Concentration of Random Determinants and Permanent Estimators

Abstract

Abstract. We show that the absolute value of the determinant of a matrix with random independent (but not necessarily iid) entries is strongly concentrated around its mean.As an application, we show that Godsil-Gutman and Barvinok estimators for the permanent of a strictly positive matrix give sub-exponential approximation ratios with high probability.A positive answer to the main conjecture of the paper would lead to polynomial approximation ratios in the above problem.

Untitled

“…To do so, we use the following beautiful characterization, which can be found (for example) in Costello and Vu [18]. Lemma 8.3 ([18]).…”

“…, x m ) of degree n. Here the x i 's can be thought of as just formal variables. 18 The standard initial state |1 n corresponds to the degree-n polynomial J m,n (x 1 , . .…”

Singular values of Gaussian matrices and permanent estimators

“…Indeed, if for some fixed constants one has , then for any , with G denoting the standard Gaussian matrix, uniformly in A ; that is, for such matrices this estimator achieves subexponentional (in n ) errors, with (arithmetic) running time. An improved analysis in presented in , where it is shown that the approximation error in the same set of matrices is only exponential in .…”

“…Thus, the interest is in obtaining algorithms that compute approximations to the permanent, and indeed a polynomial running time Markov Chain Monte Carlo randomized algorithm that evaluates per(A) (up to (1 + ) multiplicative errors, with complexity polynomial in ) is available [9]. In practice, however, the running time of such an algorithm, which is O(n 10 ), still makes it challenging to implement for large n. (An alternative, faster MCMC algorithm is presented in [3], with claimed running time of O(n 7 (log n) 4 ). )…”

“…uniformly in A; that is, for such matrices this estimator achieves subexponentional (in n) errors, with o(n 3 ) (arithmetic) running time. An improved analysis in presented in [4], where it is shown that the approximation error in the same set of matrices is only exponential in n 2/3 log n.…”