A randomized algorithm for approximating the log determinant of a symmetric positive definite matrix

Boutsidis, Christos; Drineas, Petros; Kambadur, Prabhanjan; Kontopoulou, Eugenia-Maria; Zouzias, Anastasios

doi:10.1016/j.laa.2017.07.004

Cited by 47 publications

(56 citation statements)

References 19 publications

(29 reference statements)

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“…They do not provide rigorous bounds as we provide for our method. Boutsidis et al [6] use a similar scheme based on Taylor expansion for approximating the log-determinant, and do provide rigorous bounds. Nevertheless, our experiments demonstrate that our Chebyshev interpolation based method provides superior accuracy.…”

Section: Related Workmentioning

confidence: 99%

Approximating Spectral Sums of Large-Scale Matrices using Stochastic Chebyshev Approximations

Han¹,

Malioutov²,

Avron³

et al. 2017

SIAM J. Sci. Comput.

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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. * This article is partially based on preliminary results published in the proceeding of the 32nd International Conference on Machine Learning (ICML 2015).

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Section: Related Workmentioning

confidence: 99%

Approximating Spectral Sums of Large-Scale Matrices using Stochastic Chebyshev Approximations

Han¹,

Malioutov²,

Avron³

et al. 2017

SIAM J. Sci. Comput.

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“…In contrast, our proposed entropic approach implicitly encodes the constraint that densities are necessarily positive. Given equivalent moment information, we achieve competitive results on matrices obtained from the SuiteSparse Matrix Collection [11] which consistently outperform competing approximations to the log-determinant [12,7].…”

Section: Introductionmentioning

confidence: 99%

Entropic Trace Estimates for Log Determinants

Fitzsimons

Granziol

Cutajar

et al. 2017

Lecture Notes in Computer Science

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Abstract. The scalable calculation of matrix determinants has been a bottleneck to the widespread application of many machine learning methods such as determinantal point processes, Gaussian processes, generalised Markov random fields, graph models and many others. In this work, we estimate log determinants under the framework of maximum entropy, given information in the form of moment constraints from stochastic trace estimation. The estimates demonstrate a significant improvement on state-of-the-art alternative methods, as shown on a wide variety of UFL sparse matrices. By taking the example of a general Markov random field, we also demonstrate how this approach can significantly accelerate inference in large-scale learning methods involving the log determinant.

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“…Our work has been inspired by the approach of Boutsidis, Drineas, Kambadur, Kontopoulou, and Zouzias [1] that uses Taylor expansion for log(1 + x) to approximate the log determinant. The coefficients of this expansion have the same sign and thus the Hutchinson estimator [3] can be applied to each partial sum of this expansion.…”

Section: Our Results and Related Workmentioning

confidence: 99%

Approximations of Schatten Norms via Taylor Expansions

Braverman

2019

Computer Science – Theory and Applications

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In this paper we consider symmetric, positive semidefinite (SPSD) matrix A and present two algorithms for computing the p-Schatten norm A p . The first algorithm works for any SPSD matrix A. The second algorithm works for non-singular SPSD matrices and runs in time that depends on κ = λ 1 (A) λn(A) , where λi(A) is the i-th eigenvalue of A. Our methods are simple and easy to implement and can be extended to general matrices. Our algorithms improve, for a range of parameters, recent results of Musco, Netrapalli, Sidford, Ubaru and Woodruff (ITCS 2018.) and match the running time of the methods by Han, Malioutov, Avron, and Shin (SISC 2017) while avoiding computations of coefficients of Chebyshev polynomials. in [4,5] for a range of parameters, for example, when p > 1 and ǫ = o(n

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A randomized algorithm for approximating the log determinant of a symmetric positive definite matrix

Cited by 47 publications

References 19 publications

Approximating Spectral Sums of Large-Scale Matrices using Stochastic Chebyshev Approximations

Approximating Spectral Sums of Large-Scale Matrices using Stochastic Chebyshev Approximations

Entropic Trace Estimates for Log Determinants

Approximations of Schatten Norms via Taylor Expansions

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