通过最优传输进行线性独立成分分析
Linear Independent Component Analysis via Optimal Transport
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中文总结 AI 辅助
研究如何从线性混合信号中恢复独立源信号，提出用平方瓦瑟斯坦距离衡量非高斯性，基于此构建OT - ICA算法，实验表明该算法在不同分布上优于传统方法，且可用于无分布假设的应用ICA任务。
中文摘要 AI 辅助
线性独立成分分析（ICA）从线性混合信号中恢复联合独立的源信号。经典ICA算法通过最大化由负熵衡量的非高斯性来实现，负熵通过信息理论与独立性相关。由于精确的负熵优化难以处理，它们依赖代理对比函数，如四阶累积量和参数对数似然。本文提出用与标准高斯分布的平方瓦瑟斯坦距离$W_2^2$来衡量非高斯性。证明当投影恢复独立成分时，标准正态分布与数据线性投影之间的瓦瑟斯坦距离最大。基于此提出OT - ICA算法，通过基于梯度的优化找到该投影。对模拟数据的实证评估表明，OT - ICA在潜在变量的不同分布上优于基于代理的方法。在脑电伪迹去除和计量经济学价格发现中的应用证实了OT - ICA可用于无分布假设的应用ICA任务。
英文摘要
Linear Independent Component Analysis (ICA) recovers jointly independent source signals from their linear mixtures. To achieve this, classical ICA algorithms attempt to maximize non-Gaussianity, measured by negentropy, which is linked to independence by information theory. Because exact negentropy optimization is intractable, they rely on proxy contrast functions, such as fourth-order cumulants and parametric log-likelihoods. We propose instead to use the squared $L_2$-Wasserstein distance to a standard Gaussian as the ICA contrast. We show that the Wasserstein distance between a standard normal distribution and linear projections of the data is maximized when the projection recovers an independent component, and that under a regularity condition on the sources this maximum is separated from every genuine mixture by an explicit margin. We uncover the advantageous properties of the resulting estimator: for sources with a smooth density it is $\sqrt{N}$-consistent and asymptotically normal with a closed-form variance, whereas for a source with an atom the contrast has a kink at the true direction, which makes the estimator exact with probability tending to one. The proposed OT-ICA algorithm finds this projection by gradient-based optimization. Empirical evaluation on simulated data shows that OT-ICA outperforms proxy-based methods and that the $W_2^2$ contrast provides more robust signal than proxy contrasts in measuring non-Gaussianity for different distributional mixtures of the latent variables. Application to linear causal disentanglement and EEG artifact removal, along with further applications detailed in the appendix, confirms OT-ICA can be used for applied ICA tasks without distributional assumptions.
发表机构
- University of Tübingen（图宾根大学）
- Max Planck Institute for Intelligent Systems, Tübingen（图宾根马克斯·普朗克智能系统研究所）
- Institute of Artificial Intelligence TU Braunschweig（布伦瑞克工业大学人工智能研究所）
机构由 AI 辅助整理，请以论文原文为准。
