Locality-Sensitive Hashing for f-Divergences: Mutual Information Loss and Beyond

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 Kreĭn kernel (which can be expressed as the difference of two positive definite kernels) to the problem of maximum inner product search.