  Computing approximate nearest neighbors in high dimensional spaces is a
central problem in large-scale data mining with a wide range of applications in
machine learning and data science. A popular and effective technique in
computing nearest neighbors approximately is the locality-sensitive hashing
(LSH) scheme. In this paper, we aim to develop LSH schemes for distance
functions that measure the distance between two probability distributions,
particularly for f-divergences as well as a generalization to capture mutual
information loss. First, we provide a general framework to design LHS schemes
for f-divergence distance functions and develop LSH schemes for the generalized
Jensen-Shannon divergence and triangular discrimination in this framework. We
show a two-sided approximation result for approximation of the generalized
Jensen-Shannon divergence by the Hellinger distance, which may be of
independent interest. Next, we show a general method of reducing the problem of
designing an LSH scheme for a Krein kernel (which can be expressed as the
difference of two positive definite kernels) to the problem of maximum inner
product search. We exemplify this method by applying it to the mutual
information loss, due to its several important applications such as model
compression.
