Bayesian estimation of Karhunen-Loeve expansions; A random subspace approach

One of the most widely-used procedures for dimensionality reduction of high dimensional data is Principal Component Analysis (PCA). More broadly, low-dimensional stochastic representation of random fields with finite variance is provided via the well known Karhunen-Loeve expansion (KLE). The KLE is analogous to a Fourier series expansion for a random process, where the goal is to find an orthogonal transformation for the data such that the projection of the data onto this orthogonal subspace is optimal in the L-2 sense, i.e., which minimizes the mean square error. In practice, this orthogonal transformation is determined by performing an SVD (Singular Value Decomposition) on the sample covariance matrix or on the data matrix itself. Sampling error is typically ignored when quantifying the principal components, or, equivalently, basis functions of the KLE. Furthermore, it is exacerbated when the sample size is much smaller than the dimension of the random field. In this paper, we introduce a Bayesian KLE procedure, allowing one to obtain a probabilistic model on the principal components, which can account for inaccuracies due to limited sample size. The probabilistic model is built via Bayesian inference, from which the posterior becomes the matrix Bingham density over the space of orthonormal matrices. We use a modified Gibbs sampling procedure to sample on this space and then build probabilistic Karhunen-Loeve expansions over random subspaces to obtain a set of low-dimensional surrogates of the stochastic process. We illustrate this probabilistic procedure with a finite dimensional stochastic process inspired by Brownian motion. (C) 2016 Elsevier Inc. All rights reserved.