# Identifiability of Potentially Degenerate Gaussian Mixture Models With Piecewise Affine Mixing Source: [Identifiability of Potentially Degenerate Gaussian Mixture Models With Piecewise Affine Mixing]() Collection index: [llms.txt]() Data updated: 2026-04-17T05:00:04.440Z Authors: Danru Xu, S\\'ebastien Lachapelle, Sara Magliacane Published: 2026-04-14T18:39:08.000Z ## 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. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. - [Causal Representation Learning on Synthetic Numerical Experiments](): Causal Representation Learning; Synthetic Numerical Experiments; R^2 Stage 1: 0.99. ## Original sources and implementation [Original paper]() [PDF]()