Finite-Sample Analysis of Nonlinear Independent Component Analysis:Sample Complexity and Identifiability Bounds

arXiv cs.LG / 4/13/2026

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Key Points

The paper addresses a key gap in nonlinear independent component analysis (ICA) by providing the first comprehensive finite-sample statistical characterization with neural network encoders, rather than relying on asymptotic results.
It introduces a direct link between excess risk and identification error that avoids parameter-space arguments and is designed to prevent suboptimal scaling from rate degradation.
The authors prove matching information-theoretic lower bounds, establishing that their sample complexity results are optimal.
The analysis is extended to practical training via SGD, showing that finite-iteration gradient descent can achieve comparable sample efficiency under standard optimization “landscape” assumptions.
Simulation experiments are used to validate the derived scaling laws, especially their dependence on problem dimension and diversity, and the work motivates further study of neural network training’s finite-sample behavior.

Abstract

Independent Component Analysis (ICA) is a fundamental unsupervised learning technique foruncovering latent structure in data by separating mixed signals into their independent sources. While substantial progress has been made in establishing asymptotic identifiability guarantees for nonlinear ICA, the finite-sample statistical properties of learning algorithms remain poorly understood. This gap poses significant challenges for practitioners who must determine appropriate sample sizes for reliable source recovery. This paper presents a comprehensive finite-sample analysis of nonlinear ICA with neural network encoders, providing the first complete characterization with matching upper and lower bounds. Our theoretical development introduces three key technical contributions. First, we establish a direct relationship between excess risk and identification error that bypasses parameter-space arguments, thereby avoiding the rate degradation that would otherwise yield suboptimal scaling. Second, we prove matching information-theoretic lower bounds that confirm the optimality of our sample complexity results. Third, we extend our analysis to practical SGD optimization, showing that the same sample efficiency can be achieved with finite-iteration gradient descent under standard landscape assumptions. We validate our theoretical predictions through carefully designed simulation experiments. This gap points toward valuable future research on finite-sample behavior of neural network training and highlights the importance of our validated scaling laws for dimension and diversity.