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 Rn 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$\ mathb{R ⁇ n$的概率分布, 两种操作可以通过考虑空间的商数来诱发: 包装和量化。 对于一个 tatice $\Lambda$, 和一个基本域 $D$, 它通过 $\Lambda$ 进行拼贴, 商号上的包装分布是通过对每个共位的密度进行拼凑而获得的, 而 方位的量化分布则通过对每个基本域的翻译进行整合来定义。 这些操作分别界定包装和量化的随机变量, 超过$D和$\Lambda$, 相当于原始的随机变量。 我们调查了这一分解的信息- 理论属性, 如 entropy, 共同信息 和 Fisher 信息矩阵, 并显示它自然地向更抽象的本地紧凑的表组群进行概括化 。