New formula links geometric mean quantization to probability masses
Geometric mean quantization via adaptive approximation
Information Theory
Summary
This paper studies how to measure complexity in a probability distribution by looking at how deeply one must keep dividing space where the distribution is heavy. The authors connect this depth to certain mathematical dimensions that describe the distribution's spread. They also show when and how these dimensions exist and how they relate to different averaging methods of information. Finally, they provide examples that illustrate different behaviors of these dimensions in probability mixtures.
What this means in practice
- •For data compression engineers: Identify fundamental limits on quantization methods based on probability mass distributions to optimize encoding depth.
- •For signal processing teams: Use the connection between local dimensions and quantization depth to refine adaptive sampling strategies for signals with nonuniform densities.
A theory result. No direct application yet.
Authors
Marc Kesseböhmer, Aljoscha Niemann
Abstract
Let $ν$ be a compactly supported Borel probability measure on $\mathbb R^{d}$ with $ν(B(x,r))\leq Cr^{a}$ for some $a>0$. Refine a dyadic cube exactly when its mass is at least $t$, and let $\mathcal{L}_ν(t)$ 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 $\log(1/t)/\mathcal{L}_ν(t)$. The dimension exists precisely when $(q-1)\sum_{Q}ν(Q)^{q}$, summed over all dyadic cubes, converges as $q\downarrow1$, 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.