Spectrum-Adapted Polynomial Approximation for Matrix Functions with Applications in Graph Signal Processing

Fan, Tiffany; Shuman, David I.; Ubaru, Shashanka; Saad, Yousef

doi:10.3390/a13110295

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Spectrum-Adapted Polynomial Approximation for Matrix Functions with Applications in Graph Signal Processing

^2,*

and

⁴

Institute for Computational & Mathematical Engineering, Stanford University, Stanford, CA 94305, USA

Department of Mathematics, Statistics, and Computer Science, Macalester College, St. Paul, MN 55105, USA

IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA

⁴

Department of Computer Science and Engineering, University of Minnesota, Minneapolis, MN 55455, USA

Author to whom correspondence should be addressed.

Algorithms 2020, 13(11), 295; https://doi.org/10.3390/a13110295

Received: 16 October 2020 / Revised: 16 October 2020 / Accepted: 29 October 2020 / Published: 13 November 2020

(This article belongs to the Special Issue Efficient Graph Algorithms in Machine Learning)

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Abstract

We propose and investigate two new methods to approximate

f (A) b

for large, sparse, Hermitian matrices

A

. Computations of this form play an important role in numerous signal processing and machine learning tasks. The main idea behind both methods is to first estimate the spectral density of

A

, and then find polynomials of a fixed order that better approximate the function f on areas of the spectrum with a higher density of eigenvalues. Compared to state-of-the-art methods such as the Lanczos method and truncated Chebyshev expansion, the proposed methods tend to provide more accurate approximations of

f (A) b

at lower polynomial orders, and for matrices

A

with a large number of distinct interior eigenvalues and a small spectral width. We also explore the application of these techniques to (i) fast estimation of the norms of localized graph spectral filter dictionary atoms, and (ii) fast filtering of time-vertex signals. View Full-Text

Keywords: matrix function; spectral density estimation; polynomial approximation; orthogonal polynomials; graph spectral filtering; weighted least squares polynomial regression

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Fan, T.; Shuman, D.I.; Ubaru, S.; Saad, Y. Spectrum-Adapted Polynomial Approximation for Matrix Functions with Applications in Graph Signal Processing. Algorithms 2020, 13, 295. https://doi.org/10.3390/a13110295

AMA Style

Fan T, Shuman DI, Ubaru S, Saad Y. Spectrum-Adapted Polynomial Approximation for Matrix Functions with Applications in Graph Signal Processing. Algorithms. 2020; 13(11):295. https://doi.org/10.3390/a13110295

Chicago/Turabian Style

Fan, Tiffany, David I. Shuman, Shashanka Ubaru, and Yousef Saad. 2020. "Spectrum-Adapted Polynomial Approximation for Matrix Functions with Applications in Graph Signal Processing" Algorithms 13, no. 11: 295. https://doi.org/10.3390/a13110295

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Spectrum-Adapted Polynomial Approximation for Matrix Functions with Applications in Graph Signal Processing

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