Wave-Informed Matrix Factorization with Global Optimality Guarantees

TL;DR

This paper introduces a wave-informed matrix factorization method that incorporates physical wave constraints, guarantees global optimality, and improves applications like structural health monitoring.

Contribution

It presents a novel non-convex matrix factorization approach that can be solved to global optimality with polynomial time guarantees, integrating wave physics into representation learning.

Findings

Efficient polynomial-time algorithm for globally optimal solutions

Enhanced structural health monitoring accuracy

Theoretical guarantees for convergence and physical constraint satisfaction

Abstract

With the recent success of representation learning methods, which includes deep learning as a special case, there has been considerable interest in developing representation learning techniques that can incorporate known physical constraints into the learned representation. As one example, in many applications that involve a signal propagating through physical media (e.g., optics, acoustics, fluid dynamics, etc), it is known that the dynamics of the signal must satisfy constraints imposed by the wave equation. Here we propose a matrix factorization technique that decomposes such signals into a sum of components, where each component is regularized to ensure that it satisfies wave equation constraints. Although our proposed formulation is non-convex, we prove that our model can be efficiently solved to global optimality in polynomial time. We demonstrate the benefits of our work by applications in structural health monitoring, where prior work has attempted to solve this problem using sparse dictionary learning approaches that do not come with any theoretical guarantees regarding convergence to global optimality and employ heuristics to capture desired physical constraints.