Finite-Sample Analysis of Nonlinear Independent Component Analysis:Sample Complexity and Identifiability Bounds
2026-04-10 • Machine Learning
Machine Learning
AI summaryⓘ
The authors study a way to separate mixed signals into original parts using neural networks, focusing on how much data is needed to do this well. They provide the first clear limits on how many samples are needed and prove these limits are the best possible. Their work also shows that common training methods like gradient descent can reach this efficiency. They back up their theory with experiments and highlight the practical importance of understanding sample size in this area.
Independent Component Analysisnonlinear ICAfinite-sample analysisneural network encodersexcess riskidentification errorinformation-theoretic lower boundssample complexitystochastic gradient descentoptimization landscape
Authors
Yuwen Jiang
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.