Singularity of discrete random matrices

Abstract

Let ξ be a non-constant real-valued random variable with finite support, and let Mn(ξ) denote an n × n random matrix with entries that are independent copies of ξ. We show that, if ξ is not uniform on its support, then P[Mn(ξ) is singular] = (1 + on(1))P[zero row or column, or two equal (up to sign) rows or columns].For ξ which is uniform on its support, we show that P[Mn(ξ) is singular] = (1 + on(1)) n P[two rows or columns are equal].Corresponding estimates on the least singular value are also provided.

Securing Secure Aggregation: Mitigating Multi-Round Privacy Leakage in Federated Learning

“…2. In the i.i.d random selection scheme, the probability that the participation matrix after rounds P ( ) has full rank is lower-bounded as follows [28] Pr[P ( ) has full rank] ≥ 1 − 2 (1 − ) − (1 + (1)) ( − 1) ( 2 + (1 − ) 2 ) , which converges to 1 exponentially fast if = / ∈ (0, 1/2) is a fixed constant. Hence, it follows that the probability the server can reconstruct all individual models is lower-bounded by the same probability.…”

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.…”