Approximate Bregman near neighbors in sublinear time: Beyond the triangle inequality

TL;DR

This paper introduces the first provable approximate nearest-neighbor algorithms for Bregman divergences, enabling efficient search beyond traditional metric assumptions and extending existing data structures to non-metric distances.

Contribution

The paper presents two novel algorithms for ANN search with Bregman divergences, one with sublinear query time and general applicability, and another with faster query time leveraging divergence-specific properties.

Findings

First algorithm: O(log^d n) query time, O(n log^d n) space.

Second algorithm: O(log n) query time, O(n) space.

Extends ANN search methods beyond Euclidean and metric spaces.

Abstract

In this paper we present the first provable approximate nearest-neighbor (ANN) algorithms for Bregman divergences. Our first algorithm processes queries in O(log^d n) time using O(n log^d n) space and only uses general properties of the underlying distance function (which includes Bregman divergences as a special case). The second algorithm processes queries in O(log n) time using O(n) space and exploits structural constants associated specifically with Bregman divergences. An interesting feature of our algorithms is that they extend the ring-tree + quad-tree paradigm for ANN searching beyond Euclidean distances and metrics of bounded doubling dimension to distances that might not even be symmetric or satisfy a triangle inequality.