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With the recent success of representation learning methods, which includesdeep learning as a special case, there has been considerable interest indeveloping representation learning techniques that can incorporate knownphysical constraints into the learned representation. As one example, in manyapplications that involve a signal propagating through physical media (e.g.,optics, acoustics, fluid dynamics, etc), it is known that the dynamics of thesignal must satisfy constraints imposed by the wave equation. Here we propose amatrix factorization technique that decomposes such signals into a sum ofcomponents, where each component is regularized to ensure that it satisfieswave equation constraints. Although our proposed formulation is non-convex, weprove that our model can be efficiently solved to global optimality inpolynomial time. We demonstrate the benefits of our work by applications instructural health monitoring, where prior work has attempted to solve thisproblem using sparse dictionary learning approaches that do not come with anytheoretical guarantees regarding convergence to global optimality and employheuristics to capture desired physical constraints.

Wave-Informed Matrix Factorization with Global Optimality Guarantees