Resilience Of The Rank Of Random Matrices

Let M be an n x m matrix of independent Rademacher (+/- 1) random variables. It is well known that if n <= m, then M is of full rank with high probability. We show that this property is resilient to adversarial changes to M. More precisely, if m >= n + n(1 -epsilon) , then even after changing the sign of (1 - epsilon)m/2 entries, M is still of full rank with high probability. Note that this is asymptotically best possible as one can easily make any two rows proportional with at most m/2 changes. Moreover, this theorem gives an asymptotic solution to a slightly weakened version of a conjecture made by Van Vu in [17].