Sample-Optimal Low-Rank Approximation of Distance Matrices

Piotr Indyk, Ali Vakilian, Tal Wagner, David P. Woodruff

Research output: Contribution to journalConference articlepeer-review

11 Scopus citations

Abstract

A distance matrix A ∈ Rn×m represents all pairwise distances, Aij = d(xi, yj), between two point sets x1, ..., xn and y1, ..., ym in an arbitrary metric space (Z, d). Such matrices arise in various computational contexts such as learning image manifolds, handwriting recognition, and multi-dimensional unfolding. In this work we study algorithms for low-rank approximation of distance matrices. Recent work by Bakshi and Woodruff (NeurIPS 2018) showed it is possible to compute a rank-k approximation of a distance matrix in time O((n + m)1+γ) · poly (k, 1/ε), where ε > 0 is an error parameter and γ > 0 is an arbitrarily small constant. Notably, their bound is sublinear in the matrix size, which is unachievable for general matrices. We present an algorithm that is both simpler and more efficient. It reads only O((n + m)k/ε) entries of the input matrix, and has a running time of O(n + m) · poly (k, 1/ε). We complement the sample complexity of our algorithm with a matching lower bound on the number of entries that must be read by any algorithm. We provide experimental results to validate the approximation quality and running time of our algorithm.

Original languageEnglish
Pages (from-to)1723-1751
Number of pages29
JournalProceedings of Machine Learning Research
Volume99
StatePublished - 2019
Externally publishedYes
Event32nd Conference on Learning Theory, COLT 2019 - Phoenix, United States
Duration: 25 Jun 201928 Jun 2019

Funding

FundersFunder number
Ainesh Bakshi
National Science FoundationCCF-1815840
Simons Foundation
Simons Institute for the Theory of Computing, University of California Berkeley

    Keywords

    • Distance Matrix
    • Low-rank Approximation

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