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Nonlinear independent component analysis (nICA) aims at recoveringstatistically independent latent components that are mixed by unknown nonlinearfunctions. Central to nICA is the identifiability of the latent components,which had been elusive until very recently. Specifically, Hyv\"arinen et al.have shown that the nonlinearly mixed latent components are identifiable (up tooften inconsequential ambiguities) under a generalized contrastive learning(GCL) formulation, given that the latent components are independent conditionedon a certain auxiliary variable. The GCL-based identifiability of nICA iselegant, and establishes interesting connections between nICA and popularunsupervised/self-supervised learning paradigms in representation learning,causal learning, and factor disentanglement. However, existing identifiabilityanalyses of nICA all build upon an unlimited sample assumption and the use ofideal universal function learners -- which creates a non-negligible gap betweentheory and practice.  Closing the gap is a nontrivial challenge, as there is a lack of established``textbook'' routine for finite sample analysis of such unsupervised problems.This work puts forth a finite-sample identifiability analysis of GCL-basednICA. Our analytical framework judiciously combines the properties of the GCLloss function, statistical generalization analysis, and numericaldifferentiation. Our framework also takes the learning function's approximationerror into consideration, and reveals an intuitive trade-off between thecomplexity and expressiveness of the employed function learner. Numericalexperiments are used to validate the theorems.

On Finite-Sample Identifiability of Contrastive Learning-Based Nonlinear Independent Component Analysis