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arxiv: 2606.28228 · v1 · pith:UQ3YQJFWnew · submitted 2026-06-26 · 💻 cs.LG · stat.ML

Disentangling Continuous-Time Latent Dynamics: Identifiability of Latent SDEs via Diffusion Shifts

Yuanyuan Wang , Wenjie Wang , Haoxuan Li , Mingming Gong , Kun Zhang This is my paper

Pith reviewed 2026-06-29 04:23 UTC · model grok-4.3

classification 💻 cs.LG stat.ML
keywords latent SDEsidentifiabilitydiffusion shiftscontinuous-time modelscausal representation learningOrnstein-Uhlenbeck processesadditive noise
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The pith

Two diagonal diffusion regimes identify latent coordinates of SDEs up to permutation and scaling.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that observations from two environments whose diffusion covariances are diagonal in the latent space and have distinct per-coordinate variance ratios suffice to recover the latent variables of an additive-noise SDE observed through an unknown nonlinear map. The result requires no sparsity assumption on the drift and holds first for linear Ornstein-Uhlenbeck systems before extending to the general nonlinear case. If correct, it supplies a concrete route to disentangle continuous-time latent dynamics from time-series data collected under controlled environmental shifts. The same conditions also recover the instantaneous drift-Jacobian causal graph up to the same permutation.

Core claim

Two diagonal diffusion regimes with pairwise distinct coordinate-wise variance ratios identify the latent coordinates up to permutation and scaling, without any sparsity assumption on the drift. The result is first proved for linear Ornstein-Uhlenbeck systems and then extended to general additive-noise latent SDEs. Under mild smoothness, the instantaneous drift-Jacobian causal graph is identifiable up to the same permutation.

What carries the argument

Environment-induced shifts in diagonal diffusion covariance between two regimes that produce distinct coordinate-wise variance ratios.

If this is right

  • Latent coordinates become recoverable up to permutation and scaling from data in only two environments.
  • The causal graph encoded by the drift Jacobian becomes identifiable under the same conditions.
  • A two-stage estimator recovers the latent representation and optionally the graph.
  • The predicted identifiability boundary is confirmed on synthetic systems and illustrated on real sensor trajectories.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same diffusion-shift logic could be tested on other continuous-time models whose noise structure is environment-dependent.
  • Collecting paired trajectories under deliberately altered noise variances might become a practical protocol for latent disentanglement.
  • Relaxing diagonality or increasing the number of environments would be a direct next step to widen applicability.

Load-bearing premise

The diffusion covariance is exactly diagonal in the latent coordinates and the two environments produce distinct variance ratios for each coordinate.

What would settle it

A counter-example in which two environments satisfy the distinct-ratio condition yet the latent coordinates cannot be recovered up to permutation and scaling would falsify the identifiability theorem.

Figures

Figures reproduced from arXiv: 2606.28228 by Haoxuan Li, Kun Zhang, Mingming Gong, Wenjie Wang, Yuanyuan Wang.

Figure 1
Figure 1. Figure 1: Latent disentanglement in the nonlinear d = 5 setting (three-layer leaky-tanh MLP mixing). Each column shows the 5 × 5 scatter matrix of true coordinates zj (horizontal) versus learned coordinates z˜i (vertical) for one regime condition. Distinct ratios yields the expected near￾permutation alignment; the controls do not recover a clean one-to-one alignment. shows that the corresponding |Dφ| matrix is near-… view at source ↗
Figure 2
Figure 2. Figure 2: Structural diagnostics for the same setting and run as Figure [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: shows the pairwise scatter plots of true latent coordinates zj versus learned representations z˜i for each regime condition. Under Distinct ratios, each learned coordinate aligns tightly with exactly one true coordinate, producing a near-permutation pattern. Both controls lose this structure [PITH_FULL_IMAGE:figures/full_fig_p031_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Mean absolute encoder Jacobian |Dφ| (φ = h ◦ g) for the dense linear d = 5 setting. All three regime conditions use the same representative seed. A near-monomial matrix confirms successful inversion of the mixing under Distinct ratios. Both controls yield less monomial, more mixed encoder Jacobians [PITH_FULL_IMAGE:figures/full_fig_p032_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Latent recovery in the dense linear d = 7 setting (three-layer MLP, leaky-tanh mixing). Layout follows [PITH_FULL_IMAGE:figures/full_fig_p032_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Mean absolute encoder Jacobian |Dφ| (φ = h ◦ g) for the dense linear d = 7 setting. All three regime conditions use the same representative seed. A near-monomial matrix confirms successful inversion under Distinct ratios. 32 [PITH_FULL_IMAGE:figures/full_fig_p032_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: shows the pairwise scatter plots for the sparse linear d = 5 setting [PITH_FULL_IMAGE:figures/full_fig_p033_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Structural diagnostics for the sparse linear [PITH_FULL_IMAGE:figures/full_fig_p034_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Latent recovery in the sparse linear d = 7 setting (three-layer MLP, leaky-tanh mixing). Layout follows [PITH_FULL_IMAGE:figures/full_fig_p034_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Structural diagnostics for the sparse linear [PITH_FULL_IMAGE:figures/full_fig_p035_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Latent recovery in the nonlinear d = 5 setting (three-layer MLP, LeakyReLU mixing). Each column shows the 5 × 5 scatter matrix of true latent coordinates zj (horizontal) versus learned representations z˜i (vertical) for one regime condition. Figures 13 and 14 show the d = 7 results under LeakyReLU mixing. The identifiability gap remains clear: Distinct ratios achieves near-perfect disentanglement and high… view at source ↗
Figure 12
Figure 12. Figure 12: Structural diagnostics for the nonlinear [PITH_FULL_IMAGE:figures/full_fig_p037_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Latent recovery in the nonlinear d = 7 setting (three-layer MLP, LeakyReLU mixing). Each column shows the 7 × 7 scatter matrix of true latent coordinates zj (horizontal) versus learned representations z˜i (vertical) for one regime condition. Under Distinct ratios, each z˜i aligns with exactly one zj . Both controls fail to disentangle. 37 [PITH_FULL_IMAGE:figures/full_fig_p037_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Structural diagnostics for the nonlinear [PITH_FULL_IMAGE:figures/full_fig_p038_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Drift-Jacobian diagnostics for a randomly selected seed, shared across the Hardanger [PITH_FULL_IMAGE:figures/full_fig_p041_15.png] view at source ↗
read the original abstract

Causal representation learning for time series has developed strong identifiability results in discrete-time latent causal models, but identifiability in continuous-time latent stochastic differential equation (SDE) models remains largely open. We address this gap using environment-induced shifts in diffusion covariance. We study additive-noise latent SDEs observed through an unknown nonlinear diffeomorphism, with shared drift but environment-specific diffusion covariance. We show that two diagonal diffusion regimes with pairwise distinct coordinate-wise variance ratios identify the latent coordinates up to permutation and scaling, without any sparsity assumption on the drift. We first prove this result for linear Ornstein--Uhlenbeck systems and then extend it to general additive-noise latent SDEs. Under mild smoothness, the instantaneous drift-Jacobian causal graph is identifiable up to the same permutation. We propose a two-stage estimator for latent disentanglement and optional graph recovery; experiments on synthetic systems confirm the predicted identifiability boundary, and an application to Hardanger Bridge monitoring data illustrates the approach on real sensor trajectories.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript claims that for latent SDEs with additive noise observed through an unknown nonlinear diffeomorphism, with shared drift but environment-specific diffusion covariances, two diagonal diffusion regimes with pairwise distinct coordinate-wise variance ratios identify the latent coordinates up to permutation and scaling, without sparsity assumptions on the drift. The result is first established for linear Ornstein-Uhlenbeck systems and then extended to general additive-noise latent SDEs. Under mild smoothness, the instantaneous drift-Jacobian causal graph is also identifiable up to the same equivalence. A two-stage estimator is proposed for latent disentanglement (and optional graph recovery), with experiments on synthetic systems confirming the identifiability boundary and an application to Hardanger Bridge sensor data.

Significance. If the results hold, this advances causal representation learning by moving identifiability results into continuous-time latent SDE models while removing the sparsity requirements common in discrete-time settings. The use of environment-induced diffusion shifts as the identifying signal, combined with the absence of drift sparsity and the additional graph identifiability result, provides a distinctive theoretical contribution. Empirical confirmation on both synthetic and real trajectories strengthens the practical relevance for time-series disentanglement.

major comments (1)
  1. [nonlinear extension] The extension from the linear OU case to general additive-noise latent SDEs is load-bearing for the central claim. The abstract sketches the strategy but does not detail how the argument carries over; any additional regularity conditions on the drift or diffeomorphism must be stated explicitly to confirm there are no hidden gaps.
minor comments (2)
  1. [Abstract] The precise definition of 'pairwise distinct coordinate-wise variance ratios' should be formalized with an equation or inequality in the main theorem statement to remove any ambiguity.
  2. The two-stage estimator description would benefit from pseudocode or explicit steps showing how the diffusion covariance estimates are used to recover the latent coordinates.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our contribution and for the constructive comment. We address the major comment below.

read point-by-point responses

  1. Referee: [nonlinear extension] The extension from the linear OU case to general additive-noise latent SDEs is load-bearing for the central claim. The abstract sketches the strategy but does not detail how the argument carries over; any additional regularity conditions on the drift or diffeomorphism must be stated explicitly to confirm there are no hidden gaps.

    Authors: We agree that the nonlinear extension is central and that the abstract provides only a high-level sketch. The full argument appears in the main text (following the linear OU result), where we use the fact that the diffeomorphism preserves the additive-noise structure and that the shared drift remains identifiable via the distinct diffusion ratios. The regularity conditions are the mild smoothness assumptions already stated for existence/uniqueness of solutions and applicability of Itô calculus (C² drift, C³ diffeomorphism). In the revision we will insert an explicit remark immediately after the linear theorem that (i) lists these conditions in one place and (ii) gives a concise step-by-step outline of how the linear identifiability argument lifts to the nonlinear setting, thereby removing any ambiguity about hidden gaps. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper advances a mathematical identifiability theorem for latent SDEs under environment-induced diffusion shifts. The central result states that two diagonal diffusion regimes with pairwise distinct per-coordinate variance ratios suffice to identify latent coordinates (up to permutation and scaling) and the instantaneous drift-Jacobian graph, first for linear OU processes and then for general additive-noise SDEs. This identifying condition is stated explicitly as an assumption in the theorem statement rather than derived from or presupposed by the target result. No fitted parameters are renamed as predictions, no self-citation chains are invoked to justify uniqueness, and the proof is presented as self-contained without reduction to prior author work or ansatz smuggling. The absence of sparsity assumptions on the drift is an explicit feature of the stated conditions, not a hidden circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard smoothness and diffeomorphism assumptions plus the specific diagonal-diffusion regime; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The observation map is an unknown nonlinear diffeomorphism.
    Stated in the model setup for both linear and general cases.
  • domain assumption Drift is shared across environments while diffusion covariance is environment-specific and diagonal.
    Central modeling choice that enables the variance-ratio condition.

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Reference graph

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