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Self-sufficient Independent Component Analysis via KL Minimizing Flows

Song Liu

TL;DR

This work introduces Self-sufficient Independent Component Analysis (SICA), a non-linear ICA framework that enforces a self-sufficiency density factorization and minimizes a conditional KL divergence returned by flow-based de-mixing transforms. By using iterative KL minimization with either Wasserstein gradient flows or rectified flows, SICA learns invertible de-mixing functions without requiring priors, likelihoods, or adversarial training. The approach is extended to sequence data via a time-indexed sufficiency assumption and demonstrated on autoregressive signals and MNIST, where it outperforms several nonlinear ICA baselines, especially under nonlinear mixing. The results suggest that KL-minimizing flows offer a robust, flexible path to disentangling nonlinear mixtures in both synthetic and real-world data, with potential for broader application in time-series and image separation.

Abstract

We study the problem of learning disentangled signals from data using non-linear Independent Component Analysis (ICA). Motivated by advances in self-supervised learning, we propose to learn self-sufficient signals: A recovered signal should be able to reconstruct a missing value of its own from all remaining components without relying on any other signals. We formulate this problem as the minimization of a conditional KL divergence. Compared to traditional maximum likelihood estimation, our algorithm is prior-free and likelihood-free, meaning that we do not need to impose any prior on the original signals or any observational model, which often restricts the model's flexibility. To tackle the KL divergence minimization problem, we propose a sequential algorithm that reduces the KL divergence and learns an optimal de-mixing flow model at each iteration. This approach completely avoids the unstable adversarial training, a common issue in minimizing the KL divergence. Experiments on toy and real-world datasets show the effectiveness of our method.

Self-sufficient Independent Component Analysis via KL Minimizing Flows

TL;DR

This work introduces Self-sufficient Independent Component Analysis (SICA), a non-linear ICA framework that enforces a self-sufficiency density factorization and minimizes a conditional KL divergence returned by flow-based de-mixing transforms. By using iterative KL minimization with either Wasserstein gradient flows or rectified flows, SICA learns invertible de-mixing functions without requiring priors, likelihoods, or adversarial training. The approach is extended to sequence data via a time-indexed sufficiency assumption and demonstrated on autoregressive signals and MNIST, where it outperforms several nonlinear ICA baselines, especially under nonlinear mixing. The results suggest that KL-minimizing flows offer a robust, flexible path to disentangling nonlinear mixtures in both synthetic and real-world data, with potential for broader application in time-series and image separation.

Abstract

We study the problem of learning disentangled signals from data using non-linear Independent Component Analysis (ICA). Motivated by advances in self-supervised learning, we propose to learn self-sufficient signals: A recovered signal should be able to reconstruct a missing value of its own from all remaining components without relying on any other signals. We formulate this problem as the minimization of a conditional KL divergence. Compared to traditional maximum likelihood estimation, our algorithm is prior-free and likelihood-free, meaning that we do not need to impose any prior on the original signals or any observational model, which often restricts the model's flexibility. To tackle the KL divergence minimization problem, we propose a sequential algorithm that reduces the KL divergence and learns an optimal de-mixing flow model at each iteration. This approach completely avoids the unstable adversarial training, a common issue in minimizing the KL divergence. Experiments on toy and real-world datasets show the effectiveness of our method.

Paper Structure

This paper contains 21 sections, 3 theorems, 50 equations, 4 figures, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. ICA AND SELF-SUFFICIENT ICA
  3. Independent Component Analysis
  4. Sufficiently Independent Component Analysis
  5. Sequence SICA
  6. LEARNING DE-MIXING FLOWS
  7. Minimizing KL using Wasserstein Gradient Flow
  8. Minimising KL using Rectified Flow
  9. EXPERIMENTS
  10. Autoregressive Signals
  11. MNIST Images
  12. CONCLUSIONS
  13. MISSING PROOFS
  14. Proof of Equation \ref{['eq.aica']}
  15. Proof of Theorem \ref{['thm.kl.reducing']}
  16. ...and 6 more sections

Key Result

Theorem 3.1

If ${\bm{l}}$ is invertible as defined above, the KL divergence $D_{\mathrm{KL}}^{(j)}\left({\bm{l}}^{(j)}\right)$ at the end of each iteration is non-increasing, i.e.,

Figures (4)

  • Figure 1: Two dependent signals mixed by a linear mixing function. In this case, both a linear ICA (LICA, suzuki_least-squares_2011) and a non-linear ICA (iVAE khemakhem_variational_nodate) fail to de-mix the signals (the heart is still tilted), whereas the proposed method, SICA, successfully de-mixes the two signals.
  • Figure 2: Left: Graphical models of ICA, Auxiliary Variable ICA and SICA. Right: SICA assumption on Sequence data.
  • Figure 3: AR (7) dataset. Left: Illustrative Example. Middle and Right, MCC over various mixing steps. The higher the better. MCC is measured over 20 independent runs, and error bars represent the standard error.
  • Figure 4: MNIST dataset. Left: Illustrative Example. Middle and Right: MCC over various mixing steps. MCC are measured over 20 independent runs and error bars represent the standard error.

Theorems & Definitions (7)

  • Theorem 3.1
  • Theorem 3.3
  • proof
  • proof
  • proof
  • Lemma A.1
  • proof