On the singularity probability of discrete random matrices

Abstract

Let n be a large integer and M n be an n by n complex matrix whose entries are independent (but not necessarily identically distributed) discrete random variables. The main goal of this paper is to prove a general upper bound for the probability that M n is singular. For a constant 0 < p < 1 and a constant positive integer r, we will define a property p-bounded of exponent r. Our main result shows that if the entries of M n satisfy this property, then the probability that M n is singular is at most (p 1/r + o(1)) n . All of the results in this paper hold for any characteristic zero integral domain replacing the complex numbers. In the special case where the entries of M n are "fair coin flips" (taking the values +1, −1 each with probability 1/2), our general bound implies that the probability that M n is singular is at most ( 1 √ 2 + o(1)) n , improving on the previous best upper bound of ( 3 4 + o(1)) n , proved by Tao and Vu [Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603-628]. In the special case where the entries of M n are "lazy coin flips" (taking values +1, −1 each with probability 1/4 and value 0 with probability 1/2), our general bound implies that the probability that M n is singular is at most ( 1 2 + o(1)) n , which is asymptotically sharp. Our method is a refinement of those from [Jeff Kahn, János Komlós, Endre Szemerédi, On the probability that a random ±1-matrix is singular, J. Amer. Math. Soc. 8 (1) (1995) 223-240; Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603-628]. In particular, we make a critical use of the structure theorem from [Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603-628], which was obtained using tools from additive combinatorics.

Invertibility of symmetric random matrices

“…A motivating example is for random Bernoulli matrices B whose entries are valued symmetric random variables. If all entries are independent, it is conjectured that the singularity probability of B is , while the best current bound is due to Bourgain, Vu and Wood . The typical norm of the inverse in this case is , see .…”

On the universality of the non-singularity of general Ginibre and Wigner random matrices

“…taking values −1 or 1 with probability 1/2 each, showing that the probability of singularity is bounded above by θ n for θ = .999. The value of θ has been improved by Tao and Vu [25], [26] to θ = 3/4 + o(1) and by Bourgain, Vu and Wood [5] to θ = 1/ √ 2 + o(1). Slinko [22] considered Ginibre random matrices whose entries have the same uniform distribution taking values in a finite set, proving also that the probability of singularity is O(n −1/2 ) as n → ∞.…”

Jointly clustering rows and columns of binary matrices

“…Denote the SVD of the block rating matrix B by B = UBΣBV ⊤ B , then the SVD of R is simply R = U KΣB V ⊤ , where U = UC UB and V = VCVB. When r → ∞, B has full rank almost surely [7]. We will assume B is full rank in the following proofs, which implies that UBU ⊤ B = I and VBV ⊤ B = I.…”