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Towards Identifiable Representations under Misspecified Structure

Yuke Li, Yujia Zheng, Ziyi Chen, Kun Zhang, Heng Huang

arXiv:2609.33273v1cs.LG

TL;DR

The paper addresses identifiability when latent-dependent noise violates the conditional-independence assumption used in prior results. It models structural misspecification as a perturbation, establishes precise and approximate identification guarantees, and reports improved latent recovery in experiments.

  • Problem

    Latent-dependent noise makes identifiability more challenging, while prior results assume observations are conditionally independent given the latent variables.

    Source: §1 · p. 1

  • Method

    The paper models structural misspecification as a perturbed factor analysis problem and develops precise and approximate identifiability guarantees alongside an unsupervised variational estimator.

    Source: §1 · pp. 1–2

  • Results

    The method achieves the highest MCC for component-wise recovery and the highest R2 in the approximate-identifiability regime, with experiments reporting improved latent recovery under misspecified structure.

    Source: §1 · p. 2, §5.2 · pp. 9–10

  • Takeaways & Limitations

    Experiments separately support the paper’s precise component-wise and approximate subspace-identification regimes under misspecified structure.

    Source: §1 · p. 2

    Approximate component-wise identifiability requires additional component-level information and is left for future work.

    Source: §3.3 · p. 6

Abstract

from arXiv · show

The presence of noise that depends on the latent variables poses a fundamental challenge to identifiability. Existing results rely on conditional independence among the observations given the latent variables. We study a more general \emph{misspecified structure}, where this conditional factorization does not hold, and establish both precise and approximate identifiability guarantees. We characterize structural misspecification as a perturbed factor analysis problem. For precise identifiability, we establish subspace identifiability under spectral separation and controlled perturbation, followed by component-wise identifiability under structural sparsity. When the precise condition is not guaranteed, we derive an approximate subspace-identifiability theorem. Based on these results, we develop an unsupervised variational estimator for recovering latent variables. Experiments demonstrate the effectiveness of the proposed framework.

1 INTRODUCTION

Identifiability becomes harder when noise depends on latent variables, while existing approaches rely on conditional independence or additional assumptions that may not hold in realistic systems. This work treats the resulting misspecified structure as a perturbed factor analysis problem and develops precise and approximate identifiability guarantees alongside an unsupervised estimator.

1 INTRODUCTION

  • Prior approaches either treat noise as known auxiliaries, require environment-based invariant/variant partitions, assume noise-latent independence, or require sufficiently distinct latent-variable relations.These are the approaches and assumptions summarized in the introduction.Source: §1 · p. 1
  • Dependent noise can leave observations conditionally dependent given the latent state, violating an assumption used by prior identifiability results.The paper illustrates this with lung-cancer state and inspiration level both affecting multiple chest X-ray regions.Source: §1 · p. 1
  • Structural misspecification is characterized as a perturbed factor analysis problem, providing the basis for complementary precise and approximate identifiability regimes.Precise subspace identifiability requires perturbation sufficiently small relative to spectral separation; component-wise identifiability additionally relies on structural sparsity.Source: §1 · p. 1
  • When the precise-identifiability condition is not guaranteed, the framework bounds identification error between the recovered representation and its admissible equivalence class.The authors describe this as one of the first general frameworks for latent-variable recovery under misspecified structure with identifiability guarantees.Source: §1 · p. 2
  • The proposed unsupervised variational estimator uses the same learning objective in both regimes, and experiments report improved latent recovery under misspecified structure.The experiments separately evaluate precise component-wise identification and approximate subspace identification.Source: §1 · p. 2

2 PROBLEM SETTING

The problem setting seeks to recover latent variables from observations when the usual conditional factorization across observation blocks fails. It defines subspace and component-wise identifiability as distinct recovery targets.

2 PROBLEM SETTING

  • The observation x is K-dimensional and the latent variable z is N-dimensional; the goal is to identify z from observed x.The data-generating process allows a smooth noise function to depend on z through an exogenous variable.Source: §2 · p. 2
  • Subspace identifiability allows an invertible transformation of z, whereas component-wise identifiability requires each latent coordinate to match a unique estimated coordinate up to permutation.The component-wise definition permits a coordinate-specific transformation after matching.Source: §2 · p. 2
  • The paper presents subspace-identifiability results first, followed by a component-wise identifiability result.The two targets are treated in separate stages.Source: §2 · p. 2
  • Misspecified structure means the three observation blocks do not factorize conditionally on z, unlike the conditional-independence structure used in prior identification results.The paper partitions x into three disjoint blocks and notes that this factorization can fail under misspecification.Source: §2 · p. 2

3 IDENTIFIABILITY RESULTS

The section models the failed conditional-independence decomposition as a perturbed operator factorization, then develops identifiability guarantees across perturbation regimes. Precise subspace recovery relies on spectral separation, while component-wise recovery adds structural sparsity.

3 IDENTIFIABILITY RESULTS

  • The analysis corrects the invalid conditional-independence decomposition with a perturbation operator whose upper bound quantifies structural misspecification.Equation 5 provides the basis for the precise and approximate identifiability results.Source: §3 · pp. 3–4

3.1 PRECISE SUBSPACE IDENTIFIABILITY

  • When the misspecification perturbation is below the spectral separation threshold, Theorem 1 establishes precise subspace identifiability.The result identifies the latent representation through an invertible, differentiable transformation.Source: §3.1 · pp. 4–5

3.2 APPROXIMATE SUBSPACE IDENTIFIABILITY

  • When the precise condition is not guaranteed, Theorem 2 bounds approximate identification error under the stated operator and latent-space assumptions.The bound connects spectral perturbation to latent-space discrepancy through an inverse-Lipschitz condition.Source: §3.2 · pp. 5–6

3.3 PRECISE COMPONENT-WISE IDENTIFIABILITY

  • Approximate component-wise identifiability is left for future work; the component-wise result assumes independent latent dimensions, with dependence alternatively allowed through a known auxiliary domain variable.The alternative is discussed in Proposition 3.Source: §3.3 · pp. 6–7
  • Under independent latent coordinates and structural sparsity, Proposition 2 yields component-wise recovery up to permutation.The result builds on subspace identifiability and rules out transformations that mix multiple latent coordinates.Source: §3.3 · p. 6

4 APPROACH

The approach learns latent variables and dependent noise jointly through a variational model that reconstructs observations. Its objective combines reconstruction, KL regularization, and a sparsity penalty.

4 APPROACH

  • The estimator learns the joint data-generating model with variational inference because the latent variables are unobserved and p(x|z) cannot be optimized directly.It introduces separate encoders for z and ϵ from observed x.Source: §4 · p. 7

4.1 NETWORK DESIGN

  • Separate encoders infer z and ϵ, and a decoder uses both estimates to reconstruct x while KL regularization matches posterior distributions to learned priors.The noise prior pγ(ϵ|z) uses a Gaussian transformed by an invertible normalizing flow.Source: §4.1 · p. 7, Figure 2 · p. 7

4.2 TRAINING OBJECTIVE

  • The training objective combines reconstruction and KL terms with an ℓ1 sparsity regularizer, used as a differentiable surrogate for the ℓ0 norm.The objective does not require knowledge of the theoretical observation partitions.Source: §4.2 · p. 8

5 EXPERIMENTS

Experiments assess precise component-wise recovery and approximate subspace recovery using separate synthetic datasets and metrics. The method ranks highest among compared approaches in both regimes, with ablations indicating benefits from modeling dependent noise and structural sparsity.

5.1 EXPERIMENTAL SETUP

  • Separate synthetic datasets evaluate precise component-wise recovery with MCC and approximate subspace recovery with aligned R2.Approximate component-wise identifiability is left for future work; the comparisons include representative identification approaches and ablations.Source: §5.1 · pp. 8–9

5.2 RESULTS AND DISCUSSIONS:

  • Our method achieves the highest MCC among compared approaches, and its estimated coordinates show clear one-to-one correspondence with true coordinates.Figure 3(a) visualizes coordinate correlations; Figure 3(b) compares MCC scores.Source: §5.2 · p. 9, Figure 3 · p. 9
  • Removing latent-dependent noise modeling or structural-sparsity regularization lowers MCC in the precise-identifiability experiment.IndVAE also performs substantially worse because its conditional-independence assumption is violated by the synthetic data.Source: §5.2 · p. 9
  • Our method achieves the highest R2 among compared approaches in the approximate-identifiability regime.The reported R2 indicates the estimated representation’s correspondence with the true latent variables.Source: §5.2 · p. 10
  • Removing latent-dependent noise modeling or structural sparsity also lowers R2 in the approximate-identifiability experiment.The comparison reports improved latent recovery over IndVAE, MCRL, Beta-VAE, iVAE, and SLOW-VAE outside the precise-identifiability regime.Source: §5.2 · p. 10

AI USE STATEMENT

The authors used generative AI tools only for language editing and reviewed the resulting text. The notation appendix lists symbols for variables, operators, functions, and model parameters.

AI USE STATEMENT

  • Generative AI tools assisted only with language editing, and the authors reviewed and revised all AI-assisted text.The authors state that the tools were not used for research ideas, theoretical results, experimental design, or scientific conclusions.Source: AI USE STATEMENT · p. 10

A NOTIONS

  • The notation defines symbols for observations, reconstructions, latent variables, and dependent noise and auxiliary variables.It also distinguishes true quantities from their estimates.Source: A NOTIONS · p. 15
  • The notation also specifies symbols for operators, mixing functions, Jacobians, posterior and prior parameters, and the decoder.It identifies the ℓ1 norm used in the paper.Source: A NOTIONS · p. 15

B.1 PREVIOUS RESULTS

The previous-results section derives an operator relation using conditional-density factorization and injectivity assumptions. Distinct conditional distributions yield latent identifiability up to an invertible transformation.

B.1 PREVIOUS RESULTS

  • The three-block partition is an existential structural condition; learning and inference require neither partition labels nor separate access to the blocks.The blocks are x′a, x′b, and x′c.Source: §B.1 · p. 15
  • Under the stated assumptions, the latent variable z′ is identified up to an invertible transformation h′.The assumptions include bounded joint and derived densities and injectivity of specified operators.Source: §B.1 · pp. 15–16
  • The derivation produces an operator equation whose left side uses only observable variables while its right side explicitly depends on z′.The stated step uses injectivity of an operator.Source: §B.1 · p. 16
  • Distinct conditional distributions across z′ give distinct eigenvalues, allowing a bijective permutation while preserving their values.The resulting latent recovery is therefore only up to an invertible transformation.Source: §B.1 · p. 16

B.2 PROOF OF PROPOSITION 1 AND THEOREM 1

The proof treats structural misspecification as a perturbation and establishes precise subspace identification when the perturbation is controlled relative to spectral separation. When that condition is unavailable, it bounds approximate identification error; the perturbation discussion gives a sufficient condition for the precise regime.

B.2 PROOF OF PROPOSITION 1 AND THEOREM 1

  • When the perturbation is sufficiently small relative to spectral separation, Theorem 1 establishes precise subspace identifiability.The proof constructs an operator decomposition with a perturbation term and uses spectral projections to recover the latent subspace.Source: §B.2 · pp. 17–18

B.3 PROOF OF THEOREM 2

  • When the precise-identification condition is not guaranteed, Theorem 2 bounds the error between the recovered representation and its admissible equivalence class.The bound is expressed in terms of the perturbation and a continuity constant for the conditional-distribution map.Source: §B.3 · pp. 19–20

B.4 DISCUSSIONS ON Per

  • The discussion derives a sufficient condition for the perturbation norm to remain below the threshold required for precise identifiability.The proof uses an operator-norm bound to establish the condition.Source: §B.4 · p. 20

B.5 PROOF OF PROPOSITION 2

Proposition 2 uses structural sparsity to reduce the invertible ambiguity left by Theorem 1 to component-wise transformations and permutation. An additional proposition gives a route to component-wise identification when latent dimensions are conditionally independent given a sufficiently variable domain.

B.5 PROOF OF PROPOSITION 2

  • Under Theorem 1 and the stated independence and structural-sparsity conditions, Proposition 2 identifies latent components up to permutation and component-wise invertible transformations.The proof uses the support structure of the Jacobian to rule out transformations that mix latent coordinates.Source: §B.5 · p. 21

B.6 ADDITIONAL PROPOSITION

  • Proposition 3 shows identifiability in the non-i.i.d. scenario: domain-dependent variation in p(z | u) can reduce the remaining invertible ambiguity to permutation and component-wise transformations of the latent coordinates.The sufficient-variability condition requires 2N + 1 distinct domain values u in [1, 2N + 1], with the 2N vectors w(z, u) − w(z, u0), for u ≠ u0, linearly independent.Source: §B.6 · p. 22

C.1 SYNTHETIC DATA-GENERATING PROCESS

The synthetic construction creates dependent noise and shared disturbance across three observation blocks, then varies its strength to evaluate precise and approximate regimes. The experiments also specify the training setup and compare the approach on MSMT17, where it surpasses the reported person-classification baselines.

C.1 SYNTHETIC DATA-GENERATING PROCESS

  • The synthetic process uses three disjoint three-dimensional observation blocks; each has noise ϵv = Cvz + τη0 + svηv, with τ > 0 and sv > 0, and η0 shared across blocks.The process samples 10,000 points, using 90% for training and 10% for testing.Source: §C.1 · pp. 23–24

C.2 ASSUMPTION JUSTIFICATIONS:

  • Small nonzero τ yields the precise regime, while larger τ values violate the sufficient perturbation condition and are used to evaluate approximate identifiability.The construction keeps the remaining assumptions of Theorem 2 unchanged for the approximate-regime evaluation.Source: §C.2 · pp. 24–25

C.3 IMPLEMENTATIONS & TRAINING DETAILS.

  • Synthetic models use N=3 and K=9, single-layer MLP modules, AdamW, learning rate 1 × 10^-3, batch size 64, and 10 random seeds.Reported performance is the mean ± standard deviation across runs; β1 = β2 = 0.02.Source: §C.3 · p. 26

C.4 THE OBJECTIVE OF INDVAE

  • The IndVAE baseline factors the joint log-likelihood into observation, noise-given-latent, and latent-prior terms under its conditional-independence setup.The section derives its loss from that factorization.Source: §C.4 · p. 26

D.1 REAL-WORLD EXPERIMENT FOR JUSTIFYING PROPOSITION 2

  • For person classification, inferred z is passed to an MLP classifier to predict each person’s one-hot identity label.The setup represents person identity as z and pose, gait, or clothing variation as dependent noise.Source: §D.1 · pp. 27–28
  • On MSMT17, the method surpasses IndVAE, MCRL, GTL, CLIPReID, TransReID, and AGW in reported Top-1 accuracy.Table 2 reports the comparison on the person-classification task.Source: §D.1 · p. 28, Table 2 · p. 28

D.2 ADDITIONAL ABLATION STUDIES

The additional experiments report that the method retains higher MCC than IndVAE as synthetic latent dimension increases and that the default regularization setting performs best. On domain-adaptation benchmarks, it leads across Office-Home transfer directions and most VisDA-2017 categories, with stated exceptions and average accuracies.

D.2 ADDITIONAL ABLATION STUDIES

  • As synthetic latent dimension increases across N ∈ {8, 12, 18}, the method consistently achieves higher MCC than IndVAE.The authors report that Jacobian-based sparsity regularization remains effective within the considered regime.Source: §D.2 · pp. 28–29, Table 3 · p. 29
  • The default hyperparameters (λ, β1, β2) = (1, 0.02, 0.02) obtain the best synthetic-dataset MCC among the tested settings.The sensitivity analysis varies one of the three hyperparameters at a time while keeping the others fixed.Source: §D.2 · p. 29, Table 4 · p. 29
  • On MSTM-17, a single-layer GRU reaches 96.9 ± 0.4 Top-1 accuracy versus 96.2 ± 0.6 for the original MLP model.The classifier architecture and training and evaluation protocols remain unchanged in this comparison.Source: §D.2 · p. 29

D.3 ADDITIONAL REAL-WORLD EXPERIMENTS FOR PROPOSITION 3

  • The method averages 80.17% accuracy across Office-Home transfer directions, versus 77.88% for UniMoS, and leads on all 12 directions.It improves over UniMoS by 4.6 percentage points on A →C and 3.3 points on A →P.Source: §D.3 · p. 30, Table 7 · p. 31
  • On VisDA-2017, average accuracy is 90.46% versus UniMoS at 88.05%; the method leads on 10 of 12 categories and ties on Motorcycle.It does not lead on Car: DAPrompt scores 77.9% versus the method’s 76.8%.Source: §D.3 · pp. 30–31, Table 8 · p. 31

E CONCLUSION & LIMITATIONS

The paper provides identifiability guarantees for latent variables under misspecified structure, where observations may remain dependent given the latents. It establishes precise guarantees under controlled perturbation and spectral separation, and approximate subspace identifiability beyond that regime, while noting that the approximate guarantee does not extend to components.

E CONCLUSION & LIMITATIONS

  • Structural misspecification is characterized as a perturbed factor analysis problem for studying identifiability when observations may remain dependent given the latent variables.The paper also notes that noise may itself depend on the latents.Source: E CONCLUSION & LIMITATIONS · p. 31
  • Controlled perturbation and spectral separation yield precise subspace identifiability, with component-wise identifiability further obtained under structural sparsity.Source: E CONCLUSION & LIMITATIONS · p. 31
  • When precise identifiability is not guaranteed, the theory bounds error at the subspace level; approximate component-wise guarantees require additional information about component-level ambiguity.The minimal structural assumptions or auxiliary information needed remain a direction for future work.Source: E CONCLUSION & LIMITATIONS · p. 31

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