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Signature
Reference: Signatures and Streamed Data-New Mathematics for Data Science | Terry Lyons - YouTube
The goal of Rough path - Wikipedia aka Path Signature is to provide a perspective to understand data stream, especially in multimodal multidimensional settings.
Data stream, in a broad sense, can be something like drawing with your phone (handwriting), a sequence of words (text), or even an Amazon transaction. Imagine the body pose tracking problem, where a landmark of the body moving in time/video stream, it forms a path on the screen. Writing a number could also form a path, regardless of the speed you write.
There is something about the path that is invariant, just like you can take photos of a face from different angles even though it is the same face. A “3” is a “3” no matter how you write it.
In this case, typical deep learning models are spending a lot of the time learning such invariance instead of the true feature/essence of the data (learning the definition VS. learning the similarity)
Writing a “3” can be decomposed into two paths in \(x\) and \(y\) directions against the time. (But streaming data can expand beyond 3 dimensions in reality). So we need a better description of the path to solve the challenge.
The goal of Path Signature is not just to describe a certain window (values) of the path but rather the effect it has or sometimes both (values and effects).
Given a streaming data window \(I = [s, t]\) (not the time window as this is agnostic to the time), and \(k\) samples of the data, the signature is defined by
$$ S_k(\gamma, I) := \iint_{s<u_1<...<u_k<t}d\gamma_{u_1}\otimes d\gamma_{u_2}...d\gamma_{u_k} $$
It is a feature description of sequential of effects, while some parameterization like Fourier Transformation are more time-dependently and less effective to describe the invariance.
Since this is a feature by itself, you can use it with different models you like.
Library they used to calculate the signature GitHub - bottler/iisignature: Iterated integral signature calculations