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Diverse Influence Component Analysis: A Geometric Approach to Nonlinear Mixture Identifiability

Hoang-Son Nguyen, Xiao Fu

TL;DR

DICA introduces a geometry-based framework for identifying latent components from unknown nonlinear mixtures by exploiting diverse influences of latent factors on observed features. The core idea, SDI, captures how gradients of the mixing function distribute across observed dimensions, enabling identifiability when combined with the Jacobian Volume Maximization objective (J-VolMax). The approach yields latent recovery up to a permutation and per-component invertible mappings, without relying on auxiliary signals, independence, or Jacobian sparsity. Empirical results on synthetic nonlinear mixtures and single-cell transcription factor inference demonstrate strong performance, with practical implications for disentangled representation learning and causal inference. Limitations include computational cost of the log-determinant term and theoretical analysis under noise, suggesting directions for scalable optimization and robustness in future work.

Abstract

Latent component identification from unknown nonlinear mixtures is a foundational challenge in machine learning, with applications in tasks such as disentangled representation learning and causal inference. Prior work in nonlinear independent component analysis (nICA) has shown that auxiliary signals -- such as weak supervision -- can support identifiability of conditionally independent latent components. More recent approaches explore structural assumptions, e.g., sparsity in the Jacobian of the mixing function, to relax such requirements. In this work, we introduce Diverse Influence Component Analysis (DICA), a framework that exploits the convex geometry of the mixing function's Jacobian. We propose a Jacobian Volume Maximization (J-VolMax) criterion, which enables latent component identification by encouraging diversity in their influence on the observed variables. Under reasonable conditions, this approach achieves identifiability without relying on auxiliary information, latent component independence, or Jacobian sparsity assumptions. These results extend the scope of identifiability analysis and offer a complementary perspective to existing methods.

Diverse Influence Component Analysis: A Geometric Approach to Nonlinear Mixture Identifiability

TL;DR

DICA introduces a geometry-based framework for identifying latent components from unknown nonlinear mixtures by exploiting diverse influences of latent factors on observed features. The core idea, SDI, captures how gradients of the mixing function distribute across observed dimensions, enabling identifiability when combined with the Jacobian Volume Maximization objective (J-VolMax). The approach yields latent recovery up to a permutation and per-component invertible mappings, without relying on auxiliary signals, independence, or Jacobian sparsity. Empirical results on synthetic nonlinear mixtures and single-cell transcription factor inference demonstrate strong performance, with practical implications for disentangled representation learning and causal inference. Limitations include computational cost of the log-determinant term and theoretical analysis under noise, suggesting directions for scalable optimization and robustness in future work.

Abstract

Latent component identification from unknown nonlinear mixtures is a foundational challenge in machine learning, with applications in tasks such as disentangled representation learning and causal inference. Prior work in nonlinear independent component analysis (nICA) has shown that auxiliary signals -- such as weak supervision -- can support identifiability of conditionally independent latent components. More recent approaches explore structural assumptions, e.g., sparsity in the Jacobian of the mixing function, to relax such requirements. In this work, we introduce Diverse Influence Component Analysis (DICA), a framework that exploits the convex geometry of the mixing function's Jacobian. We propose a Jacobian Volume Maximization (J-VolMax) criterion, which enables latent component identification by encouraging diversity in their influence on the observed variables. Under reasonable conditions, this approach achieves identifiability without relying on auxiliary information, latent component independence, or Jacobian sparsity assumptions. These results extend the scope of identifiability analysis and offer a complementary perspective to existing methods.
Paper Structure (21 sections, 14 theorems, 102 equations, 3 figures, 4 tables)

This paper contains 21 sections, 14 theorems, 102 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Proposed Approach: Diverse Influence Component Analysis
  4. Sufficiently Diverse Influence
  5. Proposed Learning Criterion
  6. Related Works
  7. Numerical Results
  8. Synthetic Data Experiments
  9. Inferring Transcription Factor Activities from Single-cell Gene Expressions
  10. Conclusion
  11. Preliminaries
  12. Notation
  13. Background on Convex Geometry
  14. Proof of Theorem \ref{['thm:infinitethm']}
  15. Proof of Theorem \ref{['thm:finitethm']}
  16. ...and 6 more sections

Key Result

Theorem 3.2

Denote any optimal solution of Problem eq:jvolmax as $(\widehat{\bm \theta}, \widehat{\bm \phi})$. Assume $\widehat{\bm f} =\bm f_{\widehat{\bm \theta}}$ and $\widehat{\bm g} =\bm g_{\widehat{\bm \phi}}$ are universal function representers. Suppose the model in eq:nmi_model and Assumption as:sdi hol in which $\bm \pi$ is a permutation of $\{1,\ldots,d\}$ and $\rho_i(\cdot):\mathbb{R}\rightarrow \m

Figures (3)

  • Figure 1: [Left]$\boldsymbol{s}$, $\boldsymbol{x}$, and $\nabla f_i(\boldsymbol{s})\in\mathbb{R}^d$ for $d=2$, $\forall i\in[m]$; line thickness indicates the magnitude of influence of $s_i$ on $x_j$. [Middle] Condition 1 in Assumption \ref{['as:sdi']} for $d=2$: axes represent $\partial f_i/\partial s_1$ and $\partial f_i/\partial s_2\in\mathbb{R}$; the pink region is ${\rm conv}\{\nabla f_1(\boldsymbol{s}),\ldots,\nabla f_m(\boldsymbol{s})\}$, the dashed purple ellipse is $\mathcal{E}(\mathcal{B}_{1}^{\boldsymbol{w}(\boldsymbol{s})})$, and the solid black diamond is $\mathcal{B}_{1}^{\boldsymbol{w}(\boldsymbol{s})}$. [Right] Condition 2 in Assumption \ref{['as:sdi']}: shaded region shows ${\rm conv}\{\nabla f_1(\boldsymbol{s}),\ldots,\nabla f_m(\boldsymbol{s})\}^{*}$, dashed ellipse shows $\mathcal{E}(\mathcal{B}^{\boldsymbol{w}(\boldsymbol{s})})^{}$, and solid rectangle shows $({\cal B}_{1}^{\boldsymbol{w}(\boldsymbol{s})})^\ast = {\cal B}_{\infty}^{\boldsymbol{w}(\boldsymbol{s})}$.
  • Figure 2: Heatmap of $R^2$ scores between estimated components and ground-truth mRNA concentrations of TFs.
  • Figure 3: Some resulting images obtained by varying a certain component $s_i$ by $\pm4~{\rm std}$ (increasing from left to right) from the latent vector of an anchor image. Each row corresponds to one of $10$ different anchor images sampled from test set, and each column is the resulting image by varing from the corresponding anchor image. We can see that some latent components correlate to the semantic meaning (i.e., digit) of output images: as some $s_i$ increases/decreases, the semantic digit of all $10$ anchor images change uniformly towards another digit.

Theorems & Definitions (22)

  • Theorem 3.2: Identifiability of J-VolMax
  • Theorem 3.3: Identifiability under Finite-sample SDI
  • Proposition A.1
  • proof
  • Proposition A.2
  • Proposition A.3
  • Proposition A.4
  • Proposition A.4
  • Lemma B.1
  • proof
  • ...and 12 more