独立成分分析的基础

中文摘要 AI 辅助

我们在一份自成体系的笔记中，基于标准文献阐述了线性独立成分分析（ICA）模型的数学基础，目标读者具备测度论概率论背景。我们首先建立了$\mathbb{R}^d$上概率测度的特征函数理论，包括其解析性及确定和刻画分布的方式。随后，我们聚焦于ICA模型的若干可识别性结果，对源的假设逐步强化：从仅非恒定，到非高斯，再到无高斯的独立源。在最严格的假设下，我们证明独立源可在平移、置换、缩放和符号的范围内被识别，即便存在加性高斯噪声时亦如此。此外，我们提出在线等变梯度下降ICA算法，用于在标准完整无噪非高斯ICA设定下从数据中恢复独立源。

英文摘要

We present the mathematical foundations of linear independent component analysis (ICA) models based on standard literature in a self-contained note. It is aimed at readers with a background in measure-theoretic probability theory. We first develop the theory of the characteristic functions of probability measures on $\mathbb{R}^d$, including their analyticity and the way in which they determine and characterise the distributions. We then focus on several identifiability results of ICA models with successively strengthened assumptions on the sources: from merely non-constant, to non-Gaussian, to Gaussian-free independent sources. Under the strictest assumptions, we show that the independent sources are identifiable up to translation, permutation, scales and signs, and this even in the presence of additive Gaussian noise. Furthermore, we present the online equivariant gradient descent ICA algorithm for recovering the independent sources from data, in the standard complete noiseless non-Gaussian ICA setting.

发表机构

• Korteweg-de Vries Institute for Mathematics（科特维格-德弗里斯数学研究所）
• University of Amsterdam（阿姆斯特丹大学）

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