A high order solver for signature kernels

Signature kernels are at the core of several machine learning algorithms for analysing multivariate time series. The kernel of two bounded variation paths (such as piecewise linear interpolations of time series data) is typically computed by solving a Goursat problem for a hyperbolic partial differential equation (PDE) in two independent time variables. However, this approach becomes considerably less practical for highly oscillatory input paths, as they have to be resolved at a fine enough scale to accurately recover their signature kernel, resulting in significant time and memory complexities. To mitigate this issue, we first show that the signature kernel of a broader class of paths, known as smooth rough paths, also satisfies a PDE, albeit in the form of a system of coupled equations.We then use this result to introduce new algorithms for the numerical approximation of signature kernels. As bounded variation paths (and more generally geometric p-rough paths) can be approximated by piecewise smooth rough paths, one can replace the PDE with rapidly varying coefficients in the original Goursat problem by an explicit system of coupled equations with piecewise constant coefficients derived from the first few iterated integrals of the original input paths. While this approach requires solving more equations, they do not require looking back at the complex and fine structure of the initial paths, which significantly reduces the computational complexity associated with the analysis of highly oscillatory time series

Learning on sequential data with evolution equations

Data which have a sequential structure are ubiquitous in many scientific domains such as physical sciences or mathematical finance. This motivates an important research effort in developing statistical and machine learning models for sequential data. Recently, the signature map, rooted in the theory of controlled differential equations, has emerged as a principled and systematic way to encode sequences into finite-dimensional vector representations. The signature kernel provides an interface with kernel methods which are recognized as a powerful class of algorithms for learning on structured data. Furthermore, the signature underpins the theory of neural controlled differential equations, neural networks which can handle sequential inputs, and more specifically the case of irregularly sampled time-series. This thesis is at the intersection of these three research areas and addresses key modelling and computational challenges for learning on sequential data. We make use of the well-established theory of reproducing kernels and the rich mathematical properties of the signature to derive an approximate inference scheme for Gaussian processes, Bayesian kernel methods, for learning with large datasets of multi-channel sequential data. Then, we construct new basis functions and kernel functions for regression problems where the inputs are sets of sequences instead of a single sequence. Finally, we use the modelling paradigm of stochastic partial differential equations to design a neural network architecture for learning functional relationships between spatio-temporal signals. The role of differential equations of evolutionary type is central in this thesis as they are used to model the relationship between independent and dependent signals, and provide tractable algorithms for kernel methods on sequential data

Neural signature kernels as infinite-width-depth-limits of controlled ResNets

Motivated by the paradigm of reservoir computing, we consider randomly initialized controlled ResNets defined as Euler-discretizations of neural controlled differential equations (Neural CDEs). We show that in the infinite-width-then-depth limit and under proper scaling, these architectures converge weakly to Gaussian processes indexed on some spaces of continuous paths and with kernels satisfying certain partial differential equations (PDEs) varying according to the choice of activation function. In the special case where the activation is the identity, we show that the equation reduces to a linear PDE and the limiting kernel agrees with the signature kernel of Salvi et al. (2021). In this setting, we also show that the width-depth limits commute. We name this new family of limiting kernels neural signature kernels. Finally, we show that in the infinite-depth regime, finite-width controlled ResNets converge in distribution to Neural CDEs with random vector fields which, depending on whether the weights are shared across layers, are either time-independent and Gaussian or behave like a matrix-valued Brownian motion

Optimal stopping via distribution regression: a higher rank signature approach

Distribution Regression on path-space refers to the task of learning functions mapping the law of a stochastic process to a scalar target. The learning procedure based on the notion of path-signature, i.e. a classical transform from rough path theory, was widely used to approximate weakly continuous functionals, such as the pricing functionals of path--dependent options' payoffs. However, this approach fails for Optimal Stopping Problems arising from mathematical finance, such as the pricing of American options, because the corresponding value functions are in general discontinuous with respect to the weak topology. In this paper we develop a rigorous mathematical framework to resolve this issue by recasting an Optimal Stopping Problem as a higher order kernel mean embedding regression based on the notions of higher rank signatures of measure--valued paths and adapted topologies. The core computational component of our algorithm consists in solving a family of two--dimensional hyperbolic PDEs

Distribution regression for sequential data

Distribution regression refers to the supervised learning problem where labels are only available for groups of inputs instead of individual inputs. In this paper, we develop a rigorous mathematical framework for distribution regression where inputs are complex data streams. Leveraging properties of the expected signature and a recent signature kernel trick for sequential data from stochastic analysis, we introduce two new learning techniques, one feature-based and the other kernel-based. Each is suited to a different data regime in terms of the number of data streams and the dimensionality of the individual streams. We provide theoretical results on the universality of both approaches and demonstrate empirically their robustness to irregularly sampled multivariate time-series, achieving state-of-the-art performance on both synthetic and real-world examples from thermodynamics, mathematical finance and agricultural science

Higher order kernel mean embeddings to capture filtrations of stochastic processes

Stochastic processes are random variables with values in some space of paths. However, reducing a stochastic process to a path-valued random variable ignores its filtration, i.e. the flow of information carried by the process through time. By conditioning the process on its filtration, we introduce a family of higher order kernel mean embeddings (KMEs) that generalizes the notion of KME to capture additional information related to the filtration. We derive empirical estimators for the associated higher order maximum mean discrepancies (MMDs) and prove consistency. We then construct a filtration-sensitive kernel two-sample test able to capture information that gets missed by the standard MMD test. In addition, leveraging our higher order MMDs we construct a family of universal kernels on stochastic processes that allows to solve real-world calibration and optimal stopping problems in quantitative finance (such as the pricing of American options) via classical kernel-based regression methods. Finally, adapting existing tests for conditional independence to the case of stochastic processes, we design a causal-discovery algorithm to recover the causal graph of structural dependencies among interacting bodies solely from observations of their multidimensional trajectories

The SIB Swiss Institute of Bioinformatics' resources: focus on curated databases

The SIB Swiss Institute of Bioinformatics (www.isb-sib.ch) provides world-class bioinformatics databases, software tools, services and training to the international life science community in academia and industry. These solutions allow life scientists to turn the exponentially growing amount of data into knowledge. Here, we provide an overview of SIB's resources and competence areas, with a strong focus on curated databases and SIB's most popular and widely used resources. In particular, SIB's Bioinformatics resource portal ExPASy features over 150 resources, including UniProtKB/Swiss-Prot, ENZYME, PROSITE, neXtProt, STRING, UniCarbKB, SugarBindDB, SwissRegulon, EPD, arrayMap, Bgee, SWISS-MODEL Repository, OMA, OrthoDB and other databases, which are briefly described in this article

Using a mixed methodological approach to understand prosocial and antisocial behaviors in multiplayer online video games.

Ce travail de recherche a pour objectif de mieux comprendre les déterminants psychologiques, émotionnels et motivationnels des comportements prosociaux et antisociaux dans les jeux vidéo en ligne multijoueur en associant plusieurs approches méthodologiques. L’approche quantitative, par le biais de questionnaires en ligne, permet de différencier les profils psychologiques et motivationnels des joueurs occasionnels, des joueurs réguliers et des hyperjoueurs et de proposer des modèles explicatifs des comportements sociaux dans les jeux en ligne multijoueur. Une étude complémentaire vise à évaluer l’impact des jeux en ligne sur le sentiment d’isolement, les émotions et la régulation émotionnelle en situation de confinement. L’approche expérimentale observationnelle avec mesure physiologique (suivi oculaire, expressions faciales et rythme cardiaque) fournit des informations sur les déclencheurs potentiels des émotions lors de parties en ligne et sur la façon dont les joueurs réagissent en temps réel en fonction de leur profil psychologique. L’analyse qualitative d’entretiens semi-directifs explore les liens entre les comportements dans les jeux et les comportements sur Internet et hors ligne et informe sur la manière dont les joueurs perçoivent leurs comportements dans différents contextes. Les résultats indiquent que les traits de personnalité, les processus émotionnels et les motivations à jouer influencent la manière dont les joueurs se comportent entre eux dans les jeux en ligne multijoueur. On retrouve des profils psychologiques distincts entre les joueurs qui ont des comportements prosociaux et ceux qui se comportent de façon antisociale voire agressive pendant les parties.The aim of this research work is to investigate the psychological, emotional and motivational determinants of prosocial and antisocial behaviors in multiplayer online video games by combining several, but complementary, methodological approaches. A first systematic review article provides an overview of the links between the dimensions of the Five Factor Model of personality traits and behaviors associated with video games. The article highlighted the gaps in scientific knowledge about non-problematic gamers. We used a quantitative approach, through online questionnaires, to differentiate the psychological and motivational profiles of casual, regular and hardcore gamers and to propose explanatory models of social behaviors in multiplayer online games. A complementary study aimed at evaluating the impact of online games on feelings of isolation, emotions and emotional regulation during covid-19 quarantine. The observational experimental approach with physiological measurement (i.e., eye tracking, facial expressions and heart rate), provides information on potential emotional triggers during online games and on how gamers react in real time according to their psychological profile. The qualitative analysis of semi-structured interviews explores the links between gaming behavior and online and offline behaviors and provides information on how gamers perceive their behavior in different contexts. We showed that there are distinct psychological profiles between gamers who exhibit prosocial behaviors and those who behave antisocially or aggressively while they play

Cyberbullying

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