Schrödinger principal-component analysis: On the duality between principal-component analysis and the Schrödinger equation

Sitian Qian, Ziming Liu, Yuxuan Yan

Abstract:

Principal component analysis (PCA) has been applied to analyze random fields in various scientific disciplines.
However, the explainability of PCA remains elusive unless strong domain-specific knowledge is available. This
paper provides a theoretical framework that builds a duality between the PCA eigenmodes of a random field
and eigenstates of a Schrödinger equation. Based on the duality we propose the Schrödinger PCA algorithm
to replace the expensive PCA solver with a more sample-efficient Schrödinger equation solver. We verify the
validity of the theory and the effectiveness of the algorithm with numerical experiments.