Programme

Titre : Modeling spatio temporal data : the contribution of the signature method

Résumé :

Encoding dependencies between different component of a signal or different signals in a meaningful way is an important problem because of its ubiquity in many areas of science and technology as finance, medicine, environment or climatology. There is however few approaches to encode these dependencies grouping more than two time series and providing associated data mining technics .

This suggests a move towards multilinear features including time evolution in its definition, and it is precisely within this framework that the so-called Signature Method has been introduced.

After a brief presentation of the method, we shall emphasize the main characteristics of the approach in data analysis : low computational cost, interpretability.

I shall present some preliminary works on synthetic data and textual data.

Titre : Signature estimation using Seigal's factorisation formula: anticoncentration and robustification 

Résumé : 

The theory of Signatures is a fast growing field which has demonstrated wide applicability to a large range of applications, from finance to health monitoring. Computing signatures often relies on the assumptions that the signal under study is not corrupted by noise, which is rarely the case in practice. In the present paper, we study the influence of noise on the computation of signature via the theory of anti-concentration. We then propose a median of means (MoM) approach to the estimation problem and give a bound on the estimation error using Rademacher complexity. 

Titre : Learning the Dynamics of Sparsely Observed Interacting Systems 

Résumé :

We address the problem of learning the dynamics of an unknown non-parametric system linking a target and a feature time series. The feature time series is measured on a sparse and irregular grid, while we have access to only a few points of the target time series. Once learned, we can use these dynamics to predict values of the target from the previous values of the feature time series. We frame this task as learning the solution map of a controlled differential equation (CDE). By leveraging the rich theory of signatures, we are able to cast this non-linear problem as a high dimensional linear regression. We provide an oracle bound on the prediction error which exhibits explicit dependencies on the individual-specific sampling schemes. Our theoretical results are illustrated by simulations which show that our method outperforms existing algorithms for recovering the full time series while being computationally cheap. We conclude by demonstrating its potential on real-world epidemiological data. 

Titre : Framing RNN as a kernel method through the signature

Résumé :

Building on the interpretation of a recurrent neural network (RNN) as a continuous-time neural differential equation, we show, under appropriate conditions, that the solution of a RNN can be viewed as a linear function of the signature. This connection allows us to frame a RNN as a kernel method in a suitable reproducing kernel Hilbert space. As a consequence, we obtain theoretical guarantees on generalization and stability for a large class of recurrent networks. If time allows, the interpretation of neural networks as continuous-time neural differential equations will also be discussed, with results highlighting the convergence of trained residual networks towards neural ODEs in the large-depth regime.

Titre :  Mean and Principal Components of time series, a new approach with the signature method

Résumé :

The aim of our work is twofold: average multidimensional time series and find directions of importance. We encode time series with integrals of various moment orders, constituting their signature.

First, we have developed an approach to average signatures coefficients. The space of signature coefficients is a manifold with a group structure but without a Riemannian metric, making it difficult to use classic Riemannian approaches.

Then, in the same spirit as in the averaging procedure, we look for important geodesics. Important in the sense that the signature coefficients have maximum variance along those. Thus, they describe well the data in the space of signature coefficients. Those main directions could be used for a qualitative interpretation of the data but also for dimension reduction, as it is done with the well known Principal Component Analysis when analyzing data in a Euclidean space. 

Titre : Discovering underlying topological structure on a set of time series using signatures

Résumé :

Topological Data Analysis is a field of great interest in many applications such as finance or neuroscience. Our goal is to propose a novel approach to building simplicial complexes that capture the multiway ordered interactions in the components of high-dimensional time series using the theory of Signatures. Numerical experiments on an fMRI dataset illustrates the efficiency and relevance of our approach.