SUBQUADRATIC ALGORITHMS FOR KERNEL MATRI-CES VIA KERNEL DENSITY ESTIMATION

Abstract

Kernel matrices, as well as weighted graphs represented by them, are ubiquitous objects in machine learning, statistics and other related fields. The main drawback of using kernel methods (learning and inference using kernel matrices) is efficiency -given n input points, most kernel-based algorithms need to materialize the full n × n kernel matrix before performing any subsequent computation, thus incurring Ω(n 2 ) runtime. Breaking this quadratic barrier for various problems has therefore, been a subject of extensive research efforts. We break the quadratic barrier and obtain subquadratic time algorithms for several fundamental linear-algebraic and graph processing primitives, including approximating the top eigenvalue and eigenvector, spectral sparsification, solving linear systems, local clustering, low-rank approximation, arboricity estimation and counting weighted triangles. We build on the recently developed Kernel Density Estimation framework, which (after preprocessing in time subquadratic in n) can return estimates of row/column sums of the kernel matrix. In particular, we develop efficient reductions from weighted vertex and weighted edge sampling on kernel graphs, simulating random walks on kernel graphs, and importance sampling on matrices to Kernel Density Estimation and show that we can generate samples from these distributions in sublinear (in the support of the distribution) time. Our reductions are the central ingredient in each of our applications and we believe they may be of independent interest. We empirically demonstrate the efficacy of our algorithms on low-rank approximation (LRA) and spectral sparsification, where we observe a 9x decrease in the number of kernel evaluations over baselines for LRA and a 41x reduction in the graph size for spectral sparsification.

1. Introduction

For a kernel function k : R d × R d → R and a set X = {x 1 . . . x n } ⊂ R d of n points, the entries of the n × n kernel matrix K are defined as K i,j = k(x i , x j ). Alternatively, one can view X as the vertex set of a complete weighted graph where the weights between points are defined by the kernel matrix K. Popular choices of kernel functions k include the Gaussian kernel, the Laplace kernel, exponential kernel, etc; see (Schölkopf et al., 2002; Shawe-Taylor et al., 2004; Hofmann et al., 2008) for a comprehensive overview. Despite their wide applicability, kernel methods suffer from drawbacks, one of the main being efficiency -given n input points in d dimensions, many kernel-based algorithms need to materialize the full n × n kernel matrix K before performing the computation. For some problems this is unavoidable, especially if high-precision results are required (Backurs et al., 2017) . In this work, we show that we can in fact break this Ω(n 2 ) barrier for several fundamental problems in numerical linear algebra and graph processing. We obtain algorithms that run in o(n 2 ) time and scale inverselyproportional to the smallest entry of the kernel matrix. This allows us to skirt several known lower

