Fast Parallel Estimation of High Dimensional Information Theoretical Quantities with Low Dimensional Random Projection Ensembles

Abstract

The estimation of relevant information theoretical quantities, such as entropy, mutual information, and various divergences is computationally expensive in high dimensions. However, for this task, one may apply pairwise Euclidean distances of sample points, which suits random projection (RP) based low dimensional embeddings. The Johnson-Lindenstrauss (JL) lemma gives theoretical bound on the dimension of the low dimensional embedding. We adapt the RP technique for the estimation of information theoretical quantities. Intriguingly, we find that embeddings into extremely small dimensions, far below the bounds of the JL lemma, provide satisfactory estimates for the original task. We illustrate this in the Independent Subspace Analysis (ISA) task; we combine RP dimension reduction with a simple ensemble method. We gain considerable speed-up with the potential of real-time parallel estimation of high dimensional information theoretical quantities.KeywordsIndependent subspace analysisrandom projectionpairwise distancesinformation theoretical estimations