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 language||English (US)|
|Title of host publication||High Performance Computing in Science and Engineering - 3rd International Conference, HPCSE 2017, Revised Selected Papers|
|Editors||Jakub Sistek, Petr Tichy, Tomas Kozubek, Martin Cermak, Dalibor Lukas, Jiri Jaros, Radim Blaheta|
|Number of pages||15|
|State||Published - 2018|
|Event||3rd International Conference on High Performance Computing in Science and Engineering, HPCSE 2017 - Karolinka, Czech Republic|
Duration: May 22 2017 → May 25 2017
|Name||Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)|
|Other||3rd International Conference on High Performance Computing in Science and Engineering, HPCSE 2017|
|Period||5/22/17 → 5/25/17|
Bibliographical noteFunding Information:
This work was supported byNSF under grant CCF-1318597.