Approximate Nonnegative Rank Is Equivalent to the Smooth Rectangle Bound
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 the analogue 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 cost 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.
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