Identifiability of Potentially Degenerate Gaussian Mixture Models With Piecewise Affine Mixing

Danru Xu, Sebastien Lachapelle, Sara Magliacane
Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, PMLR 300:1252-1260, 2026.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v300-xu26c, title = { Identifiability of Potentially Degenerate Gaussian Mixture Models With Piecewise Affine Mixing }, author = {Xu, Danru and Lachapelle, Sebastien and Magliacane, Sara}, booktitle = {Proceedings of The 29th International Conference on Artificial Intelligence and Statistics}, pages = {1252--1260}, year = {2026}, editor = {Khan, Emtiyaz and Li, Yingzhen and Solin, Arno and Ramdas, Aaditya}, volume = {300}, series = {Proceedings of Machine Learning Research}, month = {02--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v300/main/assets/xu26c/xu26c.pdf}, url = {https://proceedings.mlr.press/v300/xu26c.html}, 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. } }
Endnote
%0 Conference Paper %T Identifiability of Potentially Degenerate Gaussian Mixture Models With Piecewise Affine Mixing %A Danru Xu %A Sebastien Lachapelle %A Sara Magliacane %B Proceedings of The 29th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2026 %E Emtiyaz Khan %E Yingzhen Li %E Arno Solin %E Aaditya Ramdas %F pmlr-v300-xu26c %I PMLR %P 1252--1260 %U https://proceedings.mlr.press/v300/xu26c.html %V 300 %X 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.
APA
Xu, D., Lachapelle, S. & Magliacane, S.. (2026). Identifiability of Potentially Degenerate Gaussian Mixture Models With Piecewise Affine Mixing . Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 300:1252-1260 Available from https://proceedings.mlr.press/v300/xu26c.html.

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