Can we extract information from signature (rough path theory) to construct part of signal? This question is related to rough path theory. Consider we have obtained  signature obtained from a  set discrete data points postulating linear from one data point to another. Such signature are used in some machine learning applications. A link to such application is : https://sutdbrain.files.wordpress.com/2017/07/nengli_lim_sutd_technical_talk-first-half.pdf
I am looking for possibility of extracting information from the signature to construct the signal. This would similar to filtering as in digital signal processing https://en.wikipedia.org/wiki/Digital_signal_processing. 
Are anyone aware of such application of signature? More importantly is there any model (in the literature) of the original discrete data points based on the linear (or other) combination of  signature elements. 
 A: Inverting the signature is an ongoing subject of research. The basic version of your final question is "If I have a d-dimensional path starting at the origin which is N points joined by linear interpolation, and I know its signature up to level m (i.e. truncated -- I have the values of iterated integrals of depth m or less), can I recover the points?". 
The general answer to this question is no. This is clear, because the number of unknowns d(N-1) could be very large compared to the size of the truncated signature, so we have an undetermined system.
Some ideas:


*

*If N is known and small (where how small depends on d and m) then the calculation is possible. Cases are discussed by Améndola, Friz and Sturmfels in https://arxiv.org/abs/1804.08325.

*If the signature is not truncated and we have an infinite computer, i.e. if m is infinity, then we can invert the path using the method of Lyons and Xu, see https://arxiv.org/abs/1406.7833.

*Given a truncated signature, finding any path (not specifically a piecewise linear path) which has that truncated signature is not easy in general. I sometimes think of a truncated signature as a nice global summary of a shape of a path, but one that doesn't give any concrete local information. The signature isn't a magical lossless compression method. Applications where the method is useful might be where we care what a path is basically like, what it does, what its net effect is on some system which it is driving, and that kind of thing.

