Can we get back path from signatures?

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This question is related to rough path theory. A link to wikipedia is: https://en.wikipedia.org/wiki/Rough_path It appears path and signatures has one to one mapping (like Fourier transform), the question is, given a signature can we get back the path from which it is derived? It is assumed hat the initial condition is zero or known and there is no retraces or double back on itself in the entire path (never crosses itself). Any comments would be highly appreciated.