An O(n)-randomness perturbation combining a dense deterministic pattern matrix with a non-uniform sparse dependent perturbation reduces condition numbers to O(n) for any input matrix.
& Massart, P.Concentration Inequalities: A Nonasymptotic Theory of Independenceisbn: 9780199535255 (Oxford University Press, Oxford, 2013)
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Well-Conditioned Oblivious Perturbations in Linear Space
An O(n)-randomness perturbation combining a dense deterministic pattern matrix with a non-uniform sparse dependent perturbation reduces condition numbers to O(n) for any input matrix.