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The cumulative empirical spectral measure (CESM) $\Phi[\mathbf{A}] : \mathbb{R} \to [0,1]$ of a $n\times n$ symmetric matrix $\mathbf{A}$ is defined as the fraction of eigenvalues of $\mathbf{A}$ less than a given threshold, i.e., $\Phi[\mathbf{A}](x) := \sum_{i=1}^{n} \frac{1}{n} {\large\unicode{x1D7D9}}[ \lambda_i[\mathbf{A}]\leq x]$. Spectral sums $\operatorname{tr}(f[\mathbf{A}])$ can be computed as the Riemann–Stieltjes integral of $f$ against $\Phi[\mathbf{A}]$, so the task of estimating CESM arises frequently in a number of applications, including machine learning. We present an error analysis for stochastic Lanczos quadrature (SLQ). We show that SLQ obtains an approximation to the CESM within a Wasserstein distance of $t \:  \lambda_{\text{max}}[\mathbf{A}]  \lambda_{\text{min}}[\mathbf{A}] $ with probability at least $1\eta$, by applying the Lanczos algorithm for $\lceil 12 t^{1} + \frac{1}{2} \rceil$ iterations to $\lceil 4 ( n+2 )^{1}t^{2} \ln(2n\eta^{1}) \rceil$ vectors sampled independently and uniformly from the unit sphere. We additionally provide (matrixdependent) a posteriori error bounds for the Wasserstein and Kolmogorov–Smirnov distances between the output of this algorithm and the true CESM. The quality of our bounds is demonstrated using numerical experiments.more » « less

Ubaru, Shashanka ; Saad, Yousef ( , Numerical Linear Algebra with Applications)
Summary This paper addresses matrix approximation problems for matrices that are large, sparse, and/or representations of large graphs. To tackle these problems, we consider algorithms that are based primarily on coarsening techniques, possibly combined with random sampling. A multilevel coarsening technique is proposed, which utilizes a hypergraph associated with the data matrix and a graph coarsening strategy based on column matching. We consider a number of standard applications of this technique as well as a few new ones. Among standard applications, we first consider the problem of computing
partial singular value decomposition , for which a combination of sampling and coarsening yields significantly improved singular value decomposition results relative to sampling alone. We also consider thecolumn subset selection problem, a popular low‐rank approximation method used in data‐related applications, and show how multilevel coarsening can be adapted for this problem. Similarly, we consider the problem ofgraph sparsification and show how coarsening techniques can be employed to solve it. We also establish theoretical results that characterize the approximation error obtained and the quality of the dimension reduction achieved by a coarsening step, when a proper column matching strategy is employed. Numerical experiments illustrate the performances of the methods in a few applications.