Learning Non-linear Structures with Gaussian Markov Random Fields

Abstract

Nowadays, one of the most changeling points in statistics is the analysis of high dimensional data. In such cases, it is commonly assumed that the dimensionality of the data is only artificially high: although each data point is described by thousands of features, it is assumed that it can be modeled as a function of only a few underlying parameters. Formally, it is assumed that the data points are samples from a low-dimensional manifold embedded in a high-dimensional space.In this paper, we discuss a recently proposed method, known as Maximum Entropy Unfolding (MEU), for learning non-linear structures that characterize high dimensional data.This method represents a new perspective on spectral dimensionality reduction and, joined with the theory of Gaussian Markov random fields, provides a unifying probabilistic approach to spectral dimensionality reduction techniques. Parameter estimation as well as approaches to learning the structure of the GMRF are discussed