Applications of Trace Estimation Techniques

Ubaru, Shashanka; Saad, Yousef

doi:10.1007/978-3-319-97136-0_2

Cited by 16 publications

(21 citation statements)

References 30 publications

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“…which completes the proof. 9 Replacing A i 2 with A ii in equation (33), yields the same result as in equation (40).…”

Section: Prmentioning

confidence: 80%

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Stochastic diagonal estimation: probabilistic bounds and an improved algorithm

Baston¹,

Nakatsukasa²

2022

Preprint

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We study the problem of estimating the diagonal of an implicitly given matrix A. For such a matrix we have access to an oracle that allows us to evaluate the matrix vector product Av. For random variable v drawn from an appropriate distribution, this may be used to return an estimate of the diagonal of the matrix A. Whilst results exist for probabilistic guarantees relating to the error of estimates of the trace of A, no such results have yet been derived for the diagonal. We make two contributions in this regard. We analyse the number of queries s required to guarantee that with probability at least 1 − δ the estimates of the relative error of the diagonal entries is at most ε. We extend this analysis to the 2-norm of the difference between the estimate and the diagonal of A. We prove, discuss and experiment with bounds on the number of queries s required to guarantee a probabilistic bound on the estimates of the diagonal by employing Rademacher and Gaussian random variables. Two sufficient upper bounds on the minimum number of query vectors are proved, extending the work of Avron and Toledo [JACM 58(2)8, 2011], and later work of Roosta-Khorasani and Ascher [FoCM 15, 1187[FoCM 15, -1212[FoCM 15, , 2015. We first prove sufficient bounds for diagonal estimation with Gaussian vectors, and then similarly with Rademacher vectors. We find that, generally, there is little difference between the two, with convergence going as O(log(1/δ)/ε 2 ) for individual diagonal elements. However for small s, we find that the Rademacher estimator is superior. These results allow us to then extend the ideas of Meyer, Musco, Musco and Woodruff [SOSA, 142-155, 2021], suggesting algorithm Diag++, to speed up the convergence of diagonal estimation from O(1/ε 2 ) to O(1/ε) and make it robust to the spectrum of

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“…which completes the proof. 9 Replacing A i 2 with A ii in equation (33), yields the same result as in equation (40).…”

Section: Prmentioning

confidence: 80%

“…We now examine the case where A 0, as is relevant in many applications [7,8,9,12,20,23,25,33], and analysed extensively in [3,30]. For both estimators we have…”

Section: A Brief Comparisonmentioning

confidence: 99%

Stochastic diagonal estimation: probabilistic bounds and an improved algorithm

Baston¹,

Nakatsukasa²

2022

Preprint

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“…Given an updated graph 𝐺 𝑟 , estimating its connectivity 𝜆(𝐺 𝑟 ) efficiently and precisely is crucial to answering CT-Bus. We solve this problem by converting it to a matrix trace estimation problem, which can be rapidly approximated by combining the Lanczos method [19,27,45,54,55]2 with Hutchinson's stochastic trace estimator [16,26,36,42]. These techniques are often used together in the applied mathematics literature and turn out to be a winning combination for accurately estimating natural connectivity in this paper.…”

Section: Lanczos-based Connectivity Estimationmentioning

confidence: 99%

Public Transport Planning: When Transit Network Connectivity Meets Commuting Demand

Wang,

Sun,

Musco

et al. 2021

Preprint

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In this paper, we make a first attempt to incorporate both commuting demand and transit network connectivity in bus route planning (CT-Bus), and formulate it as a constrained optimization problem: planning a new bus route with 𝑘 edges over an existing transit network without building new bus stops to maximize a linear aggregation of commuting demand and connectivity of the transit network. We prove the NP-hardness of CT-Bus and propose an expansion-based greedy algorithm that iteratively scans potential candidate paths in the network. To boost the efficiency of computing the connectivity of new networks with candidate paths, we convert it to a matrix trace estimation problem and employ a Lanczos method to estimate the natural connectivity of the transit network with a guaranteed error bound. Furthermore, we derive upper bounds on the objective values and use them to greedily select candidates for expansion. Our experiments conducted on real-world transit networks in New York City and Chicago verify the efficiency, effectiveness, and scalability of our algorithms.

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“…We examine four functions in particular: f (x) = √ x (for estimating the nuclear norm), f (x) = log(x) (log determinant), f (x) = exp(x) (Estrada index), and f (x) = x 3 (triangle counting). For information on practical applications, we refer the reader to [16] or [8] and the references therein.…”

Section: Numerical Experimentsmentioning

confidence: 99%

A Multilevel Approach to Stochastic Trace Estimation

Hallman¹,

Troester²

2021

Preprint

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This article presents a randomized matrix-free method for approximating the trace of f (A), where A is a large symmetric matrix and f is a function analytic in a closed interval containing the eigenvalues of A. Our method uses a combination of stochastic trace estimation (i.e., Hutchinson's method), Chebyshev approximation, and multilevel Monte Carlo techniques. We establish general bounds on the approximation error of this method by extending an existing error bound for Hutchinson's method to multilevel trace estimators. Numerical experiments are conducted for common applications such as estimating the log-determinant, nuclear norm, and Estrada index, and triangle counting in graphs. We find that using multilevel techniques can substantially reduce the variance of existing single-level estimators.

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Applications of Trace Estimation Techniques

Cited by 16 publications

References 30 publications

Stochastic diagonal estimation: probabilistic bounds and an improved algorithm

Stochastic diagonal estimation: probabilistic bounds and an improved algorithm

Public Transport Planning: When Transit Network Connectivity Meets Commuting Demand

A Multilevel Approach to Stochastic Trace Estimation

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