Information Properties of a Random Variable Decomposition through Lattices
Abstract
A fullrank lattice in the Euclidean space is a discrete set formed by all integer linear combinations of a basis. Given a probability distribution on $\mathbb{R}^n$, two operations can be induced by considering the quotient of the space by such a lattice: wrapping and quantization. For a lattice $\Lambda$, and a fundamental domain $D$ which tiles $\mathbb{R}^n$ through $\Lambda$, the wrapped distribution over the quotient is obtained by summing the density over each coset, while the quantized distribution over the lattice is defined by integrating over each fundamental domain translation. These operations define wrapped and quantized random variables over $D$ and $\Lambda$, respectively, which sum up to the original random variable. We investigate informationtheoretic properties of this decomposition, such as entropy, mutual information and the Fisher information matrix, and show that it naturally generalizes to the more abstract context of locally compact topological groups.
 Publication:

arXiv eprints
 Pub Date:
 November 2022
 DOI:
 10.48550/arXiv.2211.03477
 arXiv:
 arXiv:2211.03477
 Bibcode:
 2022arXiv221103477M
 Keywords:

 Computer Science  Information Theory;
 94A17
 EPrint:
 11 pages, 6 figures. v2: improved some explanations in sections 2 and 4, typos, and added citation in section 3