Kinks vs. Smoothness: Identifiability of Real Analytic nICA for Laplace-like Sources

2026-09-18

AI Summary

研究解决了非线性独立成分分析(nICA)中真实潜在因素的识别问题，通过利用源分布中的断点与实解析函数的平滑性对比方法实现。

Abstract

Many machine learning systems try to explain complex data - like images or financial time series - in terms of hidden, independent factors that generated them. Recovering the true underlying factors, rather than some scrambled version of them, is the central challenge of nonlinear Independent Component Analysis (nICA). We prove identifiability (exact recovery) up to trivial ambiguities for real analytic generating functions when source probability density functions have a finite number of discontinuities in the first derivative. The Laplace distribution is the most prominent example satisfying this assumption. Our proof relies on the contrast between kinks in the source distribution and the smoothness of real analytic functions. Real analytic functions comprise a broad class of generating mechanisms, and can be approximated with Normalizing Flows or Variational Autoencoders with standard activation functions (e.g., tanh, softplus, GELU), so our result applies with minimal changes to existing training pipelines. We perform experiments on real and synthetic data with both Normalizing Flows and Variational Auto-Encoders demonstrating their identifiability properties. In experiments on CelebA data we recover several interpretable latent factors controlling unique attributes across the dataset.

Problem

• nonlinear Independent Component Analysis
• identifiability
• real analytic functions
• Laplace distribution
• latent factors

Innovation

• identifiability
• real analytic functions
• kinks
• Normalizing Flows
• Variational Autoencoders