Analysis of stochastic Lanczos quadrature for spectrum approximation
The cumulative empirical spectral measure (CESM) Ξ¦[π] : ββ [0,1] of a nΓ n symmetric matrix π is defined as the fraction of eigenvalues of π less than a given threshold, i.e., Ξ¦[π](x) := β_i=1^n1/nx1D7D9[ Ξ»_i[π]β€ x]. Spectral sums tr(f[π]) can be computed as the RiemannβStieltjes integral of f against Ξ¦[π], 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 | Ξ»_max[π] - Ξ»_min[π] | with probability at least 1-Ξ·, by applying the Lanczos algorithm for β 12 t^-1 + 1/2β iterations to β 4 ( n+2 )^-1t^-2ln(2nΞ·^-1) β vectors sampled independently and uniformly from the unit sphere. We additionally provide (matrix-dependent) a posteriori error bounds for the Wasserstein and KolmogorovβSmirnov 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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