Assessing whether two datasets are distributionally consistent has become a central theme in modern scientific analysis, particularly as generative artificial intelligence is increasingly used to produce synthetic datasets whose fidelity must be rigorously validated against the original data on which they are trained, a task made more challenging by the continued growth in data volume and problem dimensionality. In this work, we propose the use of arithmetic coding to provide a lossless and invertible compression of datasets under a physics-informed probabilistic representation. Datasets that share the same underlying physical correlations admit comparable optimal descriptions, while discrepancies in those correlations-arising from miscalibration, mismodeling, or bias-manifest as an irreducible excess in code length. This excess codelength defines an operational fidelity metric, quantified directly in bits through differences in achievable compression length relative to a physics-inspired reference distribution. We demonstrate that this metric is global, interpretable, additive across components, and asymptotically optimal in the Shannon sense. Moreover, we show that differences in codelength correspond to differences in expected negative log-likelihood evaluated under the same physics-informed reference model. As a byproduct, we also demonstrate that our compression approach achieves a higher compression ratio than traditional general-purpose algorithms such as gzip. Our results establish lossless, physics-aware compression based on arithmetic coding not as an end in itself, but as a measurement instrument for testing the fidelity between datasets.

翻译：评估两个数据集是否具有分布一致性已成为现代科学分析的核心议题，尤其在生成式人工智能被广泛用于生成合成数据集的背景下，这些合成数据集必须与训练所用的原始数据在保真度上进行严格验证。随着数据量和问题维度的持续增长，这一任务变得更具挑战性。本研究提出利用算术编码，在物理信息驱动的概率表示下实现数据集的无损可逆压缩。共享相同底层物理关联的数据集可获得相近的最优描述，而那些因校准偏差、建模错误或系统偏差导致物理关联不一致的数据集，则会表现为编码长度的不可约冗余。这种冗余编码长度定义了一种可操作的保真度度量，其通过相对于物理启发的参考分布可实现的压缩长度差异，以比特为单位直接量化。我们证明该度量具有全局性、可解释性、跨分量可加性，且在香农意义下是渐近最优的。此外，我们表明编码长度差异对应于在同一物理信息参考模型下评估的期望负对数似然差异。作为附带成果，我们还证明本压缩方法相较于传统通用算法（如gzip）实现了更高的压缩比。我们的研究结果表明，基于算术编码的无损物理感知压缩不仅是一种压缩手段，更可作为检验数据集间保真度的测量工具。