Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 47 min ago
PreprintarXiv licencearXiv

The Hellinger Conjecture for Antipodal Boolean Functions

  • Vu Khac Ky
Open at sourcePDFDOICite

Abstract

The balanced Hellinger conjecture compares the overlap of two output laws induced by a Boolean decision under binary symmetric noise. We prove that a single input coordinate minimizes this overlap among antipodal decisions, whose values reverse when all input signs are reversed, throughout the noise interval. The proof uses a normalized nuclear norm of the cross-class block of the Gram matrix of square-root channel densities. Multiplication by an antipodal decision exchanges the even and odd Fourier spaces preserved by the Gram matrix. A coordinate exchange attains the minimum cost of matching their spectra. A matrix contraction extends the sharp bound to arbitrary independent binary-input channels. Fixed coordinate flips and randomized decisions follow from the same argument and concavity. These extensions give bounds for information and testing, including linear Gaussian observations, and a weighted edge-boundary inequality.

Related

  1. A weak Hellinger inequality for noisy Boolean channels
    Polona Durcik, Marco Fraccaroli, Joris Roos
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  2. On the Fourier Entropy-Influence Conjecture for Boolean Plateaued Functions
    Vladimir N. Potapov
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  3. On a conjecture concerning antipodal labelings for cycles
    John M. Campbell
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  4. A Sharp Fisher-Information Inequality for Boolean Pairs: Proof of the Amari--Kobayashi Conjecture
    Jun Chen, Shun Watanabe, Lei Yu
    arXivPreprintarXiv licenceFull textGeneral mathematicsarXiv
  5. Tameness of transitive Boolean functions
    Xinyu Long
    arXivPreprintarXiv licenceFull textProbabilityarXiv

Cite

BibTeXRIS
BibTeX
@misc{ky2026hellinger,
  title = {{The Hellinger Conjecture for Antipodal Boolean Functions}},
  author = {Vu Khac Ky},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.10618},
  eprint = {2610.10618},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.10618},
}
RIS
TY  - UNPB
TI  - The Hellinger Conjecture for Antipodal Boolean Functions
AU  - Vu Khac Ky
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.10618
UR  - https://arxiv.org/abs/2610.10618
AB  - The balanced Hellinger conjecture compares the overlap of two output laws induced by a Boolean decision under binary symmetric noise. We prove that a single input coordinate minimizes this overlap among antipodal decisions, whose values reverse when all input signs are reversed, throughout the noise interval. The proof uses a normalized nuclear norm of the cross-class block of the Gram matrix of square-root channel densities. Multiplication by an antipodal decision exchanges the even and odd Fourier spaces preserved by the Gram matrix. A coordinate exchange attains the minimum cost of matching their spectra. A matrix contraction extends the sharp bound to arbitrary independent binary-input channels. Fixed coordinate flips and randomized decisions follow from the same argument and concavity. These extensions give bounds for information and testing, including linear Gaussian observations, and a weighted edge-boundary inequality.
ER  -