Applications of trace estimation techniques

Shashanka Ubaru, Yousef Saad

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We discuss various applications of trace estimation techniques for evaluating functions of the form tr(f(A)) where f is certain function. The first problem we consider that can be cast in this form is that of approximating the Spectral density or Density of States (DOS) of a matrix. The DOS is a probability density distribution that measures the likelihood of finding eigenvalues of the matrix at a given point on the real line, and it is an important function in solid state physics. We also present a few non-standard applications of spectral densities. Other trace estimation problems we discuss include estimating the trace of a matrix inverse tr(A-1), the problem of counting eigenvalues and estimating the rank, and approximating the log-determinant (trace of log function). We also discuss a few similar computations that arise in machine learning applications. We review two computationally inexpensive methods to compute traces of matrix functions, namely, the Chebyshev expansion and the Lanczos Quadrature methods. A few numerical examples are presented to illustrate the performances of these methods in different applications.

Original languageEnglish (US)
Title of host publicationHigh Performance Computing in Science and Engineering - 3rd International Conference, HPCSE 2017, Revised Selected Papers
EditorsJakub Sistek, Petr Tichy, Tomas Kozubek, Martin Cermak, Dalibor Lukas, Jiri Jaros, Radim Blaheta
PublisherSpringer Verlag
Pages19-33
Number of pages15
ISBN (Print)9783319971353
DOIs
StatePublished - 2018
Event3rd International Conference on High Performance Computing in Science and Engineering, HPCSE 2017 - Karolinka, Czech Republic
Duration: May 22 2017May 25 2017

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume11087 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other3rd International Conference on High Performance Computing in Science and Engineering, HPCSE 2017
CountryCzech Republic
CityKarolinka
Period5/22/175/25/17

Bibliographical note

Funding Information:
This work was supported byNSF under grant CCF-1318597.

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