Residual Component Analysis: Generalising PCA for more flexible inference in linear-Gaussian models

TL;DR

Residual Component Analysis (RCA) extends PCA by decomposing residual variance in linear-Gaussian models, enabling more flexible inference when data variance is partially explained by other factors, with applications in biology and motion capture.

Contribution

The paper introduces Residual Component Analysis (RCA), a generalization of PCA that handles residual variance from other sources, along with new algorithms for covariance decomposition.

Findings

RCA effectively recovers protein-signaling networks.

RCA models gene expression time-series data.

RCA improves human skeleton recovery from motion capture.

Abstract

Probabilistic principal component analysis (PPCA) seeks a low dimensional representation of a data set in the presence of independent spherical Gaussian noise. The maximum likelihood solution for the model is an eigenvalue problem on the sample covariance matrix. In this paper we consider the situation where the data variance is already partially explained by other actors, for example sparse conditional dependencies between the covariates, or temporal correlations leaving some residual variance. We decompose the residual variance into its components through a generalised eigenvalue problem, which we call residual component analysis (RCA). We explore a range of new algorithms that arise from the framework, including one that factorises the covariance of a Gaussian density into a low-rank and a sparse-inverse component. We illustrate the ideas on the recovery of a protein-signaling network, a gene expression time-series data set and the recovery of the human skeleton from motion capture 3-D cloud data.