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Distortion of Random Max Filters over Reflection Groups

  • Efstratios Tsoukanis
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Abstract

A max filter with Gaussian templates a1,…,aD is a G-invariant map Rd→RD, where G is a finite reflection group. As D→∞ its distortion converges a.s to the condition number of MG1/2, where MG is the second-moment matrix of a standard Gaussian vector moved into a fundamental chamber of G. We compute this limit exactly for the sign and dihedral groups, and show it grows as Θ(√d), Θ(d3/2) and Θ(k) for the sign, symmetric and dihedral groups of order 2k. The symmetric case bound uses a lower bound on the covariance of order statistics. For any finite G we also show that the infimum defining the optimal lower Lipschitz constant of a piecewise-linear G-invariant map is a minimum.

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Cite

BibTeXRIS
BibTeX
@misc{tsoukanis2026distortion,
  title = {{Distortion of Random Max Filters over Reflection Groups}},
  author = {Efstratios Tsoukanis},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.00305},
  eprint = {2611.00305},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.00305},
}
RIS
TY  - UNPB
TI  - Distortion of Random Max Filters over Reflection Groups
AU  - Efstratios Tsoukanis
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.00305
UR  - https://arxiv.org/abs/2611.00305
AB  - A max filter with Gaussian templates a_1,…,a_D is a G-invariant map R^d→R^D, where G is a finite reflection group. As D→∞ its distortion converges a.s to the condition number of M_G^1/2, where M_G is the second-moment matrix of a standard Gaussian vector moved into a fundamental chamber of G. We compute this limit exactly for the sign and dihedral groups, and show it grows as Θ(√d), Θ(d^3/2) and Θ(k) for the sign, symmetric and dihedral groups of order 2k. The symmetric case bound uses a lower bound on the covariance of order statistics. For any finite G we also show that the infimum defining the optimal lower Lipschitz constant of a piecewise-linear G-invariant map is a minimum.
ER  -