论文标题
通过晶格的随机变量分解的信息属性
Information Properties of a Random Variable Decomposition through Lattices
论文作者
论文摘要
欧几里得空间中的全级晶格是由所有基础的整数线性组合形成的离散集。给定$ \ mathbb {r}^n $上的概率分布，可以通过通过这样的晶格考虑空间的商来引起两个操作：包装和量化。对于晶格$λ$，以及一个基本域$ d $，tiles $ \ mathbb {r}^n $通过$λ$，通过将每个coset上的密度求和，而在每个caset上的量化分布可以通过lattice上的量化分布来获得，而在每个coset上的量化分布是通过在每个基本域上定义的。这些操作分别在$ d $和$λ$上定义了包装和量化的随机变量，这些变量总计为原始随机变量。我们研究了这种分解的信息理论特性，例如熵，互信息和Fisher Information矩阵，并表明它自然地将其推广到本地紧凑的拓扑组的更抽象的环境。
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 $Λ$, and a fundamental domain $D$ which tiles $\mathbb{R}^n$ through $Λ$, 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 $Λ$, 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.
