The identification of universal properties from minimally processed data sets is one goal of machine learning techniques applied to statistical physics. Here, we study how the minimum number of variables needed to accurately describe the important features of a data set—the intrinsic dimension ($I_d$)—behaves in the vicinity of phase transitions. We employ state-of-the-art nearest-neighbors-based $I_d$ estimators to compute the $I_d$ of raw Monte Carlo thermal configurations across different phase transitions: first-order, second-order, and Berezinskii-Kosterlitz-Thouless. For all the considered cases, we find that the $I_d$ uniquely characterizes the transition regime. The finite-size analysis of the $I_d$ allows us to not only identify critical points with an accuracy comparable to methods that rely on a priori identification of order parameters but also to determine the corresponding (critical) exponent $\nu$ in the case of continuous transitions. For the case of topological transitions, this analysis overcomes the reported limitations affecting other unsupervised learning methods. Our work reveals how raw data sets display unique signatures of universal behavior in the absence of any dimensional reduction scheme and suggest direct parallelism between conventional order parameters in real space and the intrinsic dimension in the data space.

中文翻译：

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通过内在维度无监督地学习普遍的批判行为

从最少处理的数据集中识别通用属性是应用于统计物理学的机器学习技术的目标之一。在这里，我们研究如何准确描述数据集的重要特征所需的最少数量的变量-内在维度（${\mathrm{一世}}_d$）—表现在相变附近。我们采用最先进的基于最近的邻居${\mathrm{一世}}_d$ 估算器来计算 ${\mathrm{一世}}_d$跨不同相变的原始蒙特卡洛热配置：一阶，二阶和Berezinskii-Kosterlitz-Thouless。对于所有考虑的情况，我们发现${\mathrm{一世}}_d$独特地描绘了过渡制度。有限元分析${\mathrm{一世}}_d$使我们不仅能够以与依赖顺序参数先验识别的方法相媲美的精度来识别关键点，而且还能确定相应的（关键）指数$\nu$在连续过渡的情况下。对于拓扑转换的情况，该分析克服了所报告的影响其他无监督学习方法的局限性。我们的工作揭示了原始数据集如何在没有任何降维方案的情况下显示出通用行为的独特特征，并提出了真实空间中常规顺序参数与数据空间中固有维度之间的直接并行性。