低秩矩阵通信复杂性的Lovett界的直接证明
A direct proof for Lovett's bound on the communication complexity of low rank matrices
摘要：
通信复杂性中的对数秩猜想表明，任何布尔秩r函数的确定性通信复杂性都有polylog（r）的界。最近，Lovett取得了重大进展，他证明了通信复杂性是O（r^1/2 * log r）的。洛维特的证明是基于已知的估计的差异低秩矩阵。我们给出了一个简单的，直接的证明的基础上，超平面舍入参数，在我们看来，揭示了更多的原因，为什么一个根因子就足够了，什么是必要的，以改善这个因素。
英文摘要：
The log-rank conjecture in communication complexity suggests that the deterministic communication complexity of any Boolean rank-r function is bounded by polylog(r). Recently, major progress was made by Lovett who proved that the communication complexity is bounded by O(r^1/2 * log r). Lovett's proof is based on known estimates on the discrepancy of low-rank matrices. We give a simple, direct proof based on a hyperplane rounding argument that in our opinion sheds more light on the reason why a root factor suffices and what is necessary to improve on this factor.
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