  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.
