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Progress on the Courtade-Kumar Conjecture: Optimal High-Noise Entropy Bounds and Generalized Coordinate-wise Mutual Information

Adel Javanmard, David P. Woodruff

TL;DR

This work advances the Courtade–Kumar conjecture by proving a universal bound on the sum of coordinate-wise mutual information for any Boolean function, thereby resolving an open question of Courtade and Kumar. It also strengthens the high-noise analysis with an optimal $O(\lambda^2)$ entropy expansion term and establishes a sharp linear Fourier concentration, extending the regime where dictatorship-like functions are optimal. The methods combine Fourier-analytic decompositions, convex optimization over level-1 energies, and refined moment bounds via hypercontractivity, yielding both structural and quantitative insights. Collectively, the results reinforce the prominence of dictatorship-like structures under severe noise and have implications for channel polarization, hardness of approximation, and noise-sensitivity theory.

Abstract

The Courtade-Kumar conjecture posits that dictatorship functions maximize the mutual information between the function's output and a noisy version of its input over the Boolean hypercube. We present two significant advancements related to this conjecture. First, we resolve an open question posed by Courtade and Kumar, proving that for any Boolean function (regardless of bias), the sum of mutual information between the function's output and the individual noisy input coordinates is bounded by $1-H(α)$, where $α$ is the noise parameter of the Binary Symmetric Channel. This generalizes their previous result which was restricted to balanced Boolean functions. Second, we advance the study of the main conjecture in the high noise regime. We establish an optimal error bound of $O(λ^2)$ for the asymptotic entropy expansion, where $λ= (1-2α)^2$, improving upon the previous best-known bounds. This refined analysis leads to a sharp, linear Fourier concentration bound for highly informative functions and significantly extends the range of the noise parameter $λ$ for which the conjecture is proven to hold.

Progress on the Courtade-Kumar Conjecture: Optimal High-Noise Entropy Bounds and Generalized Coordinate-wise Mutual Information

TL;DR

This work advances the Courtade–Kumar conjecture by proving a universal bound on the sum of coordinate-wise mutual information for any Boolean function, thereby resolving an open question of Courtade and Kumar. It also strengthens the high-noise analysis with an optimal entropy expansion term and establishes a sharp linear Fourier concentration, extending the regime where dictatorship-like functions are optimal. The methods combine Fourier-analytic decompositions, convex optimization over level-1 energies, and refined moment bounds via hypercontractivity, yielding both structural and quantitative insights. Collectively, the results reinforce the prominence of dictatorship-like structures under severe noise and have implications for channel polarization, hardness of approximation, and noise-sensitivity theory.

Abstract

The Courtade-Kumar conjecture posits that dictatorship functions maximize the mutual information between the function's output and a noisy version of its input over the Boolean hypercube. We present two significant advancements related to this conjecture. First, we resolve an open question posed by Courtade and Kumar, proving that for any Boolean function (regardless of bias), the sum of mutual information between the function's output and the individual noisy input coordinates is bounded by , where is the noise parameter of the Binary Symmetric Channel. This generalizes their previous result which was restricted to balanced Boolean functions. Second, we advance the study of the main conjecture in the high noise regime. We establish an optimal error bound of for the asymptotic entropy expansion, where , improving upon the previous best-known bounds. This refined analysis leads to a sharp, linear Fourier concentration bound for highly informative functions and significantly extends the range of the noise parameter for which the conjecture is proven to hold.
Paper Structure (22 sections, 13 theorems, 71 equations)

This paper contains 22 sections, 13 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Resolution of Open Question 1
  4. Optimal Entropy Bounds in High Noise
  5. Organization
  6. Generalized Coordinate-wise Mutual Information Bound
  7. Proof of Theorem \ref{['thm:generalized_thm1_intro']}
  8. Optimal Entropy Bound in the High Noise Regime
  9. Proof of Theorem \ref{['thm:extended_range']}
  10. Proof of Theorem \ref{['thm:linear_concentration_intro']}
  11. Potential Applications
  12. Coding Theory and Polarization
  13. Hardness of Approximation and Dictatorship Testing
  14. Noise Sensitivity and Percolation
  15. Conclusion
  16. ...and 7 more sections

Key Result

Theorem 1

If $b$ is a balanced Boolean function ($\mathbb{E}[b]=0$),

Theorems & Definitions (14)

  • Conjecture 1: Courtade-Kumar Conjecture courtade2014boolean
  • Theorem 1: Theorem 1 in courtade2014boolean
  • Theorem 2: Generalized Coordinate-wise Bound
  • Theorem 3: Optimal Asymptotic Entropy Bound
  • Theorem 4: Linear Fourier Concentration
  • Theorem 5: Extended Range for the Conjecture
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4: Entropy Decomposition
  • ...and 4 more