APPROXIMATING SPECTRAL SUMS OF LARGE-SCALE MATRICES USING STOCHASTIC CHEBYSHEV APPROXIMATIONS

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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.
Publisher
SIAM PUBLICATIONS
Issue Date
2017
Language
English
Article Type
Article
Keywords

UNCERTAINTY QUANTIFICATION; INVERSE; TRACE; DETERMINANT; ESTIMATOR

Citation

SIAM JOURNAL ON SCIENTIFIC COMPUTING, v.39, no.4, pp.A1558 - A1585

ISSN
1064-8275
DOI
10.1137/16M1078148
URI
http://hdl.handle.net/10203/226177
Appears in Collection
EE-Journal Papers(저널논문)
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