Energy landscape for large average submatrix detection problems in Gaussian random matrices

Gaussian fluctuation in random matrices.

Let N(L) be the number of eigenvalues, in an interval of length L, of a matrix chosen at random from the Gaussian Orthogonal, Unitary or Symplectic ensembles of N by N matrices, in the limit N → ∞. We prove that [N(L) − 〈N(L)〉]/√logL has a Gaussian distribution when L → ∞. This theorem, which requires control of all the higher moments of the distribution, elucidates numerical and exact results ...

Improved Submatrix Maximum Queries in Monge Matrices

We present efficient data structures for submatrix maximum queries in Monge matrices and Monge partial matrices. For n× n Monge matrices, we give a data structure that requires O(n) space and answers submatrix maximum queries in O(logn) time. The best previous data structure [Kaplan et al., SODA‘12] required O(n logn) space and O(log n) query time. We also give an alternative data structure wit...

On the average number of sharp crossings of certain Gaussian random polynomials

Gaussian random projections for Euclidean membership problems

We discuss the application of random projections to the fundamental problem of deciding whether a given point in a Euclidean space belongs to a given set. We show that, under a number of different assumptions, the feasibility and infeasibility of this problem are preserved with high probability when the problem data is projected to a lower dimensional space. Our results are applicable to any al...

Condition Numbers of Gaussian Random Matrices

Abstract. Let Gm×n be an m × n real random matrix whose elements are independent and identically distributed standard normal random variables, and let κ2(Gm×n) be the 2-norm condition number of Gm×n. We prove that, for any m ≥ 2, n ≥ 2, and x ≥ |n − m| + 1, κ2(Gm×n) satisfies 1 √ 2π (c/x)|n−m|+1 < P ( κ2(Gm×n) n/(|n−m|+1) > x) < 1 √ 2π (C/x)|n−m|+1, where 0.245 ≤ c ≤ 2.000 and 5.013 ≤ C ≤ 6.414...