A Unifying Probabilistic Perspective for Spectral Dimensionality Reduction: Insights and New Models

TL;DR

This paper presents a unifying probabilistic framework for spectral dimensionality reduction methods, introduces new models like MEU and ALLE, and demonstrates their effectiveness on real datasets.

Contribution

It unifies spectral methods under a Gaussian Markov random field perspective and proposes new models and regularization techniques for improved dimensionality reduction.

Findings

MEU generalizes PCA and relates to Laplacian eigenmaps and isomap.

ALLE performs maximum likelihood LLE exactly.

Models are competitive on robot navigation and human motion data.

Abstract

We introduce a new perspective on spectral dimensionality reduction which views these methods as Gaussian Markov random fields (GRFs). Our unifying perspective is based on the maximum entropy principle which is in turn inspired by maximum variance unfolding. The resulting model, which we call maximum entropy unfolding (MEU) is a nonlinear generalization of principal component analysis. We relate the model to Laplacian eigenmaps and isomap. We show that parameter fitting in the locally linear embedding (LLE) is approximate maximum likelihood MEU. We introduce a variant of LLE that performs maximum likelihood exactly: Acyclic LLE (ALLE). We show that MEU and ALLE are competitive with the leading spectral approaches on a robot navigation visualization and a human motion capture data set. Finally the maximum likelihood perspective allows us to introduce a new approach to dimensionality reduction based on L1 regularization of the Gaussian random field via the graphical lasso.