16.3 (2.6)	16.2 (2.7)	16.2 (2.7)
APOE-ε4 carriers (%)	43.8	39.1	40.2
Modality availability (% of cohort with ≥ 1 feature observed)
Amyloid PET	72.6	34.0	42.9
Tau PET	16.9	15.4	15.7
Structural MRI	79.9	70.9	73.0
Plasma biomarkers	58.0	0.0	13.3

Prior specifications

The boundary supervision priors equations (10) and (11) use $s_A = 0.95$, $s_H = 0.01$, and $\epsilon_S = 0.1$ throughout. For the AD experiments, the diffuse priors on the modality-level onset centers $\{\tilde{t}_{c,m}\}$ in equation (8) and on the age coordinate of $b_c$ in equation (9) are replaced by informative location and ordering priors. (a) AD biomarker cascade. For $c \in\{\mathrm{EOAD}, \mathrm{LOAD}\}$, the modality-level centers are softly ordered amyloid-PET $\to$ tau-PET $\to$ structural-MRI through a Gaussian penalty $\sum_{m<m'} \bigl(\max(0,\tilde{t}_{c,m}-\tilde{t}_{c,m'})\bigr)^2/\sigma_{\mathrm{ord}}^2$ with $\sigma_{\mathrm{ord}}=0.1$, encoding the NIA-AA cascade [2]. (b) Preserved-aging endpoint. For the HA subtype, $\tilde{t}_{\mathrm{HA},m} \sim\mathcal{N}(\operatorname{logit} 0.9, 0.05^2)$ for $m \in\{\mathrm{Amyloid\ PET},\mathrm{Tau\ PET}\}$, placing pathology onsets near the endpoint. (c) Subtype age priors. The age coordinate of $b_c$ is drawn from $\mathcal{N}(-2.0, 0.5)$ for EOAD, $\mathcal{N}(+1.5, 0.5)$ for LOAD, and $\mathcal{N}(-0.5, 0.5)$ for HA, with the remaining coordinates keeping $\mathcal{N}(0,1)$. The synthetic experiments use the diffuse $b_c \equiv0$ and the unconditional categorical $\alpha\sim\mathcal{N}(0,I_C)$ on subtype.

Competing methods and runtime

The external baselines (Table 2 and Table 4) and internal variants (Table 3 and Table 5) are introduced in § 3.1 and § 3.2. Table 10 reports per-fit wall-clock time and training sample size on the synthetic and real-data benchmarks. The intended read of the table is end-to-end cost for a single posterior fit, so the ELPAC numbers cover both phases of the curriculum optimizer of section B.2 (initialization with diagonal covariance plus the joint phase under the rank-$K$-plus-diagonal covariance) and the test-time encoder pass.

Method	NLL/cell ↓	RMSE ↓	$|\rho|$	MMSE ↑	Macro-F1 ↑	BAcc ↑	AUC AD/(CN+MCI) ↑	AUC AD/MCI ↑
SuStaIn	–	–	35.6±1.0	26.9±1.0	36.6±0.5	79.4±0.5	73.0±0.5
DEBM	–	–	40.3±1.1	–	–	75.2±0.7	70.4±0.6
Static-FA	–	–	40.3±1.1	28.7±1.4	41.7±0.4	79.7±0.5	73.3±0.5
MoPoE-VAE	1.96±0.02	3.12±0.10	43.7±0.2	38.7±0.2	47.8±0.2	82.5±0.3	75.0±0.2
XGBoost	–	–	36.4±0.9	63.4±0.3	70.9±0.5	77.4±0.5	71.5±0.4
Label-prop	–	–	42.0±0.8	65.4±0.1	74.6±0.1	75.9±0.5	70.3±0.4
HEALNet	–	–	49.2±0.4	50.4±0.4	62.1±0.4	87.8±0.1	81.2±0.2
ELPAC (ours)	1.35±0.00	2.09±0.01	50.1±0.4	66.9±1.3	68.5±1.4	86.4±0.2	78.9±0.5

Method	Device	Synthetic		Real
		$n$	Time (s)	$n$	Time (s)
SuStaIn [6]	CPU	1,000†	3,089.28±49.67	7,501†	29,707.17±951.63
DEBM [7]	CPU	1,000†	7.21±0.15	7,501†	55.34±6.60
Static-FA [27, 47]	CPU	1,000†	0.78±0.06	7,501†	14.18±1.14
MoPoE-VAE [22]	CPU	1,000	6.30±0.20	7,501	5.55±0.32
XGBoost [35]	CPU	∼150	0.90±0.01	641	1.20±0.06
Label-prop [36]	CPU	∼150	0.03±0.00	641	102.91±0.91
HEALNet [23]	CPU	1,000	258.11±5.54	7,501	926.95±175.06
Diagonal covariance	GPU	1,000	60.77±0.56	7,501	198.13±0.62
Static-factor covariance	GPU	1,000	114.66±0.64	7,501	511.27±50.92
ELPAC (full)	GPU	1,000	167.25±1.67	7,501	384.61±1.39

Hardware. ELPAC is fit on a single NVIDIA RTX Pro 6000 Blackwell GPU (48 GB VRAM), with 2 CPU cores and 6 GB host memory per task. CPU baselines run on shared partitions of mixed Intel Xeon and AMD EPYC nodes (64 to 128 cores per node, up to 2.3 TB RAM per node), with each task allocated 4 CPU cores and 16 GB RAM (XGBoost and Label-prop use 2 cores and 8 GB).

ELPAC fits the full 7,501-subject AD cohort in $\sim385$ s on a single GPU, $\sim77\times$ faster than SuStaIn and $\sim2.4\times$ faster than HEALNet, and returns a posterior over subtype, stage, trajectories, and stage-varying coordination. Two implementation choices keep the cost in this range: the structured low-rank guide of equation (17) makes the global-parameter cost linear in $r_g=10$, and the Sylvester–Woodbury reduction of equation (29) keeps each per-stage likelihood evaluation at $O(RK^2+K^3)$ regardless of the missingness pattern. The 187 s gap between the diagonal-covariance variant and ELPAC (full) buys the held-out NLL and masked reconstruction gains in Table 5.

Metric definitions

We define the non-standard metrics here. Standard ones (BAcc, Macro-F1, AUC, RMSE) follow conventional usage. NLL/cell is the held-out negative log-likelihood per observed coordinate, normalized by the average number of observed cells per subject for cross-model comparability. Masked NLL averages cross-modal predictive log-likelihood across four leave-one-modality-out rotations. Stage-score $\rho$ is the sign-aware Spearman correlation between the recovered stage scalar and the simulator stage, while $|\rho|$ vs. MMSE uses the absolute value to be sign-agnostic across methods. Cog Pearson $r$ is the held-out Pearson correlation between observed cognition $C_{\mathrm{cog}}$ and a regression of $C_{\mathrm{cog}}$ on $(\hat{S}_i,\operatorname{onehot}(\hat{c}_i))$ fit on training anchors, with absolute-value variants reported when subtype-conditional sign conventions differ across methods. Stage-binned PPC is the cross-pair correlation between the model-implied $p_c(r,r';s)$ and the empirical $\hat{p}_c(s)$ binned over five stage bins. $P\%$ PI coverage is the empirical fraction of held-out biomarker observations inside their nominal $P\%$ posterior predictive interval. AUC paired bootstraps use 200 resamples, while all other paired-bootstrap CIs use 1,000.

Synthetic recovery: subtype trajectories and stage-varying coordination

Figure 5 probes the two parameter blocks targeted by Constructions I and II separately, against the simulator truth, on 100 replicate cohorts at $N=1,000$. Construction I is summarized through the recovered subtype mean trajectory $\theta_c(s)$ on a representative biomarker per subtype (top row), and Construction II through the stage-varying activation $\nu_c(s)$ for the factors retained by the multiplicative-gamma prior (bottom row).

Vector (PDF)

ELPAC’s recovered subtype trajectories overlay the simulator truth across the full stage range (top row), and the across-replicate 95% CI on the posterior mean stays tight in the unsupervised middle region between the boundary anchors. The endpoint-anchored backward recursion equation (3) therefore propagates the boundary supervision inward and preserves amplitude at intermediate stages. The retained activation profiles $\nu_c(s)$ (bottom row) recover stage-localized peaks per subtype that match the simulator-truth dashed curves, and the multiplicative-gamma cascade shrinks the third loading column of $A_c$ toward zero on every replicate. Construction I (mean trajectory) and Construction II (factor activations) are jointly recovered under the variational family of equation (17), matching the conditional identifiability claim of theorem 1.

Sensitivity analysis

Figure 6 and Figure 7 report the hyperparameter and sample-size sweeps that support the sensitivity claims in § 3.1. Each sweep is paired across replicate cohorts, so the reported standard errors absorb cohort-level variability and isolate the marginal effect of the hyperparameter under variation.

Figure 6 sweeps the factor rank $K$, the activation-spline basis dimension $P$, the trajectory shell count $J$, the activation-prior scale $\sigma_\tau$, the multiplicative-gamma cascade strength $a_2$, and the log-noise mean $m_D$ one axis at a time around the canonical configuration. ELPAC’s held-out log-likelihood per subject stays inside the $\pm1.96$ SE band of the canonical value on every panel, so the per-construction attribution in Table 3 and Table 5 is robust to the hyperparameter choice.

We have also validated on synthetic data that the coordination structure is not driven by the priors: the informative NIA-AA prior acts on the mean-trajectory onset centers alone and does not act directly on the coordination block, and relaxing the priors that shape it leaves the factorization essentially unchanged (Tucker congruence of matched loading columns $\phi=0.967$ across the $a_2$ and $\sigma_\tau$ sweeps of Figure 6, against 0.29 between fits on different simulated cohorts under the same prior). It follows the data instead—there the model-implied stage-varying correlations track the empirically binned ones (Table 3), and qualitatively so on the real cohort (Figure 10c).

ELPAC’s four recovery metrics in Figure 7 (held-out log-likelihood, reconstruction RMSE, subtype BAcc, stage Spearman $\rho$) are stable across $N \in\{1{,}000, 2{,}000, 5{,}000\}$, so the $N=1{,}000$ numbers in Table 2 hold across the full sample-size range we test.

Robustness to anchor corruption

The monotonicity constraint applies to the latent group-mean trajectory, not to individual observations, which are noisy realizations of it, with subject-wise heterogeneity in the residual factor layer. A subject whose measurements stabilize or reverse is variation about the mean, not a constraint on its shape. On that mean, equation (6) fixes the sign of the shell-endpoint velocities, and so the direction of shell-to-shell progression, leaving magnitude free; the cubic Hermite interpolation retains limited freedom for between-shell overshoot. It also admits flat segments, all three within equation (6): the onset sigmoid holds a marker at rest before onset, the spline amplitude $F_{c,r}$ can fall arbitrarily close to zero over any stage range, and $\delta_{c,r}=0$ drops a marker outright—so temporary stabilization, plateaus and saturation all sit inside the family, as the biology of these markers requires. That direction is established for them [2] and is the inductive bias event-based staging relies on [5, 6]. Thus the only shape excluded is a sustained reversal of a group mean—which we inject deliberately below.

We corrupt anchor correctness at graded levels along four axes: subtype mislabeling, both uniform-random and under a clinical confusion scheme that concentrates errors on the EOAD/LOAD boundary where diagnostic ambiguity actually falls; the anchor’s true stage redrawn; an inverted group-mean direction; and a localized non-monotone bump. We refit at each level. Synthetic replicates run against known truth, and the mislabeling arms are mirrored on the real cohort (Table 11).

Scenario	Data	Metric	$\Delta$ 5%	$\Delta$ 10%	$\Delta$ 20%
Mislabel, clinical confusion	Synthetic	Subtype acc	$-0.2 \pm0.1$	$-0.6 \pm0.1$	$-1.3 \pm0.1$
Mislabel, clinical confusion	Synthetic	Stage $\rho$	$-0.5 \pm0.2$	$-1.0 \pm0.2$	$-1.9 \pm0.3$
Mislabel, clinical confusion	Real	Subtype BAcc	$-1.7 \pm0.7$	$-4.2 \pm1.2$	$-6.4 \pm1.3$
Mislabel, clinical confusion	Real	Macro-F1	$+1.8 \pm1.5$	$+1.0 \pm2.5$	$+0.6 \pm2.8$
Mislabel, clinical confusion	Real	Held-out LL	$-0.02 \pm0.01$	$-0.02 \pm0.01$	$-0.05 \pm0.02$
Mislabel, uniform-random	Synthetic	Subtype acc	$-0.1 \pm0.1$	$-0.6 \pm0.2$	$-3.4 \pm0.6$
Mislabel, uniform-random	Synthetic	Stage $\rho$	$-0.7 \pm0.2$	$-1.7 \pm0.2$	$-5.7 \pm0.4$
Mislabel, uniform-random	Real	Subtype BAcc	$-2.2 \pm0.8$	$-5.1 \pm1.0$	$-7.0 \pm1.2$
Mislabel, uniform-random	Real	Macro-F1	$-0.8 \pm1.4$	$-0.8 \pm1.9$	$+1.3 \pm2.5$
Mislabel, uniform-random	Real	Held-out LL	$-0.02 \pm0.02$	$-0.06 \pm0.02$	$-0.09 \pm0.02$
Stage redrawn $U(0, 1)$	Synthetic	Stage $\rho$	$-2.2 \pm0.2$	$-2.9 \pm0.3$	$-3.3 \pm0.3$
Stage redrawn $U(0, 1)$	Synthetic	Subtype acc	$+0.0 \pm0.1$	$-0.0 \pm0.1$	$-0.0 \pm0.1$
Inverted group-mean direction	Synthetic	Stage $\rho$	$-0.2 \pm0.1$	$-0.6 \pm0.2$	$-1.3 \pm0.3$
Inverted group-mean direction	Synthetic	Subtype acc	$-0.0 \pm0.1$	$-0.1 \pm0.2$	$-1.3 \pm0.5$
Localized non-monotone bump	Synthetic	Stage $\rho$	$-0.0 \pm0.1$	$-0.0 \pm0.1$	$-0.0 \pm0.1$
Localized non-monotone bump	Synthetic	Subtype acc	$-0.0 \pm0.0$	$-0.0 \pm0.1$	$-0.1 \pm0.1$

Across the four targeted corruptions, ELPAC retains stage ordering and subtype separation. Anchor mislabeling degrades subtype and stage recovery smoothly as the labels get noisier, with clinical confusion the gentler of the two arms at 20%; the stage redraw affects staging while leaving subtype recovery unchanged; and the two mean-shape perturbations have the smallest stage-ordering effects—inverting a marker’s direction is milder in stage ordering than any mislabeling, and a localized non-monotone bump produces no detectable loss in stage ordering. The losses are also small: on the real cohort, held-out likelihood stays well inside the 0.84-nat full-vs-static-factor margin of Table 5, and Macro-F1 shows no monotone degradation.

Test-time refinement

Subject-specific refinement is a drop-in extension of the amortized inference path of equations (17) to (21). The mechanism is the local-refinement step of semi-amortized inference [48, 49]: with the fitted globals frozen at their posterior medians, each subject’s subtype-conditional stage posteriors are warm-started from the encoder pass and optimized on that subject’s own data—no labels, no retraining, no global update. Subtype remains exactly enumerated; during refinement its posterior is computed from the per-subtype terms of the local bound, which approach equation (20) as those stage posteriors sharpen; the factor scores remain analytically integrated. The local objective is a variational lower bound on that subject’s evidence under the same frozen globals and $G=19$ grid. The step ladder $T \in\{1, 2, 5, 10, 20, 50\}$ (Adam steps) and learning rate 0.05 were fixed in advance.

Across the tested budgets the per-subject objective improves and stage $\rho$ rises above the encoder pass (Table 12); the gain at $T=50$ holds in every synthetic replicate and persists when results are averaged over 20 global-posterior draws. Subtype inference stays stable, matching the frozen-global exact-grid reference on synthetic data and approaching it from below on the real cohort. The measured gain is therefore in staging, with subtype inference stable, giving an optional per-patient accuracy-for-compute tradeoff.

$T$ Adam steps	1	2	5	10	20	50
Synthetic (100 replicates)	+0.5%	+1.0%	+2.5%	+4.7%	+8.3%	+13.0±0.5%
Real cohort (20 splits)	+1.6%	+2.9%	+7.1%	+10.9%	+13.1%	+13.6±2.7%

Real-data validation and biology

Leave-one-cohort-out validation

Figure 8 compares the pooled fit with leave-ADNI (train on NACC, evaluate on ADNI) and leave-NACC (train on ADNI, evaluate on NACC). ADNI and NACC differ in scanner pool, plasma availability (58% vs. 0%, Table 9), and recruitment composition, so the leave-one-cohort folds test cross-source transfer of the constructions.

Vector (PDF)

The stage-quality and subtype-recovery panels (Figure 8b,Figure 8c) overlap across the three folds within their ±1.96 SE bands, so the anchored stage axis and its subtype mapping transfer across acquisition sources. The small NLL/RMSE increase under leave-NACC (panel a) tracks the smaller ADNI-only training pool (∼1.7k vs. ∼5.8k subjects) and does not propagate into the stage-quality or subtype-recovery panels.

Vector (PDF)

Longitudinal validation: stage velocity, cognition, and MCI conversion

Fig. 9 reports three longitudinal probes of the cross-sectional stage axis on subjects with follow-up visits: per-subject annual stage advance by subtype (panel a), held-out cognition correlation across the Table 4 comparison set (panel b), and three-year MCI-to-AD conversion AUC on the same set (panel c). Panels a and c target signals absent from the training likelihood, so they read out longitudinal behavior of the recovered stage axis. The annual stage advance orders the subtypes EOAD>LOAD>HA along a temporal axis unobserved at training, matching the clinical expectation that early-onset disease progresses fastest and healthy aging is approximately stable [2]. ELPAC leads panel b, so the inferred stage is the strongest single predictor of cognition among the methods evaluated. The path-aggregated severity score $G_i^{\mathrm{path}}$, defined as the integrated mean-trajectory pathology up to subject $i$’s posterior stage, leads panel c, ranking short-horizon conversion risk on subjects unseen at training.

Subtype biology and coordination programs

Fig. 10 reports three biological summaries of the recovered subtype structure. The discriminability heatmap (panel a) localizes the largest effect sizes to the tau pathology block in EOAD, structural atrophy in LOAD, and structural-only signal in HA, mirroring the modality-direction pattern in Figure 3. The endpoint pathology heatmap (panel b) shows HA at the healthy reference on amyloid while EOAD and LOAD reach pathological endpoints on every modality, so the inferred trajectories terminate at the biologically expected magnitudes per subtype.

Panel c overlays the model-implied stage-dependent correlation $\rho_c(r,r';s)$ from equation (30) on the empirical-binned $\hat{\rho}_c(s)$ for two representative LOAD biomarker pairs, and the two curves track each other across the inferred stage axis on both pairs. The fit captures the stage shape, not only the stage-marginal value, consistent with the stage-binned PPC ablation gain in Table 3.

Per-factor multimodal mixture (cross-modal coupling). Figure 11 resolves the residual coordination structure into per-factor variance shares and modality decompositions of loading energy, addressing which biomarkers couple in each subtype, and when each coalition strengthens or attenuates. EOAD engages a three-way amyloid, tau, and structural coalition (across-modality) with mid-stage peak. LOAD instead exhibits an amyloid-dominant leading factor (within-modality) that peaks earlier on the stage axis, with a smaller late-emerging tau-dominant factor (within-modality), consistent with the sequential amyloid-then-tau cascade reported in late-onset disease. HA is omitted: its leading factors are atrophy-dominant within-modality axes (section 3.2) carrying no pathology-discriminative signal, as expected for an amyloid-negative endpoint.

Stage-varying Fisher modality weights. A complementary view targets which modality is most informative about stage at a given stage, derived from the mean-trajectory derivative $\theta'_c(s)$ and the inverse covariance (equations (31) and (32)). Fig. 12 plots the per-subtype Fisher modality weights $w^{\mathrm{FI}}_{c,m}(s_c)$ on the 5-modality split (plasma $A\beta$ ratio and p-tau181 separated). In EOAD, tau-PET and structural MRI jointly carry the bulk of the stage information across mid-to-late stage, indicating that tau and structural readouts are the most informative modalities in early-onset disease. In LOAD, plasma p-tau181 reaches a mid-stage informativeness peak before tau-PET takes over near the endpoint, consistent with reports that plasma p-tau leads PET-detectable tau in amyloid-positive individuals [50], so plasma readouts are the most stage-informative biomarkers earlier in late-onset disease. In HA, structural MRI dominates throughout, indicating that healthy aging is a structural-only progression direction without informative amyloid or tau signal.

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