  Matrix congruence extends naturally to the setting of tensors. We apply
methods from tensor decomposition, algebraic geometry and numerical
optimization to this group action. Given a tensor in the orbit of another
tensor, we compute a matrix which transforms one to the other. Our primary
application is an inverse problem from stochastic analysis: the recovery of
paths from their third order signature tensors. We establish identifiability
results, both exact and numerical, for piecewise linear paths, polynomial
paths, and generic dictionaries. Numerical optimization is applied for recovery
from inexact data. We also compute the shortest path with a given signature
tensor.
