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Which Boolean Functions Maximize Mutual Information on Noisy Inputs?
Thomas A. Courtade (University of California, Berkeley), Gowtham Ramani Kumar (Stanford University)
Abstract: We pose a simply stated conjecture regarding the maximum mutual information a Boolean function can reveal about noisy inputs. Specifically, let Xnbe independent identically distributed Bernoulli(1/2), and let Yn be the result of passing Xnthrough a memoryless binary symmetric channel with crossover probability α. For any Boolean function b : {0, 1}n→ {0, 1}, we conjecture that I(b(Xn); Yn) ≤ 1 - H(α). While the conjecture remains open, we provide substantial evidence supporting its validity. Connections are also made to discrete isoperimetric inequalities.