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. We prove that for n at least polynomial in m , 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. Applying this result to random t ‐sparse matrices, that is, 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 for . This improves on a bound proved by Ezra and Lovett showing the same bound for n at least m ^t .

中文翻译：

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关于多列随机矩阵的差异

受Komlós猜想在组合差异中的启发，我们研究了从有界格随机变量中抽取m行n独立列的随机矩阵的差异。我们证明了对ñ在至少多项式米，高概率的ℓ _∞ -discrepancy最多两倍ℓ _∞ -covering的支持下，随机变量的整数跨度的半径。将这个结果应用于随机t稀疏矩阵，即在每一列中均具有t个1和m − t零的均匀随机矩阵，我们证明ℓ_∞偏差最大为2，概率为。以兹拉（Ezra）和洛维特（Lovett）证明的界线对n至少m ^t具有相同的界线，这是一个改进。