Analysis of stochastic Lanczos quadrature for spectrum approximation

by   Tyler Chen, et al.

The cumulative empirical spectral measure (CESM) Ξ¦[𝐀] : ℝ→ [0,1] of a nΓ— n symmetric matrix 𝐀 is defined as the fraction of eigenvalues of 𝐀 less than a given threshold, i.e., Ξ¦[𝐀](x) := βˆ‘_i=1^n1/nx1D7D9[ Ξ»_i[𝐀]≀ x]. Spectral sums tr(f[𝐀]) can be computed as the Riemann–Stieltjes integral of f against Ξ¦[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 | Ξ»_max[𝐀] - Ξ»_min[𝐀] | with probability at least 1-Ξ·, by applying the Lanczos algorithm for ⌈ 12 t^-1 + 1/2βŒ‰ iterations to ⌈ 4 ( n+2 )^-1t^-2ln(2nΞ·^-1) βŒ‰ 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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