A link to wikipedia for rough pat theory is: https://en.wikipedia.org/wiki/Rough_path It appears path and signatures has one to one mapping in many cases. I understand that the signature is not dependent on initial condition and if there is any loop in the path it cancels out in the signature. Therefore we can never recover the starting point of a path or loop like path from the signature. The question is given a signature when can we get back the path from which it is derived? It is assumed that the initial condition is zero or known and there is no loop in the entire path (never crosses itself). Any comments would be highly appreciated. Any reference or work in this area would be helpful. Is there a formal procedure to get back the path from the signature?

A link to wikipedia for rough pat theory is: https://en.wikipedia.org/wiki/Rough_path It appears path and signatures has one to one mapping in many cases. I understand that the signature is not dependent on initial condition and if there is any retraces or double back on itself in the path it cancels out in the signature. Therefore we can never recover the starting point of a path or any segment where the path retraces or doubles back on itself from the signature. The question is given a signature when can we get back the path from which it is derived? It is assumed that the initial condition is zero or known and there is retraces or double back on itself in the entire path (never crosses itself). Any comments would be highly appreciated. Any reference or work in this area would be helpful. Is there a formal procedure to get back the path from the signature?