Sublinear Time Eigenvalue Approximation via Random Sampling
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We study the problem of approximating the eigenspectrum of a symmetric matrix A ∈ℝ^n × n with bounded entries (i.e., A_∞≤ 1). We present a simple sublinear time algorithm that approximates all eigenvalues of A up to additive error ±ϵ n using those of a randomly sampled Õ(1/ϵ^4) ×Õ(1/ϵ^4) principal submatrix. Our result can be viewed as a concentration bound on the full eigenspectrum of a random principal submatrix. It significantly extends existing work which shows concentration of just the spectral norm [Tro08]. It also extends work on sublinear time algorithms for testing the presence of large negative eigenvalues in the spectrum [BCJ20]. To complement our theoretical results, we provide numerical simulations, which demonstrate the effectiveness of our algorithm in approximating the eigenvalues of a wide range of matrices.
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