A full-rank 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 information-theoretic 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.

翻译：Euclidean 空间的全空格是一个由所有整数线性组合构成的离散设置。 根据$\ mathbb{R ⁇ n$ 的概率分布, 两种操作可以通过考虑空间的商数来诱导: 包装和量化。 对于一个tatice $\Lambda$, 和一个基本域$D$, 以$\mathbb{R ⁇ n$, 以$\Lambda$ 来拼写 。 将每组的密度进行组合, 并且通过对每个基本域的翻译进行量化分布定义。 这些操作分别定义了包装和四分的随机变量 $D$ 和 $\Lambda$, 这相当于原始的随机变量。 我们调查了这种分解的信息- 理论特性, 如 entropy、 相互信息 和 Fisher 信息矩阵, 并显示它自然地概括到更抽象的本地压缩的表层组的背景 。