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We study algorithms for approximating the spectral density (i.e., the eigenvalue distribution) of a symmetric matrix A ∈ ℝn×n that is accessed through matrix-vector product queries. Recent work has analyzed popular Krylov subspace methods for this problem, showing that they output an ∈ · || A||2 error approximation to the spectral density in the Wasserstein-1 metric using O (1/∈ ) matrix-vector products. By combining a previously studied Chebyshev polynomial moment matching method with a deflation step that approximately projects off the largest magnitude eigendirections of A before estimating the spectral density, we give an improved error bound of ∈ · σℓ (A) using O (ℓ log n + 1/∈ ) matrix-vector products, where σℓ (A) is the ℓth largest singular value of A. In the common case when A exhibits fast singular value decay and so σℓ (A) « ||A||2, our bound can be much stronger than prior work. We also show that it is nearly tight: any algorithm giving error ∈ · σℓ (A) must use Ω(ℓ + 1/∈ ) matrix-vector products. We further show that the popular Stochastic Lanczos Quadrature (SLQ) method essentially matches the above bound for any choice of parameter ℓ, even though SLQ itself is parameter-free and performs no explicit deflation. Our bound helps to explain the strong practical performance and observed ‘spectrum adaptive’ nature of SLQ, and motivates a simple variant of the method that achieves an even tighter error bound. Technically, our results require a careful analysis of how eigenvalues and eigenvectors are approximated by (block) Krylov subspace methods, which may be of independent interest. Our error bound for SLQ leverages an analysis of the method that views it as an implicit polynomial moment matching method, along with recent results on low-rank approximation with single-vector Krylov methods. We use these results to show that the method can perform ‘implicit deflation’ as part of moment matching. 

We study algorithms for approximating pairwise similarity matrices that arise in natural language processing. Generally, computing a similarity matrix for $$n$$ data points requires $$\Omega(n^2)$$ similarity computations. This quadratic scaling is a significant bottleneck, especially when similarities are computed via expensive functions, e.g., via transformer models. Approximation methods reduce this quadratic complexity, often by using a small subset of exactly computed similarities to approximate the remainder of the complete pairwise similarity matrix. Significant work focuses on the efficient approximation of positive semidefinite (PSD) similarity matrices, which arise e.g., in kernel methods. However, much less is understood about indefinite (non-PSD) similarity matrices, which often arise in NLP. Motivated by the observation that many of these matrices are still somewhat close to PSD, we introduce a generalization of the popular \emph{Nystrom method} to the indefinite setting. Our algorithm can be applied to any similarity matrix and runs in sublinear time in the size of the matrix, producing a rank-$$s$$ approximation with just $O(ns)$ similarity computations. We show that our method, along with a simple variant of CUR decomposition, performs very well in approximating a variety of similarity matrices arising in NLP tasks. We demonstrate high accuracy of the approximated similarity matrices in tasks of document classification, sentence similarity, and cross-document coreference.