Papers › Analysis of stochastic Lanczos quadrature for spectrum approximation

Analysis of stochastic Lanczos quadrature for spectrum approximation

13 May 2021arXiv:2105.06595links table onlyarchive 2025-07-28

Tyler Chen, Thomas Trogdon, Shashanka Ubaru

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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₌₁ⁿ 1/n x1D7D9[ λᵢ[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 | λₘₐₓ[𝐀] - λₘᵢₙ[𝐀] | with probability at least 1-η, by applying the Lanczos algorithm for ⌈12 t⁻¹ + 1/2 ⌉ iterations to ⌈4 ( n+2 )⁻¹t⁻² ln(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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