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Applications of Trace Estimation Techniques

  • Shashanka UbaruEmail author
  • Yousef Saad
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11087)

Abstract

We discuss various applications of trace estimation techniques for evaluating functions of the form \(\mathtt {tr}(f(A))\) 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 \(\mathtt {tr}(A^{-1})\), 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.

Notes

Acknowledgments

This work was supported by NSF under grant CCF-1318597.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of Minnesota at Twin CitiesMinneapolisUSA

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