Resilience of the rank of random matrices

Motivated by problems from compressed sensing, we determine the threshold behaviour of a random $n\times d \pm 1$ matrix $M_{n,d}$ with respect to the property ‘every $s$ columns are linearly independent’. In particular, we show that for every $0\lt \delta \lt 1$ and $s=(1-\delta )n$ , if $d\leq n^{1+1/2(1-\delta )-o(1)}$ then with high probability every $s$ columns of $M_{n,d}$ are linearly independent, and if $d\geq n^{1+1/2(1-\delta )+o(1)}$ then with high probability there are some $s$ linearly dependent columns.

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“…We now are in position to prove Proposition 3.1. The proof is quite similar to the proofs in [6][7][8].…”

Section: Proofsupporting

confidence: 58%

Sparse recovery properties of discrete random matrices

Ferber

Sah

Sawhney

et al. 2022

Combinator. Probab. Comp.

Self Cite

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“…A recent combinatorial innovation [6] provides a method of bypassing the inverse theorems and extracting super-polynomial probability bounds. This method has been applied successfully to many discrete random matrix questions [5,13,7,9,11]. In [10], Jain introduced a method of lattice approximations to extend least singular value bounds to matrices of superpolynomial norm perturbed by i.i.d.…”

Section: Proof Strategymentioning

confidence: 99%

“…A simple consequence of this theorem is that N n has simple spectrum with probability at most n −C for any C > 0. In the present work, we build on some recent combinatorial techniques in random matrix theory [6,7,5,9,10,11,13]. Using this method, we provide the first tail bounds for gap sizes of M n when M can have operator norm exponential in the size of the matrix.…”

Section: Introductionmentioning

confidence: 99%

Eigenvalue Gaps of Random Perturbations of Large Matrices

Luh¹,

Vogel²,

Yu³

2022

Preprint

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The current work applies some recent combinatorial tools due to Jain to control the eigenvalue gaps of a matrix Mn = M + Nn where M is deterministic, symmetric with large operator norm and Nn is a random symmetric matrix with subgaussian entries. One consequence of our tail bounds is that Mn has simple spectrum with probability at least 1 − exp(−n 2/15 ) which improves on a result of Nguyen, Tao and Vu in terms of both the probability and the size of the matrix M .

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“…Thus, Res can be seen as the resilience of the matrix against an effort to reduce its rank. It is easy to show that Res(M n ) is, with high probability, at most (1/2 + o(1))n. For a recent partial result, see [30]. A closely related question (motivated by the notion of local resilience from [70]) is the following.…”

Section: Miscellanymentioning

confidence: 99%