  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.
