Improved accuracy bounds for sums of random high-dimensional data
Improved Bounds for the Entropic Central Limit Theorem in the Log-Concave Case
Information Theory
Summary
When you add up many random quantities, their combined behavior often looks like a bell curve, and this is called the Central Limit Theorem. This paper improves the understanding of how close this combined behavior is to a perfect bell curve when the data has a special shape called log-concave. The authors show mathematically that the difference shrinks faster with more data even in high-dimensional spaces, using advanced mathematical tools. Their work means better estimates of this closeness in complex settings with many variables.
What this means in practice
- •For machine learning engineers: Improve the theoretical understanding of convergence rates when aggregating high-dimensional random features, aiding performance guarantees in learning algorithms.
- •For signal processing developers: Enhance accuracy estimates for signal distributions when combining multiple noisy high-dimensional sources with log-concave properties.
A theory result. No direct application yet.
Authors
Tom Maitre
Abstract
Let $μ$ be a log-concave isotropic probability measure on $R^d$ , and let $μ^n$ denote the law of the normalized sum of $n$ independent random vectors with law $μ$. We prove that \[ H(μ^n|γ)\leq \frac{C}{n} \min\left\{1+\log(ed)^2H(μ|γ)\ ,\ d\right\}. \] Compared to the previous best known bound, our estimate replaces the polynomial dependence on the dimension by a polylogarithmic one. The proof relies on a new combination of the De Bruijn identity with Eldan's stochastic localization process.