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Diverse Dictionary Learning

Yujia Zheng, Zijian Li, Shunxing Fan, Andrew Gordon Wilson, Kun Zhang

arXiv:2604.17568v1cs.LGmath.STstat.ML

TL;DR

Recovering latent variables from observational data is generally ill-posed without assumptions that may be unverifiable in practice. This paper introduces diverse dictionary learning, showing that set-theoretic latent structures remain identifiable under basic conditions and can yield full identifiability when dependency diversity is sufficient.

  • Problem

    Existing latent-recovery methods rely on strong assumptions whose validity is rarely verifiable and whose guarantees often fail under mild violations.

  • Method

    Diverse dictionary learning uses set algebra and Jacobian dependency sparsity to identify structured latent supports without parametric constraints or auxiliary supervision.

  • Results

    Intersections, complements, symmetric differences, and latent-to-observed dependency structures are identifiable up to appropriate indeterminacies, with full recovery possible under sufficient structural diversity.

  • Takeaways & Limitations

    Identifiability can provide actionable partial views of hidden processes and a broadly applicable estimation bias across general settings.

  • Takeaways & Limitations

    The guarantees require basic conditions such as invertibility and differentiability, along with sufficient nonlinearity for the formal theory.

Abstract

from arXiv · show

Given only observational data $X = g(Z)$, where both the latent variables $Z$ and the generating process $g$ are unknown, recovering $Z$ is ill-posed without additional assumptions. Existing methods often assume linearity or rely on auxiliary supervision and functional constraints. However, such assumptions are rarely verifiable in practice, and most theoretical guarantees break down under even mild violations, leaving uncertainty about how to reliably understand the hidden world. To make identifiability actionable in the real-world scenarios, we take a complementary view: in the general settings where full identifiability is unattainable, what can still be recovered with guarantees, and what biases could be universally adopted? We introduce the problem of diverse dictionary learning to formalize this view. Specifically, we show that intersections, complements, and symmetric differences of latent variables linked to arbitrary observations, along with the latent-to-observed dependency structure, are still identifiable up to appropriate indeterminacies even without strong assumptions. These set-theoretic results can be composed using set algebra to construct structured and essential views of the hidden world, such as genus-differentia definitions. When sufficient structural diversity is present, they further imply full identifiability of all latent variables. Notably, all identifiability benefits follow from a simple inductive bias during estimation that can be readily integrated into most models. We validate the theory and demonstrate the benefits of the bias on both synthetic and real-world data.

1 INTRODUCTION

Diverse dictionary learning asks what remains identifiable from observational data when full recovery is ill-posed and common assumptions cannot be verified. It guarantees recovery of set-theoretic latent relations and dependency structure, with sufficiently diverse structure enabling full identifiability.

  • Motivation: General dictionary learning models observations as X = f(Z), but recovering the latent generative process is fundamentally ill-posed without additional assumptions.The formulation encompasses independent component analysis, factor analysis, and causal representation learning.
  • Motivation: Prior approaches obtain identifiability by imposing parametric constraints, auxiliary variables, interventions, or counterfactual views whose validity or availability may be difficult to ensure.These strategies respectively constrain the solution space, provide weak supervision, or require control over the data-generating process.
  • Diverse dictionary learning: Diverse dictionary learning instead studies which aspects of the latent process remain recoverable in general settings without specific parametric constraints or auxiliary supervision.The framework is designed to provide meaningful guarantees across a wide range of scenarios and make identifiability actionable.
  • Diverse dictionary learning: Intersections, complements, and symmetric differences of latent variables linked to arbitrary observations, together with latent-to-observed dependency structure, remain identifiable up to appropriate indeterminacies.These guarantees are defined through basic set-theoretic operations and can be composed using set algebra.
  • Diverse dictionary learning: When dependency structure is sufficiently diverse across the full observed-variable set, the framework can recover all latent variables and yields a generalized structural criterion for full identifiability.This result is identified as Theorem 3 in the supplied passage.

2 BACKGROUND AND PROBLEM SETUP

The paper formulates latent-variable recovery in the nonlinear observational model X = g(Z), defines latent–observed dependency through the Jacobian support, and avoids restrictive assumptions used for full nonlinear identifiability.

  • Latent-variable model: Observed variables X and latent variables Z are modeled as supports in R^dx and R^dz, respectively, linked by a hidden generative process.This adopts the standard latent-variable-model perspective in which observations arise from latent variables through an unknown process.
  • Connection to linear dictionary learning: Unlike classical dictionary learning’s linear model X = DZ, the paper studies the nonlinear generative setting X = g(Z).Classical approaches represent observations as linear combinations of dictionary atoms, whereas this task extends the setup nonlinearly.
  • Connection to nonlinear identifiability results: Injectivity of g prevents information loss, but prior nonlinear-identifiability guarantees additionally constrain g or use auxiliary domain, temporal, interventional, or counterfactual information.The paper deliberately targets general real-world settings without relying on these restrictive assumptions.
  • Structure: The dependency structure is defined as the support of g’s Jacobian, identifying which latent variables functionally influence which observed variables.This structure is nonparametric and captures functional rather than statistical dependencies, without requiring latent-variable independence.

3 THEORY

This section defines generalized identifiability through set-theoretic indeterminacy, showing that latent intersections, complements, symmetric differences, and dependency structure remain recoverable under observational equivalence. These guarantees extend to atomic regions and, with sufficient diversity, can yield element-level identifiability.

  • 3.1 Generalized identifiability: Generalized identifiability evaluates local correspondence between latent components influencing specific observed-variable sets rather than requiring global recovery.Latent index sets record which latent variables influence each observed-variable set, while set-theoretic indeterminacy specifies the permissible mismatches between models.
  • 3.2 Implications: Set-theoretic indeterminacy guarantees disentanglement across object-centric, individual-centric, and shared-centric latent regions.These regions correspond to object-specific latents, latents unique to one observed-variable group, and shared latents separated from symmetric-difference latents.
  • 3.2 Implications: If the union of latent index sets covers the full latent space, generalized identifiability extends to every atomic region in the corresponding Venn diagram.Because intersections, complements, and symmetric differences compose through set algebra, they can construct diverse structured views of latent variables.
  • 3.3 Formal guarantees: Under sufficient nonlinearity and the theorem assumptions, observational equivalence implies generalized identifiability, while the latent-to-observed dependency structure is identifiable up to column permutation.Theorem 1 establishes set-theoretic identifiability, and Theorem 2 identifies the support of the Jacobian up to relabeling.
  • 3.3 Formal guarantees: With the additional sufficient-diversity condition, the theory further guarantees element identifiability up to element-wise indeterminacy.Sufficient diversity is more flexible than structural sparsity and can hold in nearly fully connected structures when connectivity patterns vary.

4 EXPERIMENT

Experiments evaluate generalized and element identifiability in nonlinear synthetic settings and assess dependency sparsity across standard disentangled-representation benchmarks. Across most datasets and backbone methods, adding dependency sparsity improves understanding of the hidden world and often outperforms latent sparsity.

  • Setup: The experiments use a variational autoencoder with dependency sparsity regularization, 10, 000 samples, α = β = 0.05, and nonlinear MLP generation processes.The MLPs use Leaky ReLU activations.
  • Generalized Identifiability: Synthetic experiments evaluate generalized identifiability across observed-variable groups using R2, where lower scores indicate more disentanglement.Datasets have dimensionality in {3, 4, 5}; comparisons cover intersections, symmetric differences, and complements, alongside a reference comparing Z and ˆZ.
  • Element Identifiability: Element identifiability experiments test whether multiple variable-pair comparisons recover latent variables up to element-wise indeterminacy using MCC.The proposed method is tested under Sufficient Diversity, while the baseline uses fully dense dependencies.
  • Complex Settings: Benchmark experiments use Cars3D, Shapes3D, and MPI3D, whose known generative factors include color, shape, scale, orientation, and viewpoint.The study incorporates the proposed sparsity loss into FactorVAE, DisCo, and EncDiff, spanning VAE, GAN, and diffusion-model backbones.
  • Latent or dependency sparsity?: Across most datasets and backbone methods, dependency sparsity consistently improves results and often benefits generative models more than latent sparsity.The comparison includes original methods and sparsity-augmented variants; the passage characterizes the finding as support for understanding the hidden world.

5 CONCLUSION

The paper introduces diverse dictionary learning to characterize what aspects of the hidden world remain recoverable under basic conditions and which inductive biases may help universally. Its set-algebra guarantees provide a complementary local perspective and unify structural conditions for full identifiability.

  • 5 CONCLUSION: Diverse dictionary learning studies recoverable aspects of the hidden world and universally beneficial inductive biases under basic conditions.The framework is designed for settings where stronger assumptions may not hold.
  • 5 CONCLUSION: Set-algebra-based guarantees provide a complementary local view to prior identifiability results based on global assumptions.The conclusion positions the guarantees as a local alternative to globally assumed structure.
  • 5 CONCLUSION: The framework unifies existing structural conditions that support full identifiability.This unification is presented as another consequence of the set-algebra guarantees.

Diverse Dictionary Learning Supplementary Material · A PROOFS

The supplied material identifies Table 2 as the notation reference used throughout the paper. No proof claims or additional supplementary findings are included in the provided passage.

  • Diverse Dictionary Learning Supplementary Material: Table 2 presents the notation used throughout the paper.
  • Diverse Dictionary Learning Supplementary Material: The passage identifies Table 2 as a reference for the paper’s notation.
  • Diverse Dictionary Learning Supplementary Material: The supplied excerpt contains a notation table rather than a stated proof result.
  • Diverse Dictionary Learning Supplementary Material: The notation in Table 2 is described as applying throughout the paper.
  • Diverse Dictionary Learning Supplementary Material: The provided passage does not specify individual notation symbols or definitions.
  • A PROOFS: The provided passage does not state any theorem, lemma, or proof argument.

A.1 PROOF OF PROPOSITION 1

Proposition 1 shows that generalized set-identifiability implies disentanglement across latent variables associated with arbitrary observed-variable sets. The guarantee covers object-centric, individual-centric, and complement relationships up to a latent-coordinate permutation.

  • Proposition 1: Under θ ∼set ˆθ, a latent variable Zi cannot be a function of ˆZπ(j) for specified cross-set index pairs, for some permutation π.This is the core implication stated for any two models and observed-variable sets.
  • Object-centric disentanglement: Object-centric disentanglement separates indices shared with one latent set from indices unique to the other.The forbidden pairs are i ∈ IK, j ∈ IV \ IK and symmetrically i ∈ IV, j ∈ IK \ IV.
  • Individual-centric disentanglement: Individual-centric disentanglement separates indices unique to one latent set from every index in the other set.The stated pairs include i ∈ (IK \ IV), j ∈ IV and the symmetric case.
  • Complement disentanglement: Complement disentanglement rules out functional dependence between indices unique to IK and indices unique to IV.This covers both directions between IK \ IV and IV \ IK.
  • Proof: The proof obtains these three cases by combining intersection, symmetric-difference, and complement set-theoretic indeterminacy under the same permutation π.The argument concludes that θ ∼set ˆθ implies all stated goals.

A.2 PROOF OF THEOREM 1

The proof establishes generalized identifiability from observational equivalence under Assumption 1 and positive latent density, showing that latent dependency sparsity patterns correspond up to permutation. It then derives the set-theoretic indeterminacy relations, including complements, and proves shared latent variables are linked by invertible componentwise transformations.

  • Theorem conditions: Under Assumption 1 and positive density of Z, observationally equivalent models satisfy generalized identifiability: θ ∼set ˆθ.The result is stated for models following the process in Section 2, with θ ∼obs ˆθ implying generalized identifiability.
  • Jacobian matching: An invertible reparameterization ϕ links the models’ Jacobians, while linear independence and Hall’s marriage theorem yield a permutation matching their latent coordinates.The proof uses D ˆZˆg = DZgD ˆZϕ−1 and obtains a permutation π with nonzero matched entries.
  • Sparsity correspondence: Every nonzero dependency in one model has a corresponding nonzero dependency in the other, and the relations can be restricted to equivalence of sparsity patterns.This establishes correspondence between the supports of DZg and D ˆZˆg.
  • Set-theoretic cases: The proof analyzes intersections, complements, and symmetric differences of latent index sets to establish the set-theoretic indeterminacy relations.The complement case is explicitly characterized by indices lying in IK \ IV or IV \ IK, while the proof separately treats shared and non-shared variables.
  • Componentwise indeterminacy: For a shared latent variable, invertibility and dependency restrictions imply Zt depends only on ˆZπ(t), so Zt = h(ˆZπ(t)) for an invertible h.The argument then uses independence to constrain variables in the symmetric difference and completes the remaining case analysis.

A.3 PROOF OF THEOREM 2

Under the assumptions of Theorem 1, observationally equivalent models have identical Jacobian support patterns up to a permutation of latent-coordinate columns. The proof derives this through an invertible reparameterization, linear independence, and a perfect matching argument.

  • Theorem 2: Observational equivalence implies that the support of D_z g is preserved in D_ẑ ĝ up to a permutation of column indices.This is the structure-identifiability conclusion of Theorem 2.
  • Proof strategy: The proof relates the two Jacobians through the chain rule, D_ẑ ĝ = D_z g D_ẑ ϕ^-1, where ϕ = ĝ^-1 ◦ g is invertible.The invertibility of ϕ ensures that its inverse exists and that the reparameterization matrix is invertible.
  • Proof strategy: Assumption 1 provides linear independence of Jacobian vectors, allowing nonzero support coordinates in one model to be mapped into the corresponding support span of the other.The constructed matrix M_ϕ transfers each supported coordinate relation between the two Jacobians.
  • Proof strategy: A bipartite graph built from the invertible reparameterization matrix admits a perfect matching by Hall’s marriage theorem, yielding a coordinate permutation.Invertibility gives linearly independent rows and the matching identifies a nonzero entry for each matched coordinate.
  • Conclusion: The matching and support-span relations show that every nonzero Jacobian element has a corresponding nonzero element in the other model, establishing support-pattern equivalence.The result is then expressed using a permutation matrix P.

A.4 PROOF OF THEOREM 3 … B.6 GENERAL NOISE AND NON-INVERTIBILITY

The paper proves element-wise identifiability under Theorem 3’s assumptions and illustrates how diverse set relations disentangle all atomic Venn-diagram regions. It further connects dependency sparsity to nonlinear identifiability, discusses practical scaling and limitations under noise or non-invertibility.

  • A.4 PROOF OF THEOREM 3: Theorem 3 establishes identifiability up to element-wise indeterminacy when the assumptions of Theorem 1 and Assumption 2 hold.The proof uses all three conditions in Assumption 2 and concludes the stated identifiability result.
  • B.1 THE VENN DIAGRAM.: Each atomic region in the three-variable Venn diagram is disentangled from all other latent indices by selecting observed-variable pairs satisfying the set-theoretic conditions.Under invertibility, covering every index outside an atomic region yields block-wise identifiability.
  • B.2 FURTHER DISCUSSION ON THE CONNECTION OF CONDITIONS.: Structural diversity extends sparsity ideas to the nonlinear setting, while injectivity serves as the nonlinear analogue of linear Restricted Isometry Property conditions.The discussion distinguishes diversity from sparsity and explains that injectivity prevents latent information from being lost through the generative map.
  • B.3 FURTHER DISCUSSION ON THE CONNECTION WITH SAES.: Unlike Sparse Autoencoders, diverse dictionary learning targets dependency sparsity through Jacobian sparsity and provides identifiability guarantees for nonlinear generative models.The paper argues that latent sparsity can cause feature splitting and absorption, whereas dependency sparsity addresses these limitations.
  • B.4 FURTHER DISCUSSION ON SUFFICIENT NONLIENARITY: Sufficient nonlinearity is often mild because each observed variable requires only as many samples as the number of latent variables influencing it.For smooth g and continuous latent densities, independently sampled Jacobian rows are generally in position; five influencing coordinates typically require roughly five samples.
  • B.5 DEPENDENCY SPARSITY REGULARIZATION ON LARGE MODELS: Large-model dependency sparsity regularization can be made practical by restricting Jacobian computation to active latent coordinates and using efficient closed-form Jacobian expressions.These strategies reduce computation and memory, and exploit factorizations available in residual-attention and feedforward architectures.
  • B.5 DEPENDENCY SPARSITY REGULARIZATION ON LARGE MODELS: Training with dependency sparsity regularization is reported to be only about twice as slow as training with standard ℓ1 latent regularization.The comparison is attributed to Farnik et al. (2025).
  • B.6 GENERAL NOISE AND NON-INVERTIBILITY: General noise complicates identifiability by entangling noise and latent variables, while partial non-invertibility can destroy information that generation cannot recover without additional assumptions.Temporal information combined with sufficiently changing nonstationary transitions is proposed as one way to address partial non-invertibility.

B.7 FURTHER DISCUSSION ON POTENTIAL IMPACT. · C ADDITIONAL EXPERIMENTS

The paper frames identifiability as a foundation for truthful and efficient learning, with applications spanning interpretability, transfer, controllable generation, multimodal alignment, and scientific discovery. Additional experiments on synthetic and real-world data further examine these implications.

  • B.7 FURTHER DISCUSSION ON POTENTIAL IMPACT.: Recovering the true generative process enables principled, domain-agnostic inductive biases for predictive and non-predictive tasks.These insights can be embedded in architectures, training objectives, and evaluation protocols.
  • B.7 FURTHER DISCUSSION ON POTENTIAL IMPACT.: Without identifiability, observationally equivalent models may fit distributions perfectly while failing to recover the underlying data-generating process.This limitation affects transfer learning, controllable generation, compositional generalization, and mechanistic interpretability.
  • B.7 FURTHER DISCUSSION ON POTENTIAL IMPACT.: Recovering essential generative factors improves efficiency by avoiding irrelevant information, arbitrary noise, and entanglement with unrelated latents.The goal is to model only representations relevant to the task rather than the entire high-dimensional space.
  • B.7 FURTHER DISCUSSION ON POTENTIAL IMPACT.: Diverse Dictionary Learning addresses Sparse Autoencoder limitations by enforcing Jacobian dependency sparsity and supporting nonlinear generative processes with identifiability guarantees.Sparse Autoencoders are described as linear and as requiring extremely high-dimensional latent vectors to capture real-world concepts.
  • B.7 FURTHER DISCUSSION ON POTENTIAL IMPACT.: Generalized identifiability results recover shared and private latent parts, supporting disentanglement of invariant content from changing styles in transfer learning.This directly targets reliable and efficient domain adaptation.
  • B.7 FURTHER DISCUSSION ON POTENTIAL IMPACT.: Identifiability supports controllable generation by helping models leverage true causation rather than correlations, such as adding eyeglasses without unintentionally increasing a child’s age.The passage contrasts correlation-based prediction with precise control guided by the underlying process.
  • B.7 FURTHER DISCUSSION ON POTENTIAL IMPACT.: The framework extends naturally to multimodal alignment and scientific discovery by treating shared concepts and hidden causes as latent variables recoverable from observations.Examples include modality-specific concepts such as image texture and audio volume, and the formulation X = f(Z).
  • C ADDITIONAL EXPERIMENTS: Further experiments evaluate the approach on both synthetic and real-world data.This section presents additional experiments beyond the applications discussed above.

C.1 ADDITIONAL SYNTHETIC EXPERIMENTS

Additional synthetic experiments show that dependency sparsity improves latent recovery over Jacobian/Hessian regularizers, remains robust to additive noise, and is stable across regularization strengths. These benefits arise from using sparsity as an estimation bias rather than assuming the data-generating process itself is sparse.

  • Additional baselines: Dependency sparsity outperforms OroJAR and the Hessian Penalty on MCC for d ∈ {3, 4, 5}, with the gap widening as dimensionality increases.Neither alternative regularizer provides identifiability guarantees in the nonparametric setting.
  • Noise robustness: Under additive noise, MCC remains essentially unchanged with only minor drops, whereas the base model degrades sharply.The result supports dependency sparsity as a stabilizing bias for latent recovery under noise.
  • Regularization weight: MCC increases steadily from λ = 0 and plateaus around λ ∈ [0.03, 0.05], indicating stability beyond the under-regularized regime.The regularization weight is not overly sensitive once it passes the under-regularized range.
  • Regularization weight: Sparsity functions only as an inductive bias during estimation, while the theory relies on structural diversity rather than sparse data-generating processes.The data-generating process itself need not be sparse for the theory to apply.

C.2 ADDITIONAL VISUAL EXPERIMENTS

Visual experiments show that dependency sparsity improves disentanglement metrics while preserving training stability and scaling to larger images. Qualitative traversals and latent swaps further demonstrate interpretable, localized control of semantic factors with minimal interference.

  • Scalability: At 128 × 128 resolution, Cars3D performance remains consistent with the 64 × 64 setting, supporting robust scaling from structural regularization.The scalability experiment reran FactorVAE with dependency sparsity after upsampling Cars3D.
  • Quantitative evaluation: Dependency sparsity improves Cars3D FactorVAE from 0.708 to 0.752 and DCI from 0.135 to 0.144, outperforming latent sparsity.On Cars3D, dependency sparsity improves both reported metrics; Table 8 extends the comparison to MPI3D with OroJAR and the Hessian Penalty.
  • Quantitative evaluation: Dependency sparsity gives the best results on Cars3D and MPI3D while maintaining backbone training stability.The evaluation reports improvements in both FactorVAE and DCI across these datasets.
  • Qualitative evaluation: Flow traversals on Fashion align individual latent coordinates with gender, heel height, and upper width, with minimal interference across factors.These qualitative results test whether dependency sparsity yields more interpretable and disentangled visual representations.
  • Qualitative evaluation: EncDiff traversals and swaps isolate semantic factors across Shapes3D, Cars3D, and MPI3D, enabling localized edits without unintended side effects.Single-factor swaps control floor or wall color, azimuth or color, and rotation or background, respectively.
  • Qualitative evaluation: Together, the results connect improved disentanglement scores with interpretability, practical usability, single-attribute manipulation, and semantically meaningful latent arithmetic.The experiments are presented as empirical validation of the theory’s identifiability benefits.
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