The Lanczos Algorithm Under Few Iterations: Concentration and Location of the Ritz Values
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We study the Lanczos algorithm where the initial vector is sampled uniformly from S^n-1. Let A be an n × n Hermitian matrix. We show that when run for few iterations, the output of the algorithm on A is almost deterministic. For instance, we show that there exists c >0 depending only on a certain global property of the spectrum of A (in particular, not depending on n) such that when Lanczos is run for at most c n iterations, the Jacobi coefficients and the Ritz values deviate from their medians by t with probability at most e^-√(n) t^2, for t< A _op. Furthermore, we show that the Lanczos algorithm fails with high probability to identify outliers of the spectrum when run for at most c' n iterations, where again c' depends only on the same global property of the spectrum of A. Classical results imply that the bound c' n is tight up to a constant factor. Our techniques also yield asymptotic results: Suppose we have a sequence of Hermitian matrices A_n ∈ M_n(C) whose spectral distributions converge in Kolmogorov distance with rate O(n^-ε) to a density, for some ε > 0. Then we show that for large enough n, and for k=O(√( n)), the Ritz values after k iterations concentrate around the roots of the kth orthogonal polynomial with respect to the limiting density.
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