Conference Agenda

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.

Rough Paths theory allows to describe the response to differential equations driven by oscillatory signals. The universality theorem ensures that the solution depends continuously on the control, provided we utilise the correct topology, namely the p-variation distance. The latter can be defined for paths parameterised on the same time interval. We extend this topology to a warping p-variation metric on the space of un-parameterised paths. Computing this metric is difficult in practice as it is defined as a min-max problem. Therefore standard dynamic programming techniques fail in this case. We overcome this difficulty and propose a branch-and-bound algorithm to compute it effectively. Finally we apply K-nearest-neighbours types of algorithm to time series classification and clustering using both the warping p-variation distance and the universally used dynamic time warping distance, showing a striking superiority of the former in both tasks.

When introducing the iterated-integral signature of a path, we often give the following fact as an illuminating example: The information that level 2 adds beyond the total displacement given by level 1, is the "signed area" of each two-dimensional projection of the path. Given any two one-dimensional paths on the same interval, we can construct another as the cumulative signed area. It is natural to ask about all the paths we can get starting with all projections of a path and iteratively taking signed area. In particular the collection of their final values, which we call the areas-of-areas. What signature elements do they correspond to? Do they contain the same information as the signature (yes). How might we find a minimum subset of them which we can take which determine the signature?

Persistent homology is a tool used to analyse topological features of data. In this talk, I will describe a new feature map for barcodes that arise in persistent homology computation. The main idea is to first realize each barcode as a path in a convenient vector space, and to then compute its path signature which takes values in the tensor algebra of that vector space. The composition of these two operations — barcode to path, path to tensor series — results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness, and achieves high performance on several classification benchmarks.

Character groups of Hopf algebras arise naturally in a variety of applications. For example, they appear in numerical analysis, control theory and the theory of rough paths and stochastic analysis. In the talk we will review the geometry and some main examples of these (infinite-dimensional) groups. Then we will report on some recent progress for character groups associated to so called combinatorial Hopf algebras. In the combinatorial setting, certain subgroups of the character group are closely connected to locally convergent Taylor series like expansions which are of interest in the applications mentioned above.