The Signature transform is a way to encode paths and distributions of random paths in Rn. It is a collection of all iterated integrals of a path.
The theory for piecewise differentiable paths follows naturally from the classical analysis and the Riemann-Stieltjes integral. Path space of realizations of stochastic processes of finite quadratic variation can have signatures with respect to Ito or Stratonovich Integral. Even wider path space can be characterized with signatures in the Rough Path Theory.
Special consideration goes to the question of signature uniqueness - what kind of infor- mation cannot be captured by integrating?
The signature of a path can index its terms with words and thus is identified with an ele- ment of the Tensor Algebra over Rn. We look the Shuffle operator on words and algebraic connection between the signature terms of the shuffled words.
We look at the newest advances in applying signatures with statistical and Machine Learn- ing methods. The natural approach is simply using the signature transform of sequential data as a feature encoding method which can then be used with almost any model.
There is a restricted space of paths on which the signature transform defines a kernel , un- locking the doors to hypothesis testing samples of paths on this space. It is a characteristic transform for distributions on this path space.
Lastly, we see usage of Deep Signature Transforms. Signatures are used as layers in deep neural networks, intertwined with non-linear transforms of the paths. Lifts are used to preserve the path structure, outputting paths of signatures of prefixes of previous paths.
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