Identifiability of Potentially Degenerate Gaussian Mixture Models With Piecewise Affine Mixing

• URL: http://arxiv.org/abs/2604.13218v1
• Date: Tue, 14 Apr 2026 18:39:08 GMT
• Title: Identifiability of Potentially Degenerate Gaussian Mixture Models With Piecewise Affine Mixing
• Abstract summary: Causal representation learning aims to identify the underlying latent variables from high-dimensional observations.<n>We study this problem for latent variables that follow a potentially degenerate Gaussian mixture distribution.<n>We propose a two-stage method to estimate the latent variables by enforcing sparsity and Gaussianity in the learned representations.
• Score: 11.762466703930512
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: Causal representation learning (CRL) aims to identify the underlying latent variables from high-dimensional observations, even when variables are dependent with each other. We study this problem for latent variables that follow a potentially degenerate Gaussian mixture distribution and that are only observed through the transformation via a piecewise affine mixing function. We provide a series of progressively stronger identifiability results for this challenging setting in which the probability density functions are ill-defined because of the potential degeneracy. For identifiability up to permutation and scaling, we leverage a sparsity regularization on the learned representation. Based on our theoretical results, we propose a two-stage method to estimate the latent variables by enforcing sparsity and Gaussianity in the learned representations. Experiments on synthetic and image data highlight our method's effectiveness in recovering the ground-truth latent variables.

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