Theoretical derivation of principal component analysis results for the two-dimensional Ising model
In the Ising model,principal component analysis(PCA)serves as an unsupervised learning method for identifying phase transitions.The PCA results of the two-dimensional square-lattice Ising model with periodic boundary conditions are theoretically deduced.By utilizing the Hamiltonian and lattice structure of the Ising model,the structure of the sample covariance matrix can be determined,enabling the realization of PCA by solving the eigenvalue problem of the sample covariance matrix.All symmetries of the lattice are implicit in the sample covariance matrix,with translational symmetry determining the eigenvector corresponding to the Fourier mode.Moreover,based on translational symmetry,other symme-tries determine the degeneracy of eigenvalues.When combined with knowledge regarding the Ising model,the meaning of the PCA results becomes clarified.Our theoretical derivation elucidates the PCA results from previous studies on the Ising model.Based on theoretical derivation,we also investigated the ability of PCA to identify phase transitions in the Ising model.From the machine learning(ML)perspective,the theoretical derivation of PCA results sheds light on the mecha-nism through which PCA identifies phase transitions.This clarity contributes to removing opacity in ML and increasing trust in the application of ML in physics.
principal component analysisIsing modelphase transitionFourier mode
万方数据
维普
