发表机构
Kyoto University（京都大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出一种基于KL散度的表示感知传输信息度量，用于非包含支撑集的离散分布，通过UCN框架保留表示变换信息，以区分相同离散概率但来源不同的表示。
AI 中文摘要
比较概率分布的信息论度量在物理学及其他领域中被广泛使用。然而，当两个离散分布具有非包含支撑集时，Kullback-Leibler（KL）散度通常不能直接适用，因此人们引入了各种替代的散度和距离。这些度量比较的是由此得到的分布本身，但通常不保留关于表示变换的信息，即离散分布是由底层连续分布通过何种表示变换生成的。在此，我们引入一种用于具有非包含支撑集的离散分布的表示感知的传输信息度量，该度量基于标准KL散度构建。我们考虑两个连续的参考分布，每个分布通过各自的离散化方案转换为离散表示。我们不仅比较得到的离散分布或其连续参考，还额外保留与表示变换方案相关的局部信息。因此，所得到的度量能够区分那些可能具有相同离散概率景观但源自不同连续参考或离散化方案的离散表示。该构建基于不可避免的规范非线性（UCN）框架中的连续到离散表示的传输信息成本。UCN提供了离散化作为外部几何操作的传输成本与附近连续分布的信息论不可区分性之间的非任意对应关系，从而允许将离散表示与统计流形上底层连续分布的局部族相关联。
英文摘要
Information-theoretic measures for comparing probability distributions are widely used across physics and other fields. When two discrete distributions have non-inclusive supports, however, the Kullback-Leibler (KL) divergence is in general not directly applicable, and various alternative divergences and distances have been introduced. These measures compare the resulting distributions themselves, but do not generally retain information about the representation transformations by which the discrete distributions are generated from underlying continuous ones. Here we introduce a representation-aware transport-information measure for discrete distributions with non-inclusive supports, formulated based on the standard KL divergence. We consider two continuous reference distributions, each transformed into a discrete representation through its own discretization scheme. Rather than comparing only the resulting discrete distributions or their continuous references, we additionally retain local information associated with the representation-change schemes. The resulting measure can therefore distinguish discrete representations that may have identical discrete probability landscapes but originate from different continuous references or discretization schemes. The construction is based on the transport-information cost of continuous-to-discrete representation in the framework of unavoidable canonical nonlinearity (UCN). UCN provides a non-arbitrary correspondence between the transport cost of discretization as an extrinsic geometric operation and the information-theoretic indistinguishability of nearby continuous distributions, thereby allowing a discrete representation to be associated with a local family of underlying continuous distributions on the statistical manifold.
Comments6 pages. Notification of the parameter on submanifold is added. Application to the case where one of the two distributions is the continuous case is added
