Remarks on the Most Informative Function Conjecture at fixed mean

Kindler, Guy; O’Donnell, Ryan; Witmer, David R.

doi:10.48550/arxiv.1506.03167

Cited by 7 publications

(10 citation statements)

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“…A weaker version of this result with ρ 0 replaced by a sequence that vanishes as n → ∞ was proven in [15], [17]. In the Gaussian setting, the analogous Courtade-Kumar conjecture and the analogous Li-Médard conjecture were respectively confirmed by Kindler, O'Donnell, and Witmer [18] and by Eldan [19]; these results are also further generalized to the Φ-stability in [18].…”

Section: Introductionmentioning

confidence: 82%

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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Section: Introductionmentioning

confidence: 82%

On the $Φ$-Stability and Related Conjectures

Lei¹

2021

Preprint

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“…Even in this simple case, determining the most informative function seems to be non-trivial. Further investigations of this problem was done in [46][47][48][49]. In particular, [46] studies a related problem in a continuous setting by considering that X and Y are Gaussian random vectors.…”

Section: Related Workmentioning

confidence: 99%

“…Further investigations of this problem was done in [46][47][48][49]. In particular, [46] studies a related problem in a continuous setting by considering that X and Y are Gaussian random vectors. Recently, Samorodnitsky [50] presented a proof of the conjecture in the high noise regime.…”

Section: Related Workmentioning

confidence: 99%

Principal Inertia Components and Applications

Calmon

Makhdoumi

Médard

et al. 2017

IEEE Trans. Inform. Theory

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We explore properties and applications of the Principal Inertia Components (PICs) between two discrete random variables X and Y . The PICs lie in the intersection of information and estimation theory, and provide a fine-grained decomposition of the dependence between X and Y . Moreover, the PICs describe which functions of X can or cannot be reliably inferred (in terms of MMSE) given an observation of Y . We demonstrate that the PICs play an important role in information theory, and they can be used to characterize information-theoretic limits of certain estimation problems. In privacy settings, we prove that the PICs are related to fundamental limits of perfect privacy.

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“…In the memoryless deterministic quantizer case, the quantizer is the sign of the received signal. Using Kindler [23], the sign of the received signal is the optimal one bit memoryless deterministic quantizer, and not necessarily the optimal entropy constrained deterministic quantizer. The curve in Fig.…”

Section: B Demonstrating the Advantages Of The Stochastic Quantizermentioning

confidence: 99%

Oblivious Fronthaul-Constrained Relay for a Gaussian Channel

Homri

Peleg

Shamai

2018

IEEE Trans. Commun.

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We consider systems in which the transmitter conveys messages to the receiver through a capacity-limited relay station. The channel between the transmitter and the relay station is assumed to be a frequency-selective additive Gaussian noise channel. It is assumed that the transmitter can shape the spectrum and adapt the coding technique so as to optimize performance. The relay operation is oblivious (nomadic transmitters), that is, the specific codebooks used are unknown. We find the reliable information rate that can be achieved with Gaussian signaling in this setting, and to that end, employ Gaussian bottleneck results combined with Shannon's incremental frequency approach. We also prove that, unlike classical water pouring, the allocated spectrum (power and bit rate) of the optimal solution could frequently be discontinuous. These results can be applied also to a MIMO transmission scheme. We also investigate the case of an entropy-limited relay. We show that the optimal relay function is always deterministic, present lower and upper bounds on the optimal performance (in terms of mutual information), and derive an analytical approximation.

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Remarks on the Most Informative Function Conjecture at fixed mean

Cited by 7 publications

References 0 publications

On the $Φ$-Stability and Related Conjectures

On the $Φ$-Stability and Related Conjectures

Principal Inertia Components and Applications

Oblivious Fronthaul-Constrained Relay for a Gaussian Channel

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