Inversion of signatures

We show below a simple example of signature inversion. A crucial parameter is the depth chosen for the signature: the function signatory.invert_signature() reconstructs a piecewise linear path with depth + 1 points that approximates the original path the signature was computed on.

import math
import torch
import signatory

# Create a path consisting in a half circle
time = torch.linspace(0, 1, 10)
path = torch.stack([torch.cos(math.pi * time), torch.sin(math.pi * time)]).T.unsqueeze(0)

# Compute the signature
depth = 11
signature = signatory.signature(path, depth)

# Reconstruct the path by inverting the signature
reconstructed_path = signatory.invert_signature(signature, depth, path.shape[2], initial_position=path[:, 0, :])

Note that the signature being translation invariant, we have given the first position of the path as argument to signatory.invert_signature(). Otherwise, the reconstructed_path would begin at zero.

We show below the original path (blue curve) and its reconstruction (orange curve).