Provable Subspace Identification Under Post-Nonlinear Mixtures

TL;DR

This paper introduces a new criterion for identifying and removing unknown nonlinear functions in post-nonlinear mixture models, relaxing previous assumptions and supported by theoretical analysis and numerical experiments.

Contribution

It presents a novel UML criterion based on null space properties that relaxes prior structural assumptions for PNL identifiability.

Findings

The proposed method guarantees identifiability under weaker conditions.

Finite-sample analysis demonstrates robustness in realistic settings.

Numerical experiments validate theoretical claims.

Abstract

Unsupervised mixture learning (UML) aims at identifying linearly or nonlinearly mixed latent components in a blind manner. UML is known to be challenging: Even learning linear mixtures requires highly nontrivial analytical tools, e.g., independent component analysis or nonnegative matrix factorization. In this work, the post-nonlinear (PNL) mixture model -- where unknown element-wise nonlinear functions are imposed onto a linear mixture -- is revisited. The PNL model is widely employed in different fields ranging from brain signal classification, speech separation, remote sensing, to causal discovery. To identify and remove the unknown nonlinear functions, existing works often assume different properties on the latent components (e.g., statistical independence or probability-simplex structures). This work shows that under a carefully designed UML criterion, the existence of a nontrivial null space associated with the underlying mixing system suffices to guarantee identification/removal of the unknown nonlinearity. Compared to prior works, our finding largely relaxes the conditions of attaining PNL identifiability, and thus may benefit applications where no strong structural information on the latent components is known. A finite-sample analysis is offered to characterize the performance of the proposed approach under realistic settings. To implement the proposed learning criterion, a block coordinate descent algorithm is proposed. A series of numerical experiments corroborate our theoretical claims.