Which Boolean Functions Maximize Mutual Information on Noisy Inputs?

Courtade, Thomas A.; Kumar, Gowtham

doi:10.1109/tit.2014.2326877

Cited by 38 publications

(71 citation statements)

References 17 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: 60%

“…Proof. The proof is inspired by a shifting technique in [11] and a convex combination argument in [6] (Theorem 3). Suppose that f is supported on S. Let S n 2 be the projection of S on the last n − 1 bits, i.e., x n 2 ∈ S n 2 if (0, x n 2 ) ∈ S or (1, x n 2 ) ∈ S. We define the following partition of S n 2 :…”

Section: Resultsmentioning

confidence: 99%

“…Let f be a Boolean function. It is conjectured by Courtade and Kumar [6] that the dictator function maximizes the mutual information I(X; f (Y )) between X and f (Y ). Recall that…”

Section: The Most Informative Boolean Functionmentioning

confidence: 99%

“…Analogous to (6), we define the random flipping operatorσ j as follows σ j (x 1 , · · · , x j , · · · , x n ) = (x 1 , · · · , x j +Z j , · · · , x n ).…”

Section: Discrete Torusunclassified

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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: 60%

Section: Resultsmentioning

confidence: 99%

“…Let f be a Boolean function. It is conjectured by Courtade and Kumar [6] that the dictator function maximizes the mutual information I(X; f (Y )) between X and f (Y ). Recall that…”

Section: The Most Informative Boolean Functionmentioning

confidence: 99%

“…Analogous to (6), we define the random flipping operatorσ j as follows σ j (x 1 , · · · , x j , · · · , x n ) = (x 1 , · · · , x j +Z j , · · · , x n ).…”

Section: Discrete Torusunclassified

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

Médard

2018

2018 IEEE International Symposium on Information Theory (ISIT)

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“…Besides Conjecture 1, some other related conjectures are also addressed in [1]. One conjecture is that,…”

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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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Rényi Resolvability and Its Applications to the Wiretap Channel

Tan

2017

Lecture Notes in Computer Science

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The conventional channel resolvability problem refers to the determination of the minimum rate required for an input process so that the output distribution approximates a target distribution in either the total variation distance or the relative entropy. In contrast to previous works, in this paper, we use the (normalized or unnormalized) Rényi divergence (with the Rényi parameter in [0, 2] ∪ {∞}) to measure the level of approximation. We also provide asymptotic expressions for normalized Rényi divergence when the Rényi parameter is larger than or equal to 1 as well as (lower and upper) bounds for the case when the same parameter is smaller than 1. We characterize the Rényi resolvability, which is defined as the minimum rate required to ensure that the Rényi divergence vanishes asymptotically. The Rényi resolvabilities are the same for both the normalized and unnormalized divergence cases. In addition, when the Rényi parameter smaller than 1, consistent with the traditional case where the Rényi parameter is equal to 1, the Rényi resolvability equals the minimum mutual information over all input distributions that induce the target output distribution. When the Rényi parameter is larger than 1 the Rényi resolvability is, in general, larger than the mutual information. The optimal Rényi divergence is proven to vanish at least exponentially fast for both of these two cases, as long as the code rate is larger than the Rényi resolvability. The optimal exponential rate of decay for i.i.d. random codes is also characterized exactly. We apply these results to the wiretap channel, and completely characterize the optimal tradeoff between the rates of the secret and non-secret messages when the leakage measure is given by the (unnormalized) Rényi divergence. This tradeoff differs from the conventional setting when the leakage is measured by the traditional mutual information.

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Which Boolean Functions Maximize Mutual Information on Noisy Inputs?

Cited by 38 publications

References 17 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

Rényi Resolvability and Its Applications to the Wiretap Channel

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