Dictator functions maximize mutual information

Pichler, Georg; Piantanida, Pablo; Matz, Gerald

doi:10.1214/18-aap1384

Cited by 12 publications

(8 citation statements)

References 7 publications

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“…Furthermore, it is sometimes the case that the optimal biclustering problem is easier to solve then the standard single-sided clustering problem. For example, Courtade-Kumar conjecture [13] for the standard single-sided clustering setting was ultimately proved for the biclustering setting [14]. A particular case, where (X, Y) are drawn from DSBS distribution, and the mappings f and g are restricted to be boolean function, was addressed in [14].…”

Section: Introductionmentioning

confidence: 99%

“…For example, Courtade-Kumar conjecture [13] for the standard single-sided clustering setting was ultimately proved for the biclustering setting [14]. A particular case, where (X, Y) are drawn from DSBS distribution, and the mappings f and g are restricted to be boolean function, was addressed in [14]. The bound I(f (X n ); g(Y n )) ≤ I(X; Y) was established, which is tight if and only if f and g are dictator functions.…”

Section: Introductionmentioning

confidence: 99%

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The Double-Sided Information-Bottleneck Function

Dikshtein

Ordentlich

Shitz

2021

2021 IEEE International Symposium on Information Theory (ISIT)

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We consider a two-terminal variant (double-sided) of the information bottleneck problem, which is related to biclustering. In our setup, X and Y are dependent random variables and the problem is to find two independent channels P U|X and P V|Y (setting the Markovian structure U → X → Y → V) that maximize I(U; V) subject to constraints on the relevant mutual information expressions: I(U; X) and I(V; Y). For jointly Gaussian X and Y, we show that Gaussian channels are optimal in the low-SNR regime, but not for general SNR. Similarly, it is shown that for a doubly symmetric binary source, binary symmetric channels are optimal when the correlation is low, and are suboptimal for high correlation. We conjecture that Z and S channels are optimal when the correlation is 1 (i.e., X = Y), and provide supporting numerical evidence.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Double-Sided Information-Bottleneck Function

Dikshtein

Ordentlich

Shitz

2021

2021 IEEE International Symposium on Information Theory (ISIT)

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“…The analogous question in the Gaussian setting was verified by Kindler, O'Donnell and Witmer [12]. Pichler, Piantanida and Matz [18] proved the variant that the dictator function maximizes the mutual information I(f (X); g(Y )) among all Boolean functions f and g. The original conjecture is still wide open. Courtade and Kumar [6] has observed that their conjecture holds in extremal scenarios ǫ = ǫ(n) → 0, 1/2.…”

Section: Introductionmentioning

confidence: 83%

“…Recall our definition of Zj in (18). Under the noise in (27), Zj is a Bernoulli random variable taking 1 and −1 with equal probability.…”

Section: Noise: Type IImentioning

confidence: 99%

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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“…It was subsequently generalized to the cases of α > 2, α = 1, and 1 < α < 2 in [4], [8], [10], and corresponding conjectures (mentioned above) were posed. In face, a weaker version (the two-function version) of Courtade-Kumar conjecture was solved by Pichler, Piantanida, and Matz [14] by using Fourier analysis. Ordentlich, Shayevitz, and Weinstein [15] proved a new bound for the Courtade-Kumar conjecture which improves the well-known bound ρ 2 and turns out to be asymptotically sharp in the limiting case ρ → 0.…”

Section: Introductionmentioning

confidence: 99%

On the $Φ$-Stability and Related Conjectures

Lei¹

2021

Preprint

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Let X be a random variable uniformly distributed on the discrete cube {−1, 1}n , and let T ρ be the noise operator acting on Boolean functions f : {−1, 1} n → {0, 1}, where ρ ∈ [0, 1] is the noise parameter, representing the correlation coefficient between each coordination of X and its noise-corrupted version. Given a convex function Φ and the mean Ef (X) = a ∈ [0, 1], which Boolean function f maximizes the Φ-stability E [Φ (T ρ f (X))] of f ? Special cases of this problem include the (symmetric and asymmetric) α-stability problems and the "Most Informative Boolean Function" problem. In this paper, we provide several upper bounds for the maximal Φ-stability. Considering specific Φ's, we partially resolve Mossel and O'Donnell's conjecture on α-stability with α > 2, Li and Médard's conjecture on α-stability with 1 < α < 2, and Courtade and Kumar's conjecture on the "Most Informative Boolean Function" which corresponds to a conjecture on α-stability with α = 1. Our proofs are based on discrete Fourier analysis, optimization theory, and improvements of the Friedgut-Kalai-Naor (FKN) theorem. Our improvements of the FKN theorem are sharp or asymptotically sharp for certain cases.

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Dictator functions maximize mutual information

Cited by 12 publications

References 7 publications

The Double-Sided Information-Bottleneck Function

The Double-Sided Information-Bottleneck Function

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

On the $Φ$-Stability and Related Conjectures

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