Physical meaning of principal component analysis for classical lattice systems with translational invariance

TL;DR

This paper reveals that principal component analysis (PCA) applied to translationally invariant classical lattice systems relates to reciprocal lattice vectors and Fourier transforms, offering a physical interpretation and a computational shortcut.

Contribution

It establishes a physical meaning for PCA components in classical lattice systems and connects them to Fourier analysis, providing a new perspective and practical approach.

Findings

PCA components correspond to reciprocal lattice vectors.

Eigenvalues relate to Fourier transforms of configurations.

Provides a method to approximate principal components without diagonalization.

Abstract

We explore the physical implications of applying principal component analysis (PCA) to translationally invariant classical systems defined on a $d$-dimensional hypercubic lattice. Using Rayleigh-Schr\"odinger perturbation theory, we demonstrate that the principal components are related to the reciprocal lattice vectors of the hypercubic lattice, and the corresponding eigenvalues are connected to the discrete Fourier transform of the sampled configurations. From a different perspective, we show that the PCA in question can be viewed as a numerical method for computing the ensemble average of the squared moduli of the Fourier transform of physical quantities. Our results also provide a way to determine approximately the principal components of a classical system with translational invariance without the need for matrix diagonalization.