We develop a framework for dualizing the Kolmogorov structure function $h_x(\alpha)$, which then allows using computable complexity proxies. We establish a mathematical analogy between information-theoretic constructs and statistical mechanics, introducing a suitable partition function and free energy functional. We explicitly prove the Legendre-Fenchel duality between the structure function and free energy, showing detailed balance of the Metropolis kernel, and interpret acceptance probabilities as information-theoretic scattering amplitudes. A susceptibility-like variance of model complexity is shown to peak precisely at loss-complexity trade-offs interpreted as phase transitions. Practical experiments with linear and tree-based regression models verify these theoretical predictions, explicitly demonstrating the interplay between the model complexity, generalization, and overfitting threshold.

翻译：我们提出了一个用于对偶化柯尔莫哥洛夫结构函数 $h_x(\alpha)$ 的框架，该框架进而允许使用可计算的复杂度代理。我们在信息论构造与统计力学之间建立了一个数学类比，引入了合适的配分函数和自由能泛函。我们明确证明了结构函数与自由能之间的勒让德-芬切尔对偶性，展示了Metropolis核的细致平衡，并将接受概率解释为信息论散射振幅。模型复杂度的一个类磁化率方差被证明在损失-复杂度权衡处达到峰值，这些权衡被解释为相变。使用线性和基于树的回归模型进行的实际实验验证了这些理论预测，明确展示了模型复杂度、泛化能力与过拟合阈值之间的相互作用。