The iterated-integral signature, or rough-path signature, of a path has proved useful in several machine learning applications in the last few years. This work is extended in a number of ways. Algorithms for computing the signature and log signature efficiently are investigated and evaluated, which is useful for many applications of signatures when working with large datasets. Online Chinese character recognition using signature features with recurrent neural networks is investigated. A recurrent neural network cell which stores its memory as the signature of a path is suggested and demonstrated on a toy problem.
There is an essentially unique element of the signature of a path in space which, under transformations of the space, scales with volume. That element is characterised geometrically.
Given two features of curves, you can make a new one by taking the signed area of the 2d curve those two features make as a curve is traced out. A simple algebraic description of those features (which turn out to be signature elements) which can be formed from linear combinations of such combinations of total displacements is conjectured and worked towards. This is know as “areas of areas”
