Singularity of biased discrete random matrix

Abstract

We study the singularity probability of n*n random matrices with i.i.d. entries from highly biased discrete distributions. We obtain sharp non-asymptotic bounds for this probability and derive estimates on the least singular values. Our method combines combinatorial, geometric, and probabilistic techniques such as sphere decomposition and anticoncentration inequalities. The results extend classical invertibility theory to biased discrete settings and resolve an open problem by characterizing the dominant causes of singularity in biased discrete random matrices, namely the presence of zero columns or linearly dependent column pairs.