DeepAI AI Chat
Log In Sign Up

Analysis of stochastic Lanczos quadrature for spectrum approximation

by   Tyler Chen, et al.

The cumulative empirical spectral measure (CESM) ฮฆ[๐€] : โ„โ†’ [0,1] of a nร— n symmetric matrix ๐€ is defined as the fraction of eigenvalues of ๐€ less than a given threshold, i.e., ฮฆ[๐€](x) := โˆ‘_i=1^n1/nx1D7D9[ ฮป_i[๐€]โ‰ค x]. Spectral sums tr(f[๐€]) can be computed as the Riemannโ€“Stieltjes integral of f against ฮฆ[๐€], so the task of estimating CESM arises frequently in a number of applications, including machine learning. We present an error analysis for stochastic Lanczos quadrature (SLQ). We show that SLQ obtains an approximation to the CESM within a Wasserstein distance of t | ฮป_max[๐€] - ฮป_min[๐€] | with probability at least 1-ฮท, by applying the Lanczos algorithm for โŒˆ 12 t^-1 + 1/2โŒ‰ iterations to โŒˆ 4 ( n+2 )^-1t^-2ln(2nฮท^-1) โŒ‰ vectors sampled independently and uniformly from the unit sphere. We additionally provide (matrix-dependent) a posteriori error bounds for the Wasserstein and Kolmogorovโ€“Smirnov distances between the output of this algorithm and the true CESM. The quality of our bounds is demonstrated using numerical experiments.


page 9

page 17

page 18

โˆ™ 11/28/2022

A posteriori error bounds for the block-Lanczos method for matrix function approximation

We extend the error bounds from [SIMAX, Vol. 43, Iss. 2, pp. 787-811 (20...
โˆ™ 06/17/2021

Error bounds for Lanczos-based matrix function approximation

We analyze the Lanczos method for matrix function approximation (Lanczos...
โˆ™ 12/07/2020

The Spectral-Domain ๐’ฒ_2 Wasserstein Distance for Elliptical Processes and the Spectral-Domain Gelbrich Bound

In this short note, we introduce the spectral-domain ๐’ฒ_2 Wasserstein dis...
โˆ™ 12/09/2019

Error control for statistical solutions

Statistical solutions have recently been introduced as a an alternative ...
โˆ™ 06/15/2021

Non-asymptotic convergence bounds for Wasserstein approximation using point clouds

Several issues in machine learning and inverse problems require to gener...
โˆ™ 08/30/2012

An Improved Bound for the Nystrom Method for Large Eigengap

We develop an improved bound for the approximation error of the Nystrรถm ...
โˆ™ 04/12/2019

The Lanczos Algorithm Under Few Iterations: Concentration and Location of the Ritz Values

We study the Lanczos algorithm where the initial vector is sampled unifo...