The wide applicability of kernels makes the problem of max-kernel search ubiquitous and more general than the usual similarity search in metric spaces. We focus on solving this problem efficiently. We begin by characterizing the inherent hardness of
the max-kernel search problem with a novel notion of directional concentration. Following that, we present a method to use an $O(n log n)$ algorithm to index any set of objects (points in $Real^dims$ or abstract objects) directly in the Hilbert space without any explicit feature representations of the objects in this space. We present the first provably $O(log n)$ algorithm for exact max-kernel search using this index. Empirical results for a variety of data sets as well as abstract objects demonstrate up to 4 orders of magnitude speedup in some cases. Extensions for approximate max-kernel search are also presented.
The problem of {em efficiently} finding the best match for a query in a given set with respect to the Euclidean distance or the cosine similarity has been extensively studied in literature. However, a closely related problem of efficiently finding th
e best match with respect to the inner product has never been explored in the general setting to the best of our knowledge. In this paper we consider this general problem and contrast it with the existing best-match algorithms. First, we propose a general branch-and-bound algorithm using a tree data structure. Subsequently, we present a dual-tree algorithm for the case where there are multiple queries. Finally we present a new data structure for increasing the efficiency of the dual-tree algorithm. These branch-and-bound algorithms involve novel bounds suited for the purpose of best-matching with inner products. We evaluate our proposed algorithms on a variety of data sets from various applications, and exhibit up to five orders of magnitude improvement in query time over the naive search technique.
