Unsupervised Learning Universal Critical Behavior via the Intrinsic Dimension

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