The smallest singular value of random rectangular matrices with no moment assumptions on entries

By Konstantin E. Tikhomirov. 2014

49 citations recorded by the source corpus.

Let δ > 1 and β > 0 be some real numbers. We prove that there are positive u, v, N0 depending only on β and δ with the following property: for any N,n such that N ≥ max(N0, δn), any N × n random matrix A = (aij) with i.i.d. entries satisfying $${\sup _{\lambda \in \mathbb{R}}}P\left\{ {\left| {{a_{11}} - \lambda } \right| \leqslant 1} \right\} \leqslant 1 - \beta $$ and any non-random N × n matrix B, the smallest singular value sn of A + B satisfies $$P\left\{ {{s_n}\left( {A + B} \right) \leqslant u\sqrt N } \right\} \leqslant \exp \left( { - vN} \right)$$. The result holds without any moment assumptions on the distribution of the entries of A.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/bfecb2320564654923dafebebc5980c15b0c830d

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Citations in the corpus

The smallest singular value of inhomogeneous square random matrices (2019, background): “The behavior of the smallest singular values of random matrices have been extensively studied [2, 4, 5, 11, 15, 16, 21, 19, 23, 24, 25, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42].”
Concentration inequalities for random tensors (2019, unknown): “For d = 1, this problem has been extensively studied in random matrix theory [16, 31, 23, 32, 25, 3, 26, 34, 27, 13, 10, 35, 36].”
Sample covariance matrices of heavy-tailed distributions (2016, result): “lassical works [33] and [3] regarding almost sure convergence of appropriately normalized singular values when the coordinates of the underlying distributions are i.i.d.; as well as more recent works [22, 14, 18, 25, 9, 1, 2, 29, 19, 26, 20, 17, 28, 11, 10, 31, 32, 21, 27, 8]. For a more comprehensi”