Geometric mean quantization via adaptive approximation
Abstract
Let ν be a compactly supported Borel probability measure on ℝd with ν(B(x,r))≤ Cra for some a>0. Refine a dyadic cube exactly when its mass is at least t, and let ℒν(t) be the mean depth at which this refinement stops. We show that the lower and upper geometric-mean quantization dimensions of ν are the lower and upper limits of log(1/t)/ℒν(t). The dimension exists precisely when (q−1)∑Qν(Q)q, summed over all dyadic cubes, converges as q↓1, and it is then determined by this limit. The mass-threshold formula yields harmonic integral bounds in terms of the local dimensions and encloses entropy and quantization dimensions in a common spectral interval. Convergence in law of the local information rates is equivalent to convergence of the rescaled spectra in a window of width 1/k around q=1; the two dimensions are then the arithmetic and the harmonic mean of the limit law, and we quantify their difference by sharp bounds and variance identities. Without any convergence assumption, vanishing threshold variance still forces equality of the corresponding lower and upper dimensions. Bernoulli mixtures realise every local-dimension law with compact support in (0,1], and a regime-switching example separates convergence in law from almost-everywhere convergence.
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BibTeX
@misc{kessebohmer2026geometric,
title = {{Geometric mean quantization via adaptive approximation}},
author = {Marc Kesseböhmer and Aljoscha Niemann},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26950},
eprint = {2609.26950},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26950},
}
RIS
TY - UNPB
TI - Geometric mean quantization via adaptive approximation
AU - Marc Kesseböhmer
AU - Aljoscha Niemann
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26950
UR - https://arxiv.org/abs/2609.26950
AB - Let ν be a compactly supported Borel probability measure on ℝ^d with ν(B(x,r))≤ Cr^a for some a>0. Refine a dyadic cube exactly when its mass is at least t, and let ℒ_ν(t) be the mean depth at which this refinement stops. We show that the lower and upper geometric-mean quantization dimensions of ν are the lower and upper limits of log(1/t)/ℒ_ν(t). The dimension exists precisely when (q−1)∑_Qν(Q)^q, summed over all dyadic cubes, converges as q↓1, and it is then determined by this limit. The mass-threshold formula yields harmonic integral bounds in terms of the local dimensions and encloses entropy and quantization dimensions in a common spectral interval. Convergence in law of the local information rates is equivalent to convergence of the rescaled spectra in a window of width 1/k around q=1; the two dimensions are then the arithmetic and the harmonic mean of the limit law, and we quantify their difference by sharp bounds and variance identities. Without any convergence assumption, vanishing threshold variance still forces equality of the corresponding lower and upper dimensions. Bernoulli mixtures realise every local-dimension law with compact support in (0,1], and a regime-switching example separates convergence in law from almost-everywhere convergence.
ER -