Tree-dependent Component Analysis

TL;DR

Tree-dependent Component Analysis extends ICA by finding transforms that produce data fitting a tree-structured graphical model, enabling efficient multivariate density estimation through mutual information minimization.

Contribution

It introduces a novel framework that generalizes ICA to tree-structured models, with practical approximations for mutual information-based contrast functions.

Findings

Provides two practical approximations of the contrast function

Enables efficient multivariate density estimation using bivariate densities

Generalizes ICA to tree-structured graphical models

Abstract

We present a generalization of independent component analysis (ICA), where instead of looking for a linear transform that makes the data components independent, we look for a transform that makes the data components well fit by a tree-structured graphical model. Treating the problem as a semiparametric statistical problem, we show that the optimal transform is found by minimizing a contrast function based on mutual information, a function that directly extends the contrast function used for classical ICA. We provide two approximations of this contrast function, one using kernel density estimation, and another using kernel generalized variance. This tree-dependent component analysis framework leads naturally to an efficient general multivariate density estimation technique where only bivariate density estimation needs to be performed.