On the Discrepancy of Random Matrices with Many Columns

Motivated by the Koml\'os conjecture in combinatorial discrepancy, we study the discrepancy of random matrices with $m$ rows and $n$ independent columns drawn from a bounded lattice random variable. It is known that for $n$ tending to infinity and $m$ fixed, with high probability the $\ell_\infty$-discrepancy is at most twice the $\ell_\infty$-covering radius of the integer span of the support of the random variable. However, the easy argument for the above fact gives no concrete bounds on the failure probability in terms of $n$. We prove that the failure probability is inverse polynomial in $m, n$ and some well-motivated parameters of the random variable. We also obtain the analogous bounds for the discrepancy in arbitrary norms. We apply these results to two random models of interest. For random $t$-sparse matrices, i.e. uniformly random matrices with $t$ ones and $m-t$ zeroes in each column, we show that the $\ell_\infty$-discrepancy is at most $2$ with probability $1 - O(\sqrt{ \log n/n})$ for $n = \Omega(m^3 \log^2 m)$. This improves on a bound proved by Ezra and Lovett (Ezra and Lovett, Approx+Random, 2016) showing that the same is true for $n$ at least $m^t$. For matrices with random unit vector columns, we show that the $\ell_\infty$-discrepancy is $O(\exp(\sqrt{ n/m^3}))$ with probability $1 - O(\sqrt{ \log n/n})$ for $n = \Omega(m^3 \log^2 m).$ Our approach, in the spirit of Kuperberg, Lovett and Peled (G. Kuperberg, S. Lovett and R. Peled, STOC 2012), uses Fourier analysis to prove that for $m \times n$ matrices $M$ with i.i.d. columns, and $m$ sufficiently large, the distribution of $My$ for random $y \in \{-1,1\}^n$ obeys a local limit theorem.