A Hierarchical Online-Compression Approach to Proportional Subsets of Bounded Orthonormal Systems
Abstract
Let (ϕi)i=1n be an orthonormal system on a probability space with |ϕi|∞≤ L. We prove that, for every 2≤ k<n, there is a set I of at least n−k indices such that |f|2≤ CMk√(log(e+Mk))|f|1 for every f in the span of (ϕi)i∈ I, where Mk=L√(n/klog(2k)). This replaces the logarithmic exponent 5/2 in the corresponding theorem of Guédon, Mendelson, Pajor, and Tomczak-Jaegermann by 1/2. We also obtain a partition into two sets of cardinality n/2+O(√n) for which both spans satisfy the resulting norm comparison, and for uniformly bounded systems the proportional estimate is C√(log nloglog n) for sufficiently large n. This is the third paper in a series systematically using gradient descent as a proof mechanism in harmonic analysis, following the Λ(p) and Bernoulli restricted isometry papers of Burstein, Iosevich, and Krause, and the descent here is entropic mirror descent. We represent each function in an L1 ball by a probability measure of controlled relative entropy, and this entropy pays for the corrections to the predictions of its coefficients. Processing the thresholds from large to small, we obtain a single weighted bound for all the corrections, which we turn into a finite description of approximate coefficient squares and then into a conditional Rademacher estimate, and symmetrization completes the argument directly on the L1 ball. We also record consequences for concentration and for recovery from coefficient measurements.
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BibTeX
@misc{burstein2026hierarchical,
title = {{A Hierarchical Online-Compression Approach to Proportional Subsets of Bounded Orthonormal Systems}},
author = {Will Burstein and Alex Iosevich and Ben Krause},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39025},
eprint = {2610.39025},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39025},
}
RIS
TY - UNPB
TI - A Hierarchical Online-Compression Approach to Proportional Subsets of Bounded Orthonormal Systems
AU - Will Burstein
AU - Alex Iosevich
AU - Ben Krause
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.39025
UR - https://arxiv.org/abs/2610.39025
AB - Let (ϕ_i)_i=1^n be an orthonormal system on a probability space with |ϕ_i|_∞≤ L. We prove that, for every 2≤ k<n, there is a set I of at least n−k indices such that |f|_2≤ CM_k√(log(e+M_k))|f|_1 for every f in the span of (ϕ_i)_i∈ I, where M_k=L√(n/klog(2k)). This replaces the logarithmic exponent 5/2 in the corresponding theorem of Guédon, Mendelson, Pajor, and Tomczak-Jaegermann by 1/2. We also obtain a partition into two sets of cardinality n/2+O(√n) for which both spans satisfy the resulting norm comparison, and for uniformly bounded systems the proportional estimate is C√(log nloglog n) for sufficiently large n. This is the third paper in a series systematically using gradient descent as a proof mechanism in harmonic analysis, following the Λ(p) and Bernoulli restricted isometry papers of Burstein, Iosevich, and Krause, and the descent here is entropic mirror descent. We represent each function in an L_1 ball by a probability measure of controlled relative entropy, and this entropy pays for the corrections to the predictions of its coefficients. Processing the thresholds from large to small, we obtain a single weighted bound for all the corrections, which we turn into a finite description of approximate coefficient squares and then into a conditional Rademacher estimate, and symmetrization completes the argument directly on the L_1 ball. We also record consequences for concentration and for recovery from coefficient measurements.
ER -