Analysis of stochastic Lanczos quadrature for spectrum approximation
The cumulative empirical spectral measure (CESM) ฮฆ[๐] : โโ [0,1] of a nร n symmetric matrix ๐ is defined as the fraction of eigenvalues of ๐ less than a given threshold, i.e., ฮฆ[๐](x) := โ_i=1^n1/nx1D7D9[ ฮป_i[๐]โค x]. Spectral sums tr(f[๐]) can be computed as the RiemannโStieltjes integral of f against ฮฆ[๐], 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 | ฮป_max[๐] - ฮป_min[๐] | with probability at least 1-ฮท, by applying the Lanczos algorithm for โ 12 t^-1 + 1/2โ iterations to โ 4 ( n+2 )^-1t^-2ln(2nฮท^-1) โ vectors sampled independently and uniformly from the unit sphere. We additionally provide (matrix-dependent) a posteriori error bounds for the Wasserstein and KolmogorovโSmirnov distances between the output of this algorithm and the true CESM. The quality of our bounds is demonstrated using numerical experiments.
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