A Unified Framework for Probabilistic Component Analysis

TL;DR

This paper introduces a unified probabilistic framework for various component analysis methods, enabling the creation of new models and probabilistic versions of existing techniques through latent neighborhood selection.

Contribution

It unifies many component analysis algorithms under a single probabilistic framework using Markov Random Fields, and develops novel EM algorithms for these models.

Findings

The framework successfully unifies PCA, LDA, LPP, and SFA.

Proposed methods outperform state-of-the-art techniques in experiments.

New probabilistic models are derived for previously deterministic methods.

Abstract

We present a unifying framework which reduces the construction of probabilistic component analysis techniques to a mere selection of the latent neighbourhood, thus providing an elegant and principled framework for creating novel component analysis models as well as constructing probabilistic equivalents of deterministic component analysis methods. Under our framework, we unify many very popular and well-studied component analysis algorithms, such as Principal Component Analysis (PCA), Linear Discriminant Analysis (LDA), Locality Preserving Projections (LPP) and Slow Feature Analysis (SFA), some of which have no probabilistic equivalents in literature thus far. We firstly define the Markov Random Fields (MRFs) which encapsulate the latent connectivity of the aforementioned component analysis techniques; subsequently, we show that the projection directions produced by all PCA, LDA, LPP and SFA are also produced by the Maximum Likelihood (ML) solution of a single joint probability density function, composed by selecting one of the defined MRF priors while utilising a simple observation model. Furthermore, we propose novel Expectation Maximization (EM) algorithms, exploiting the proposed joint PDF, while we generalize the proposed methodologies to arbitrary connectivities via parameterizable MRF products. Theoretical analysis and experiments on both simulated and real world data show the usefulness of the proposed framework, by deriving methods which well outperform state-of-the-art equivalents.