The Log-Approximate-Rank Conjecture is False

Arkadev Chattopadhyay
We construct a simple and total XOR function F on 2n variables that has only O(n) spectral norm, O(n^2) approximate rank and O(n^{2.5}) approximate nonnegative rank. We show it has polynomially large randomized bounded-error communication complexity of Omega(sqrt(n)). This yields the first exponential gap between the logarithm of the approximate rank and randomized communication complexity for total functions. Thus, F witnesses a refutation of the Log-Approximate-Rank Conjecture which was posed by Lee and Shraibman (2007)...
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