Representation-Aware Transport-Information Measure for Non-inclusive Discrete Supports
Abstract
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.
Related
- A Transport-Information Framework for Continuous-Fiber-Aligned Coarse-Graining Representation Sets
- A Phase-Space Inclusive Figure of Merit Based in Optimal Transport for Validating Monte Carlo Reweightings
- Stellar Representation of Gottesman Kitaev Preskill States on the Double Discrete Cylinder
- Caputo–FRFT Coupling Induced by Compatible Fractional Convolution: Representation and Discrete Realization
- On representation of macroscopic crack in periodic fine-scale discrete mechanical models
BibTeX
@misc{yuge2026representationaware,
title = {{Representation-Aware Transport-Information Measure for Non-inclusive Discrete Supports}},
author = {Koretaka Yuge},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.30944},
eprint = {2609.30944},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30944},
}
RIS
TY - UNPB
TI - Representation-Aware Transport-Information Measure for Non-inclusive Discrete Supports
AU - Koretaka Yuge
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2609.30944
UR - https://arxiv.org/abs/2609.30944
AB - 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.
ER -