How to quantize n outputs of a binary symmetric channel to n − 1 bits?

Huleihel, Wasim; Ordentlich, Or

doi:10.1109/isit.2017.8006496

Cited by 7 publications

(3 citation statements)

References 11 publications

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“…As a consequence of Theorem 3.2, it suffices to study Courtade-Kumar's conjecture for monotone functions. This has been observed by Courtade and Kumar [6], and Huleihel and Ordentlich [9].…”

Section: The Most Informative Boolean Functionsupporting

confidence: 58%

Boolean Functions: Noise Stability, Non-Interactive Correlation, and Mutual Information

Médard

2018

2018 IEEE International Symposium on Information Theory (ISIT)

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Let T ǫ be the noise operator acting on Boolean functions f : {0, 1} n → {0, 1}, where ǫ ∈ [0, 1/2] is the noise parameter. Given α > 1 and fixed mean Ef , which Boolean function f has the largest α-th moment E(T ǫ f ) α ? This question has close connections with noise stability of Boolean functions, the problem of non-interactive correlation distillation, and Courtade-Kumar's conjecture on the most informative Boolean function. In this paper, we characterize maximizers in some extremal settings, such as low noise (ǫ = ǫ(n) is close to 0), high noise (ǫ = ǫ(n) is close to 1/2), as well as when α = α(n) is large. Analogous results are also established in more general contexts, such as Boolean functions defined on discrete torus (Z/pZ) n and the problem of noise stability in a tree model. * These results were presented in part at the 2018 International Symposium on Information Theory, Colorado, USA. Both authors are with the Research

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Section: The Most Informative Boolean Functionsupporting

confidence: 58%

Boolean Functions: Noise Stability, Non-Interactive Correlation, and Mutual Information

Médard

2018

2018 IEEE International Symposium on Information Theory (ISIT)

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“…A complementary problem concerning Conjecture 1 is posed and proved by Huleihel and Ordentlich [9] as follows.…”

Section: Theorem 3 ( [5]mentioning

confidence: 99%

On the Most Informative Boolean Functions of the Very Noisy Channel

Yang

Wesel

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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Let X n be a uniformly distributed n-dimensional binary vector, and Y n is the result of passing X n through a binary symmetric channel (BSC) with crossover probability α. A recent conjecture postulated by Courtade and Kumar states thatAlthough the conjecture has been proved to be true in high noise regime by Samorodnitsky, here we present a calculus-based approach to show the same result by examining the second derivative of H(α) − H(f (X n )|Y n ) at α = 1/2. In this way, one can clearly see that the dictator function is the most informative function in high noise regime.

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“…Weinberger and Shayevitz [ 4 ] considered the optimal Boolean function under quadratic loss. Huleihel and Ordentlich [ 5 ] considered the complementary case and showed that

for all

. Nazer et al focused on information distilling quantizers [ 6 ], which can be seen as a generalized version of the above problem.…”

Section: Introductionmentioning

confidence: 99%

Information Geometric Approach on Most Informative Boolean Function Conjecture

2018

Entropy

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Let X n be a memoryless uniform Bernoulli source and Y n be the output of it through a binary symmetric channel. Courtade and Kumar conjectured that the Boolean function f : { 0 , 1 } n → { 0 , 1 } that maximizes the mutual information I ( f ( X n ) ; Y n ) is a dictator function, i.e., f ( x n ) = x i 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.

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How to quantize n outputs of a binary symmetric channel to n − 1 bits?

Cited by 7 publications

References 11 publications

Boolean Functions: Noise Stability, Non-Interactive Correlation, and Mutual Information

Boolean Functions: Noise Stability, Non-Interactive Correlation, and Mutual Information

On the Most Informative Boolean Functions of the Very Noisy Channel

Information Geometric Approach on Most Informative Boolean Function Conjecture

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