具备物理意识的、通过算术编码实现的香农最优压缩以实现分布一致性
Physics-Aware, Shannon-Optimal Compression via Arithmetic Coding for Distributional Fidelity
AI总结：
本文提出一种基于信息论的方法，利用无损压缩评估数据集的分布一致性，通过物理指导的概率表示提供绝对的信度标准，同时提升压缩效率。
摘要：
评估两个数据集是否在分布上一致是现代科学分析的核心，尤其在生成式人工智能产生合成数据时，其保真度需与真实观测在高维设置中验证。现有方法通常是相对的，只能确定一个数据集比另一个更一致于参考，但无法提供基于物理的绝对保真度标准。本文提出一种信息论方法，通过算术编码实现无损压缩，为数据集保真度提供基于物理的概率表示的操作性度量。共享相同物理相关性的数据集可以得到相似的最优描述，而差异（源于校准错误、建模错误或偏差）则表现为相对于由物理本身定义的香农最优极限的不可约编码长度过剩。这种编码长度过剩定义了绝对保真度度量，直接以比特数量化。不同于传统度量，缺乏内在尺度，零过剩提供了明确且具有物理意义的目标，对应于与基础分布的一致性。本文证明该度量是全局的、可解释的、在组件间可加的，并且渐近最优，差异在编码长度上对应于在共同参考模型下的预期负对数似然差异。作为副产品，本文方法在压缩效率上优于标准通用算法如gzip。这些结果使算术编码不仅成为压缩工具，更成为绝对、基于物理的分布保真度评估的测量仪器。
英文摘要：
Assessing whether two datasets are distributionally consistent is central to modern scientific analysis, particularly as generative artificial intelligence produces synthetic data whose fidelity must be validated against real observations in increasingly high-dimensional settings. Existing approaches are typically relative: they determine whether one dataset is more consistent with a reference than another, but do not provide a physically grounded absolute standard for fidelity. We propose an information-theoretic approach in which lossless compression via arithmetic coding provides an operational measure of dataset fidelity under a physics-informed probabilistic representation. Datasets sharing the same underlying physical correlations admit comparable optimal descriptions, while discrepancies-arising from miscalibration, mismodeling, or bias-manifest as an irreducible excess in codelength relative to the Shannon-optimal limit defined by the physics itself. This excess codelength defines an absolute fidelity metric, quantified directly in bits. Unlike conventional measures, which lack an intrinsic scale, zero excess provides a well-defined and physically meaningful target corresponding to consistency with the underlying distribution. We show that this metric is global, interpretable, additive across components, and asymptotically optimal, with differences in codelength corresponding to differences in expected negative log-likelihood under a common reference model. As a byproduct, our approach achieves improved compression relative to standard general-purpose algorithms such as gzip. These results establish arithmetic coding not merely as a compression tool, but as a measurement instrument for absolute, physics-grounded assessment of distributional fidelity.
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