Tight Sampling Bounds for Eigenvalue Approximation

David Woodruff develops sampling bounds for estimating the spectrum of symmetric matrices with bounded entries. Principal-submatrix sampling achieves epsilon times n additive error using roughly 1/epsilon² samples, eliminating dependence on n and improving prior epsilon dependence up to logarithmic factors. Squared row-norm sampling gives epsilon times the Frobenius norm accuracy with roughly 1/epsilon² samples, improving a previous 1/epsilon⁸ bound. For bounded-entry positive-semidefinite matrices, O(1/epsilon) sampled columns permit nonadaptive approximation of the leading eigenvector with epsilon times n additive error. Applications include faster dense-matrix spectral sketches and improved sample complexity. Joint work with William Swartworth.