• Corpus ID: 233296442

On the $\Phi$-Stability and Related Conjectures

@inproceedings{Yu2021OnT,
  title={On the \$\Phi\$-Stability and Related Conjectures},
  author={Lei Yu},
  year={2021}
}
  • Lei Yu
  • Published 18 April 2021
  • Mathematics
Let X be a random variable uniformly distributed on the discrete cube {−1, 1}, and let Tρ be the noise operator acting on Boolean functions f : {−1, 1} → {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 (Tρf(X))] of f? Special cases of this problem include the (symmetric and… 

Figures from this paper

References

SHOWING 1-10 OF 49 REFERENCES

On the Entropy of a Noisy Function

If a Boolean function f is close to a characteristic function g of a subcube of dimension n-1, then the entropy of T<sub>ϵ</sub>f is at most that of Tϵg.

Which Boolean Functions Maximize Mutual Information on Noisy Inputs?

This work poses a simply stated conjecture regarding the maximum mutual information a Boolean function can reveal about noisy inputs and provides substantial evidence supporting its validity.

Which Boolean functions are most informative?

  • G. KumarT. Courtade
  • Computer Science
    2013 IEEE International Symposium on Information Theory
  • 2013
This work introduces a simply stated conjecture regarding the maximum mutual information a Boolean function can reveal about noisy inputs and provides substantial evidence supporting its validity.

Noise sensitivity of Boolean functions and applications to percolation

It is shown that a large class of events in a product probability space are highly sensitive to noise, in the sense that with high probability, the configuration with an arbitrary small percent of

On the Most Informative Boolean Functions of the Very Noisy Channel

  • Hengjie YangR. Wesel
  • Computer Science
    2019 IEEE International Symposium on Information Theory (ISIT)
  • 2019
A calculus-based approach is presented to show a dimension-dependent result by examining the second derivative of H(α) − H(f(X n)|Y n) at α = 1/2, and it is shown that the dictator function is the most informative function in the high noise regime.

An improved upper bound for the most informative boolean function conjecture

A new upper bound is derived that holds for all balanced functions, and improves upon the best known previous bound for α > 1 over 3.

Remarks on the Most Informative Function Conjecture at fixed mean

A continuous version of the conjecture on the sphere is proved and it implies the previously-known analogue for Gaussian space and Courtade and Kumar's stronger Lex Conjecture fails for small noise rates.

Non-interactive correlation distillation, inhomogeneous Markov chains, and the reverse Bonami-Beckner inequality

NICD, a generalization of noise sensitivity previously considered in [5, 31, 39], is extended to trees and the use of thereverse Bonami-Beckner inequality is used to prove a new isoperimetric inequality for the discrete cube and a new result on the mixing of short random walks on the cube.

ON SEQUENCES OF PAIRS OF DEPENDENT RANDOM VARIABLES

The generalized random variables $( {x,y} )$ have a given joint distribution. Pairs $( {x_i ,y_i } )$ are drawn independently. The observer of $( {x_1 , \cdots ,x_n } )$ and the observer of $( {y_1 ,

A polynomial bound in Freiman's theorem

.Earlier bounds involved exponential dependence in αin the second estimate. Ourargument combines I. Ruzsa’s method, which we improve in several places, as well asY. Bilu’s proof of Freiman’s