Singularity of discrete random matrices

Abstract

Let p ∈ (0, 1/2) be fixed, and let Bn(p) be an n × n random matrix with i.i.d. Bernoulli random variables with mean p. We show that for all t ≥ 0,where sn(Bn(p)) denotes the least singular value of Bn(p) and Cp, ǫp > 0 are constants depending only on p.

Singularity of discrete random matrices

“…In the first paper [7] of this two-part work, we settled the above conjecture when ξ = Ber(p) for fixed p ∈ (0, 1/2), where Ber(p) denotes the random variable attaining the value 0 with probability 1 − p and the value 1 with probability p. Previously, works of Basak and Rudelson [1], Huang [5], and Litvak and Tikhomirov [10] established such a result for ξ = Ber(p n ), for all (log n−ω n (1))/n ≤ p n ≤ c, where c > 0 is a small universal constant and ω n (1) is some specific slowly growing function of n.…”

Rank deficiency of random matrices

“…The proof of Proposition 2.7 follows the usual format of taking a dyadic decomposition of possible values for the threshold function, performing randomized rounding on potential kernel vectors at the correct scale, and then tensorizing the resulting small ball probabilities. The difference in the statement above compared to the versions in [8,9,17] is that we are missing k rows as opposed to 1 row. Additionally, we are considering the independent threshold model rather than the "multislice" models considered in [9], which actually simplifies the proof.…”

“…Remark. A modification of our proof, with Proposition 2.7 replaced by the corresponding versions in [8,9], shows that for any fixed ξ which is supported on finitely many points,…”

“…By considering the probability of a row of the matrix being 0, one sees that the result of Tikhomirov is optimal up to the o n (1) term. Recently, several works [1,[7][8][9]12] have addressed the more refined question of determining the probability of singularity of M n (ξ) up to a (1 + o n (1)) factor; in contrast, the aforementioned result of Tikhomirov determines this probability only up to a subexponential (in n) factor. While these works have succeeded in the case of sparse Bernoulli matrices (with sparsity allowed to depend on n) [1,7,12], as well as in the case of a fixed ξ which is not uniform on its support [8,9], we note that for the case of Ber(1/2), the estimate of Tikhomirov remains essentially the best known.…”

The singularity probability of a random symmetric matrix is exponentially small

“…Another major advance on the problem was made recently by Jain, Sah and Sawhney [17,18], who (building on the recent work of Litvak and Tikhomirov [26]), proved the natural analogue of (1) for random matrices with independent entries chosen from a finite set S, for any non-uniform distribution on S. For the case of {−1, 1}-matrices, however, they were unable to improve on the bound of Tikhomirov.…”