Randomized algorithms for estimating the trace of an implicit symmetric positive semi-definite matrix

Avron, Haim; Toledo, Sivan

doi:10.1145/1944345.1944349

Cited by 274 publications

(429 citation statements)

References 20 publications

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“…Here, if we define 2) and define the DOS φ(ω) using the square root of the eigenvalues of the Hessian associated with a molecular potential function with respect to atomic coordinates of the molecule, we have…”

Section: Applicationmentioning

confidence: 99%

Approximating Spectral Densities of Large Matrices

Saad²,

2016

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Abstract. In physics, it is sometimes desirable to compute the so-called Density Of States (DOS), also known as the spectral density, of a real symmetric matrix A. The spectral density can be viewed as a probability density distribution that measures the likelihood of finding eigenvalues near some point on the real line. The most straightforward way to obtain this density is to compute all eigenvalues of A. But this approach is generally costly and wasteful, especially for matrices of large dimension. There exists alternative methods that allow us to estimate the spectral density function at much lower cost. The major computational cost of these methods is in multiplying A with a number of vectors, which makes them appealing for large-scale problems where products of the matrix A with arbitrary vectors are relatively inexpensive. This paper defines the problem of estimating the spectral density carefully, and discusses how to measure the accuracy of an approximate spectral density. It then surveys a few known methods for estimating the spectral density, and proposes some new variations of existing methods. All methods are discussed from a numerical linear algebra point of view.

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Section: Applicationmentioning

confidence: 99%

Approximating Spectral Densities of Large Matrices

Saad²,

2016

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“…The Hutchinson estimator has lower variance than Girard's but requires more samples to achieve a given relative precision (Avron and Toledo, 2011). In addition, using the Gaussian distribution is more natural in data assimilation and allows Tr(HK) to be computed as a by-product of an ensemble of variational assimilations 192 Y. Michel: Diagnostics on the cost-function (Desroziers et al, 2009).…”

Section: Application Of the Randomized Trace Estimatormentioning

confidence: 99%

Diagnostics on the cost-function in variational assimilations for meteorological models

Michel

2014

Nonlin. Processes Geophys.

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Abstract. Several consistency diagnostics have been proposed to evaluate variational assimilation schemes. The "Bennett-Talagrand" criterion in particular shows that the cost-function at the minimum should be close to half the number of assimilated observations when statistics are correctly specified. It has been further shown that sub-parts of the cost function also had statistical expectations that could be expressed as traces of large matrices, and that this could be exploited for variance tuning and hypothesis testing.The aim of this work is to extend those results using standard theory of quadratic forms in random variables. The first step is to express the sub-parts of the cost function as quadratic forms in the innovation vector. Then, it is possible to derive expressions for the statistical expectations, variances and cross-covariances (whether the statistics are correctly specified or not). As a consequence it is proven in particular that, in a perfect system, the values of the background and observation parts of the cost function at the minimum are positively correlated. These results are illustrated in a simplified variational scheme in a one-dimensional context.These expressions involve the computation of the trace of large matrices that are generally unavailable in variational formulations of the assimilation problem. It is shown that the randomization algorithm proposed in the literature can be extended to cover these computations, yet at the price of additional minimizations. This is shown to provide estimations of background and observation errors that improve forecasts of the operational ARPEGE model.

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“…The numerator of (5.16) is the trace of an implicit matrix, and hence can be approximated by Hutchinson's [2,24] Monte-Carlo method. To apply L † , we can utilize nearly-linear time solvers for Laplacian systems [47,13].…”

Section: Inputmentioning

confidence: 99%

Kirchhoff Index as a Measure of Edge Centrality in Weighted Networks: Nearly Linear Time Algorithms

Li¹,

Zhang²

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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Estimating the relative importance of vertices and edges is a fundamental issue in the analysis of complex networks, and has found vast applications in various aspects, such as social networks, power grids, and biological networks. Most previous work focuses on metrics of vertex importance and methods for identifying powerful vertices, while related work for edges is much lesser, especially for weighted networks, due to the computational challenge. In this paper, we propose to use the well-known Kirchhoff index as the measure of edge centrality in weighted networks, called θ-Kirchhoff edge centrality. The Kirchhoff index of a network is defined as the sum of effective resistances over all vertex pairs. The centrality of an edge e is reflected in the increase of Kirchhoff index of the network when the edge e is partially deactivated, characterized by a parameter θ. We define two equivalent measures for θ-Kirchhoff edge centrality. Both are global metrics and have a better discriminating power than commonly used measures, based on local or partial structural information of networks, e.g. edge betweenness and spanning edge centrality. Despite the strong advantages of Kirchhoff index as a centrality measure and its wide applications, computing the exact value of Kirchhoff edge centrality for each edge in a graph is computationally demanding. To solve this problem, for each of the θ-Kirchhoff edge centrality metrics, we present an efficient algorithm to compute its -approximation for all the m edges in nearly linear time in m. The proposed θ-Kirchhoff edge centrality is the first global metric of edge importance that can be provably approximated in nearly-linear time. Moreover, according to the θ-Kirchhoff edge centrality, we present a θ-Kirchhoff vertex centrality measure, as well as a fast algorithm that can compute -approximate Kirchhoff vertex centrality for all the n vertices in nearly linear time in m.

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Randomized algorithms for estimating the trace of an implicit symmetric positive semi-definite matrix

Cited by 274 publications

References 20 publications

Approximating Spectral Densities of Large Matrices

Approximating Spectral Densities of Large Matrices

Diagnostics on the cost-function in variational assimilations for meteorological models

Kirchhoff Index as a Measure of Edge Centrality in Weighted Networks: Nearly Linear Time Algorithms

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