Field Theory of Data: Anomaly Detection via the Functional Renormalization Group. The 2D Ising Model as a Benchmark

TL;DR

This paper introduces a physics-inspired framework using the functional renormalization group to improve anomaly detection in noisy data, demonstrating high accuracy with the 2D Ising model as a benchmark.

Contribution

It establishes a novel connection between phase transition detection and anomaly detection, applying the functional renormalization group to non-equilibrium systems.

Findings

Achieves less than 4% error in critical threshold detection.

Outperforms Kullback-Leibler divergence in identifying phase transitions.

Provides a universal approach for data structure resolution near criticality.

Abstract

We establish a correspondence between anomaly detection in high-noise regimes and the renormalization group flow of non-equilibrium field theories. We provide a physical grounding for this framework by proving that the detection of phase transitions in interacting non-equilibrium systems maps to the study of an effective equilibrium field theory near its Gaussian fixed point, which we identify with the universal Marchenko-Pastur distribution. Applying the Functional Renormalization Group to the two-dimensional Model A, we demonstrate that the noise-to-signal ratio acts as a physical temperature, where the signal emerges as ordered domains within a thermalized background of fluctuations. Using the exact Onsager solution as a benchmark, we show that this approach identifies critical thresholds with an error below 4%, significantly outperforming standard information-theoretic metrics such as the Kullback-Leibler divergence. Our results provide a universal strategy for resolving structures in complex datasets near criticality, bridging the gap between statistical mechanics and statistical inference.