Approximating spectral sums of large-scale matrices using stochastic Chebyshev approximations

Insu Han, Dmitry Malioutov, Haim Avron, Jinwoo Shin

Research output: Contribution to journalArticlepeer-review


Computation of the trace of a matrix function plays an important role in many scientific computing applications, including applications in machine learning, computational physics (e.g., lattice quantum chromodynamics), network analysis, and computational biology (e.g., protein folding), just to name a few application areas. We propose a linear-time randomized algorithm for approximating the trace of matrix functions of large symmetric matrices. Our algorithm is based on coupling function approximation using Chebyshev interpolation with stochastic trace estimators (Hutchinson's method), and as such requires only implicit access to the matrix, in the form of a function that maps a vector to the product of the matrix and the vector. We provide rigorous approximation error in terms of the extremal eigenvalue of the input matrix, and the Bernstein ellipse that corresponds to the function at hand. Based on our general scheme, we provide algorithms with provable guarantees for important matrix computations, including log-determinant, trace of matrix inverse, Estrada index, Schatten p-norm, and testing positive definiteness. We experimentally evaluate our algorithm and demonstrate its effectiveness on matrices with tens of millions dimensions.

Original languageEnglish
Pages (from-to)A1558-A1585
JournalSIAM Journal on Scientific Computing
Issue number4
StatePublished - 2017


  • Chebyshev approximation
  • Hutchinson's method
  • Matrix computation
  • Spectral function

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Applied Mathematics


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