Identifiability-Guaranteed Simplex-Structured Post-Nonlinear Mixture Learning via Autoencoder

TL;DR

This paper introduces a new autoencoder-based method for identifying nonlinearly mixed latent components in data, with relaxed conditions, finite sample guarantees, and demonstrated effectiveness on synthetic and real datasets.

Contribution

It provides new relaxed identifiability conditions, finite sample complexity results, and a practical autoencoder framework for learning simplex-structured post-nonlinear mixtures.

Findings

New identifiability conditions under relaxed assumptions

First finite sample complexity results for this problem

Autoencoder-based algorithm effectively learns the mixture in experiments

Abstract

This work focuses on the problem of unraveling nonlinearly mixed latent components in an unsupervised manner. The latent components are assumed to reside in the probability simplex, and are transformed by an unknown post-nonlinear mixing system. This problem finds various applications in signal and data analytics, e.g., nonlinear hyperspectral unmixing, image embedding, and nonlinear clustering. Linear mixture learning problems are already ill-posed, as identifiability of the target latent components is hard to establish in general. With unknown nonlinearity involved, the problem is even more challenging. Prior work offered a function equation-based formulation for provable latent component identification. However, the identifiability conditions are somewhat stringent and unrealistic. In addition, the identifiability analysis is based on the infinite sample (i.e., population) case, while the understanding for practical finite sample cases has been elusive. Moreover, the algorithm in the prior work trades model expressiveness with computational convenience, which often hinders the learning performance. Our contribution is threefold. First, new identifiability conditions are derived under largely relaxed assumptions. Second, comprehensive sample complexity results are presented -- which are the first of the kind. Third, a constrained autoencoder-based algorithmic framework is proposed for implementation, which effectively circumvents the challenges in the existing algorithm. Synthetic and real experiments corroborate our theoretical analyses.