We consider two known lower bounds on randomized communication complexity: the smooth rectangle bound and the logarithm of the approximate nonnegative rank. Our main result is that they are the same up to a multiplicative constant and a small additive term.The logarithm of the nonnegative rank is known to be a nearly tight lower bound on the deterministic communication complexity. Our result indicates that proving an analogous result for the randomized case, namely that the log approximate nonnegative rank is a nearly tight bound on randomized communication complexity, would imply the tightness of the information complexity bound.Another corollary of our result is the existence of a Boolean function with a quasipolynomial gap between its approximate rank and approximate nonnegative rank.We also show that our method yields an alternative simple proof of the equivalence between the approximate rank and the approximate μ norm, first shown by Lee and Shraibman.

中文翻译：

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近似非负秩等价于平滑矩形边界

我们考虑随机通信复杂度的两个已知下限：平滑矩形边界和近似非负秩的对数。我们的主要结果是它们在乘法常数和小的加法项之前是相同的。众所周知，非负秩的对数是确定性通信复杂性的近乎严格的下界。我们的结果表明，证明随机情况的类似结果，即对数近似非负秩是随机通信复杂性的近乎严格的界限，这意味着信息复杂性界限的紧密性。我们结果的另一个推论是存在在其近似秩和近似非负秩之间具有拟多项式间隙的布尔函数。