Identifiability of Potentially Degenerate Gaussian Mixture Models With Piecewise Affine Mixing

2026-04-16 · Source: cs.AI updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics · Depth: Expert, extended

Summary

This research addresses the identifiability of underlying latent variables from high-dimensional observations, specifically when these variables follow a potentially degenerate Gaussian mixture distribution (pdGMM) and are observed through a piecewise affine mixing function. The authors present a series of progressively stronger theoretical identifiability results, culminating in identifiability up to permutation and scaling, even in challenging scenarios where probability density functions are ill-defined due to degeneracy. A key aspect involves leveraging sparsity regularization on the learned representation. Based on these theoretical findings, a two-stage method is proposed to estimate latent variables by enforcing sparsity and Gaussianity. Experimental evaluations on both synthetic and image datasets, including a "Multiple Balls" scenario, demonstrate the method's effectiveness in recovering ground-truth latent variables, outperforming baselines like VaDE in specified settings.

Key takeaway

For AI Scientists and Research Scientists working on causal representation learning with potentially degenerate latent variables, this work provides a robust theoretical and practical framework. You should consider implementing the proposed two-stage autoencoder method, particularly when dealing with piecewise affine mixing functions and sparse, low-rank latent structures. The method's ability to achieve identifiability up to permutation and scaling, even with ill-defined PDFs, offers a significant advantage for developing interpretable and disentangled representations in complex, high-dimensional data.

Key insights

Identifies degenerate Gaussian mixture latent variables from piecewise affine observations using sparsity regularization for disentanglement.

Principles

• Sparsity regularization enables stronger identifiability for degenerate latent variables.
• Identifiability of pdGMMs can be established from an open subset of their domain.
• Piecewise affine mixing functions can approximate complex nonlinear transformations.

Method

A two-stage autoencoder approach: first, minimize reconstruction error and enforce Gaussianity for affine identifiability; second, apply an inner autoencoder with L1-norm sparsity constraint for permutation and scaling identifiability.

In practice

• Use a two-stage autoencoder for latent variable recovery from complex observations.
• Apply L1-norm regularization to encourage sparsity in learned representations.
• Consider over-parameterizing latent dimensions for empirical robustness.

Topics

• Causal Representation Learning
• Degenerate Gaussian Mixture Models
• Piecewise Affine Mixing
• Identifiability Theory
• Sparsity Regularization

Code references

• danrux/Identi-pdGMM

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by cs.AI updates on arXiv.org.