Efficient metric representations for big data

Abstract

Contemporary datasets are often represented as points in a high-dimensional metric space. To deal with increasingly larger datasets, many algorithms rely on efficient or compressed representations of the induced metric. In this thesis, we study several fundamental aspects of efficient metric representations. Our results include: -- Fully determining the minimal number of bits required to represent all distances, up to a given precision, in a finite Euclidean or Manhattan metric space. -- A space-efficient data structure for Euclidean approximate nearest neighbor search in high dimensions. -- A sublinear time algorithm for low-rank approximation of distance matrices, which is optimal in the number of entries it reads of the input matrix. -- A fast algorithm for nearest neighbor search in the Optimal Transport distance. Previous bounds on Euclidean metric compression have been restricted to discretizing a classical dimensionality reduction theorem of Johnson and Lindenstrauss (1984). Our results improve over those bounds, thereby establishing an asymptotic advantage of generic sketching over dimension reduction. All of our algorithms are both proven analytically and implemented and validated empirically.