Optimization flow for approximating a matrix state involving orthonormal constraints

Abstract

In this work, we introduce a continuous-time dynamical flow. The purpose of this flow is to approximate a matrix state while precisely adhering to orthonormal constraints. Additionally, we apply restrictions on the probability distribution that expand beyond these constraints. Our work contributes in two ways. Firstly, we demonstrate in theory that our proposed flow guarantees convergence to the stationary point of the objective function. It consistently reduces the value of this function for almost any initial value. Secondly, we show that our approach can retrieve the decomposition of a given matrix. Even if the matrix is not inherently decomposable, our results illustrate that our approach remains reliable in obtaining optimal solutions.