On the discrepancy of random matrices with many columns

Abstract

Motivated by the Komlós 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 ℓ ∞ -discrepancy is at most twice the ℓ ∞ -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 ℓ ∞ -discrepancy is at most 2 with probability 1 − O( log n/n) for n = Ω(m 3 log 2 m). This improves on a bound proved by Ezra and Lovett (Ezra and Lovett, Approx+Random, 2015) showing that the same is true for n at least m t . For matrices with random unit vector columns, we show that the ℓ ∞ -discrepancy is O(exp( n/m 3 )) with probability 1 − O( log n/n) for n = Ω(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 × n matrices M with i.i.d. columns, and n sufficiently large, the distribution of M y for random y ∈ {−1, 1} n obeys a local limit theorem.

Discrepancy in random hypergraph models

“…Another result that Ezra and Lovett proved, in [EL15], that if n ≫ m t , then the discrepancy of the random t-regular hypergraph is at most O(1). This has recently been improved due to Franks and Saks [FS18], who prove that as long as n = Ω m 3 log 2 m , the discrepancy of the random t-regular hypergraph is almost surely at most 1.…”

“…The o(1) in the above theorem is O 1 √ log n . One thing that is common to [FS18] and [HR18] is a multivariate local limit theorem on the number of colorings that give discrepancy at most 1, inspired by [KLP12]. Our approach for this is more direct and essentially just amounts to a careful second moment computation of the number of colorings which give discrepancy at most 1, and is the content of Section 4.…”

A spectral bound on hypergraph discrepancy

“…We argue that this is an interesting regime for random hypergraphs, as this kind of discrepancy bound is not implied by the Beck-Fiala conjecture. The case where n = Ω(m log m), is of particular interest, and we conclude with a conjecture, building on an open problem (open problem 1) in [FS18]: Conjecture 3.1. There is an absolute constant K > 0 such that the following holds.…”

“…However, in this regime, we believe that with high probability, the discrepancy is much lower than √ t (in contrast to λ growing). Recently, Franks and Saks [FS18] showed that for n = Ω(m 3 ), the discrepancy is O(t). We argue that this is an interesting regime for random hypergraphs, as this kind of discrepancy bound is not implied by the Beck-Fiala conjecture.…”

The Discrepancy of Random Rectangular Matrices

“…Ezra and Lovett [19] consider a regular model in which a binary matrix A is chosen uniformly at random conditioned on each column having exactly t ones for some sparsity parameter t. They show when n > m t , then disc(A) = O(1) with high probability. Franks and Saks [21] removed this exponential dependency on t: they showed that if n = Ω m 3 log 2 m , then disc(A) ≤ 2 with high probability. In an independent and simultaneous work, Hoberg and Rothvoß [24] consider A drawn from the (m, n, p)-Bernoulli Ensemble and gave the improved bound that disc(A) ≤ 1 with high probability if n = Ω m 2 log m and mp = Ω(log n).…”