Abstract
Let be a memoryless uniform Bernoulli source and be the output of it through a binary symmetric channel. Courtade and Kumar conjectured that the Boolean function that maximizes the mutual information is a dictator function, i.e., for some i. We propose a clustering problem, which is equivalent to the above problem where we emphasize an information geometry aspect of the equivalent problem. Moreover, we define a normalized geometric mean of measures and interesting properties of it. We also show that the conjecture is true when the arithmetic and geometric mean coincide in a specific set of measures.
Highlights
Let X n be an independent and identically distributed (i.i.d.) uniform Bernoulli source andY n be an output of it through a memoryless binary symmetric channel with crossover probability p < 1/2
The cluster center μ A is an arithmetic mean of measures in the set { PY n | xn : x n ∈ A}
The following theorem shows that the KL divergence on { PY n | xn : x n ∈ Ω} corresponds to the Hamming distance on Ω
Summary
Let X n be an independent and identically distributed (i.i.d.) uniform Bernoulli source and. It can be shown that the maximizing mutual information is equivalent to clustering probability measures under. In the equivalent clustering problem, the center of the cluster is an arithmetic mean of measures. We provide the role of the geometric mean of measures (with appropriate normalization) in this. We propose an equivalent formulation of the conjecture using the geometric mean of measures. Note that the geometric mean allows us to connect Conjecture 1 to the other well-known clustering problem. The arithmetic mean of measures in the set { PY n | xn : x n ∈ A} is denoted by μ A.
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