Analysis of stochastic Lanczos quadrature for spectrum approximation
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The cumulative empirical spectral measure (CESM) Φ[𝐀] : ℝ→ [0,1] of a n× n symmetric matrix 𝐀 is defined as the fraction of eigenvalues of 𝐀 less than a given threshold, i.e., Φ[𝐀](x) := ∑_i=1^n1/nx1D7D9[ λ_i[𝐀]≤ x]. Spectral sums tr(f[𝐀]) can be computed as the Riemann–Stieltjes integral of f against Φ[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 | λ_max[𝐀] - λ_min[𝐀] | with probability at least 1-η, by applying the Lanczos algorithm for ⌈ 12 t^-1 + 1/2⌉ iterations to ⌈ 4 ( n+2 )^-1t^-2ln(2nη^-1) ⌉ vectors sampled independently and uniformly from the unit sphere. We additionally provide (matrix-dependent) 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.
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