Wave-Informed Matrix Factorization with Global Optimality Guarantees
This work addresses the need for physically constrained representation learning in domains like structural health monitoring, offering a method with theoretical guarantees where prior approaches relied on heuristics.
The paper tackles the problem of incorporating wave equation constraints into representation learning for signals propagating through physical media, proposing a matrix factorization method that ensures each component satisfies these constraints and proving it can be efficiently solved to global optimality in polynomial time.
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.