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Matrix congruence can be extended to the setting of tensors. I describe how to apply methods from tensor decomposition, algebraic geometry and numerical optimization to this group action. Given a tensor in the orbit of another tensor, we can compute a matrix which transforms one to the other. The application is an inverse problem from stochastic analysis: the recovery of paths from their signature tensors of order three. I give identifiability results and recovery algorithms for piecewise linear paths, polynomial paths, and generic dictionaries. This is based on joint work with Max Pfeffer and Bernd Sturmfels.