# 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.

READ FULL TEXT