3:00pm - 5:00pmSignature tensors of paths
Chair(s): Carlos Améndola (TU Munich), Joscha Diehl (MPI Leipzig), Francesco Galuppi (MPI Leipzig), Anna Seigal (UC Berkeley)
Given a path X in R^n, it is possible to naturally associate an infinite list of tensors, called the iterated-integral signature of X. These tensors were introduced in the 1950s by Kuo-Tsai Chen, who proved that every (smooth enough) path is uniquely determined by its signature. Over the years this topic became central in control theory, stochastic analysis and, lately, in time series analysis.
In applications the following inverse problem appears: given a finite collection of tensors, can we find a path that yields them as its signature? One usually introduces additional requirements, like minimal length, or a parameterized class of functions (say, piecewise linear). It then becomes crucial to know when there are only finitely many paths having a given signature that satisfy the constraints. This problem, called identifiability, can be tackled with an algebraic-geometric approach.
On the other hand, by fixing a class of paths (polynomial, piecewise linear, lattice paths, ..), one can look at the variety carved out by the signatures of those paths inside the tensor algebra. Besides identifiability, the geometry of these signature varieties can give a lot of information on paths of that class. One important class is that of rough paths. Apart from applications to stochastic analysis, its signature variety has a strong geometric significance and it exhibits surprising similarities with the classical Veronese variety.
In time series analysis, it is often necessary to extract features that are invariant under some group action of the ambient space. The signature of iterated signals is a general way of feature extraction; one can think of it as a kind of nonlinear Fourier transform. Understanding its invariant elements relates to classical invariant theory but poses new algebraic questions owing to the particularities of iterated integrals.
Recent developments in these aspects will be explored in this minisymposium.
(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)
Varieties of signature tensors
Carlos Améndola
TU Munich
The signature of a parametric curve is a sequence of tensors whose entries are iterated integrals. This construction is central to the theory of rough paths in stochastic analysis. It is here examined through the lens of algebraic geometry. We introduce varieties of signature tensors for both deterministic paths and random paths. For the former, we focus on piecewise linear paths, on polynomial paths, and on varieties derived from free nilpotent Lie groups. For the latter, we focus on Brownian motion and its mixtures. Joint work with Peter Fritz and Bernd Sturmfels.
Learning paths from signature tensors
Max Pfeffer
MPI Leipzig
We aim to recover paths from their third order signature tensors. For this, we apply methods from tensor decomposition, algebraic geometry and numerical optimization to the group action of matrix congruence. Given a signature tensor in the orbit of another tensor, we compute a matrix which transforms one to the other. We establish identifiability results, both exact and numerical, for piecewise linear paths, polynomial paths, and generic dictionaries. Numerical optimization is applied for recovery from inexact data. We also compute the shortest path with a given signature tensor.
Signatures of paths: an algebraic perspective
Laura Colmenarejo
MPI Leipzig
Coming from stochastic analysis, the signature of a path is the collection of all the iterated integrals of the path. It can be seen in terms of tensors or as formal power series in words, which make them more relevant in other areas such as algebraic geometry or combinatorics. In this talk, I would like to look at the signatures of paths from an algebra perspective. For that, we will look at the work done by C. Améndola, P. Friz, and B. Sturmfels about the variety defined by the signature of piecewise linear paths, as well as the work done by F. Galuppi about the variety of rough paths. As a continuation, I would like to present our work on the signature varieties of two very different classes of paths: rough paths and axis-parallel paths. This is joint work with F. Galuppi and M. Michalek.
Signatures of paths transformed by polynomial maps
Rosa Preiss
TU Berlin
We characterize the signature of piecewise continuously differentiable paths transformed by a polynomial map in terms of the signature of the original path. For this aim, we define recursively an algebra homomorphism between two shuffle algebras on words. This homomorphism does not depend on the path and behaves well with respect to composition and homogeneous maps. Joint work with Laura Colmenarejo.