On Low Distortion Embeddings of Statistical Distance Measures into Low Dimensional Spaces

TL;DR

This paper explores methods for embedding statistical distance measures like Mahalanobis, Bhattacharyya, and Kullback-Leibler into low-dimensional spaces with minimal distortion, facilitating efficient high-dimensional data retrieval.

Contribution

It introduces a dimensionality reduction technique based on the Johnson-Lindenstrauss Lemma for these measures, providing low-distortion embeddings and analyzing embeddability into metric spaces.

Findings

Mahalanobis distance admits arbitrarily low distortion embeddings.

Bhattacharyya distance can be reduced with low additive error.

Certain divergences cannot be embedded into metric spaces without large distortion.

Abstract

Statistical distance measures have found wide applicability in information retrieval tasks that typically involve high dimensional datasets. In order to reduce the storage space and ensure efficient performance of queries, dimensionality reduction while preserving the inter-point similarity is highly desirable. In this paper, we investigate various statistical distance measures from the point of view of discovering low distortion embeddings into low-dimensional spaces. More specifically, we consider the Mahalanobis distance measure, the Bhattacharyya class of divergences and the Kullback-Leibler divergence. We present a dimensionality reduction method based on the Johnson-Lindenstrauss Lemma for the Mahalanobis measure that achieves arbitrarily low distortion. By using the Johnson-Lindenstrauss Lemma again, we further demonstrate that the Bhattacharyya distance admits dimensionality reduction with arbitrarily low additive error. We also examine the question of embeddability into metric spaces for these distance measures due to the availability of efficient indexing schemes on metric spaces. We provide explicit constructions of point sets under the Bhattacharyya and the Kullback-Leibler divergences whose embeddings into any metric space incur arbitrarily large distortions. We show that the lower bound presented for Bhattacharyya distance is nearly tight by providing an embedding that approaches the lower bound for relatively small dimensional datasets.