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

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Publication Type pre-print
School or College College of Engineering
Department Computing, School of
Creator Venkatasubramanian, Suresh
Other Author Abdullah, Amirali; Moeller, John
Title Approximate Bregman near neighbors in sublinear time: beyond the triangle inequality
Date 2012-01-01
Description Bregman divergences are important distance measures that are used extensively in data-driven applications such as computer vision, text mining, and speech processing, and are a key focus of interest in machine learning. Answering nearest neighbor (NN) queries under these measures is very important in these applications and has been the subject of extensive study, but is problematic because these distance measures lack metric properties like symmetry and the triangle inequality. In this paper, we present the first provably approximate nearest-neighbor (ANN) algorithms. These process queries in O(logn) time for Bregman divergences in fixed dimensional spaces. We also obtain polylogn bounds for a more abstract class of distance measures (containing Bregman divergences) which satisfy certain structural properties . Both of these bounds apply to both the regular asymmetric Bregman divergences as well as their symmetrized versions. To do so, we develop two geometric properties vital to our analysis: a reverse triangle inequality (RTI) and a relaxed triangle inequality called m-defectiveness where m is a domain-dependent parameter. Bregman divergences satisfy the RTI but not m-defectiveness. However, we show that the square root of a Bregman divergence does satisfy m-defectiveness. This allows us to then utilize both properties in an efficient search data structure that follows the general two-stage paradigm of a ring-tree decomposition followed by a quad tree search used in previous near-neighbor algorithms for Euclidean space and spaces of bounded doubling dimension. Our first algorithm resolves a query for a d-dimensional (1+e)-ANN in O ( logne )O(d) time and O (nlogd-1 n) space and holds for generic m-defective distance measures satisfying a RTI. Our second algorithm is more specific in analysis to the Bregman divergences and uses a further structural constant, the maximum ratio of second derivatives over each dimension of our domain (c0). This allows us to locate a (1+e)-ANN in O(logn) time and O(n) space, where there is a further (c0)d factor in the big-Oh for the query time.
Type Text
Publisher Association for Computing Machinery
First Page 31
Last Page 40
Dissertation Institution University of Utah
Language eng
Bibliographic Citation Abdullah, A., Moeller, J., & Venkatasubramanian, S. (2012). Approximate Bregman near neighbors in sublinear time: beyond the triangle inequality. Proceedings of the Annual Symposium on Computational Geometry, 31-40.
Rights Management © ACM, 2012. This is the authors version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published in Proceedings of the Annual Symposium on Computational Geometry (SoCG), 31-40, Oct. 2012. http://doi.acm.org/10.1145/nnnnnn.nnnnnn ; doi>10.1145/2261250.2261255
Format Medium application/pdf
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Reference URL https://collections.lib.utah.edu/ark:/87278/s6vq3mfx