Abstract : Rough path theory is an emerging mathematical technology that captures macroscopically interactions of highly oscillatory streamed data. Formally, it extends the domain of definition for the calculus of deterministic controlled differential equations, allowing them to be driven by complex signals, potentially rougher than Brownian motion. This area has built bidirectional connections with data science and machine learning, enabling the development of novel, mathematics-informed methods for efficiently analyzing time series data, e.g. PDE-based Signature kernel, path development layer with Lie group representation. This minisymposia series facilitates the discussion of new methodological innovations on this interface between rough paths and data science.
Abstract : We introduce classical (convenient) concepts of real analytic functions on path spaces and apply them to the
approximation of path space functionals. We also provide an invariance theory perspective on the Hambly-Lyons
theorem that signatures characterize paths up to tree like equivalences.
[01366] PCF-GAN: generating sequential data via the characteristic function of measures on the path space
Format : Talk at Waseda University
Author(s) :
Hao Ni (UCL)
Hang Lou (UCL)
Siran Li (Shanghai Jiao Tong University )
Abstract : Implicit Generative Models(IGMs) are powerful tools for generating high-fidelity synthetic data. However, they struggle to capture the temporal dependence of time-series data. To tackle this issue, we directly compare the path distributions via the characteristic function of measures on the path space(PCF) from rough path theory, which uniquely characterises the law of stochastic processes. We then develop a novel PCF-GAN model by incorporating PCF with IGM for time series generation, leading to significant performance boost.
[01336] Nyström approximation and convex kernel quadrature
Format : Talk at Waseda University
Author(s) :
Satoshi Hayakawa (University of Oxford)
Abstract : We will discuss a refined analysis of Nyström approximation for an integral operator based on statistical learning theory, and demonstrate how we can use these results to obtain an improved estimate of the performance of convex kernel quadrature rules that are given by the low-rank kernel of Nyström approximation.
[01374] Taylor remainder estimate for rough differential equations
Format : Online Talk on Zoom
Author(s) :
Danyu Yang (Chongqing University)
Abstract : We consider a remainder estimate for truncated Taylor expansion for differential equations driven by inhomogeneous geometric rough paths. The estimate can be applied to differential equations driven by general stochastic processes with a regular drift term. It can also be useful when dealing with differential equations driven by branched rough paths and quasi-geometric rough paths which are isomorphic to inhomogeneous geometric rough paths.
Abstract : We propose a new method for solving optimal stopping problems under minimal assumptions on the underlying stochastic process $X$. We consider stopping times represented by functionals of the rough path signature $\mathbb{X}^{<\infty}$, and prove that maximizing over these classes of signature stopping times solves the original optimal stopping problem. Using the algebraic properties of the signature, we can then recast the problem as a deterministic optimization problem on the expected signature.
[01439] Analysis on unparameterised path space: towards a coherent mathematical theory
Format : Talk at Waseda University
Author(s) :
Thomas Cass (Imperial College London)
William Turner (Imperial College London)
Abstract : The signature is a non-commutative exponential that appeared in the foundational work of K-T Chen in the 1950s. It is also a fundamental object in the theory of rough paths (Lyons, 1998). More recently, it has been proposed, and used, as part of a practical methodology to give a way of summarising multimodal, possibly irregularly sampled, time-ordered data in a way that is insensitive to its parameterisation. A key property underpinning this approach is the ability of linear functionals of the signature to approximate arbitrarily any compactly supported and continuous function on (unparameterised) path space. We present some new results on the properties of a selection of topologies on the space of unparameterised paths. We discuss various applications in this context. Based on joint work with Will Turner.
[01370] On some stability results in mathematical finance via rough path theory
Format : Talk at Waseda University
Author(s) :
CHONG LIU (ShanghaiTech University)
Abstract : In this talk I will present some recent progress on establishing some stability results in mathematical finance, e.g., in portfolio theory and utility maximization problems, via an approach of rough path theory.
[01438] Kernels Methods for Stochastic Processes
Format : Talk at Waseda University
Author(s) :
Harald Oberhauser (University of Oxford)
Abstract : Kernels provide a powerful approach to learning from structured data. An important case of structured data arises when there is a natural sequential order in a data point; classic examples are time series or text. I will talk about the signature kernel that allows the computation of inner products of the signature of paths after they've been lifted to paths evolving in an infinite-dimensional Hilbert space.
[01994] Optimal approximation with path signatures
Format : Talk at Waseda University
Author(s) :
Emilio Ferrucci (University of Oxford)
Abstract : Path signatures have been used extensively as a way of representing the information encoded in multimodal data streams. This choice of a feature set is motivated by the famous universality result of Hambly & Lyons, 2010 and the fact that a broad class of parametrisation-invariant functions of a stream can be arbitrarily well approximated as linear functions of its signature. However, not much is known on the quantitative aspects of such approximations, and, just like Taylor approximations of smooth functions, they can converge slowly. Moreover, just like monomials in a real variable, the signature may fail to be a basis, meaning a function on paths does not have a canonical coefficient corresponding to each coordinate iterated integral. In this talk we explore ways of addressing these issues.
[01279] Using AI to Accelerate (S)PDE Solving
Format : Online Talk on Zoom
Author(s) :
Qi Meng (Microsoft Research)
Abstract : Partial differential equations play an important role in science and engineering. Recently, AI emerged as a disruptive technique on scientific computing and could break the computational bottleneck on solving complex PDE systems via training deep neural network models. In this talk, I will introduce our recent work on AI accelerated PDE solving including DeepVortexNet and DLR-Net, which uses probabilistic representation and regularity features to achieve robust supervision signal and well-generalized neural network architecture, respectively.
[00769] Markov Chain Cubature for Bayesian Inference
Format : Online Talk on Zoom
Author(s) :
James Foster (University of Bath)
Abstract : Markov Chain Monte Carlo is widely regarded as the "go-to" approach for Bayesian inference and, due to the theory of stochastic differential equations, many physics-inspired MCMC algorithms can scale to high dimensions.
In this talk, we consider an alternative to Monte Carlo for SDE simulation known as "Cubature on Wiener Space". In particular, by applying SDE cubature and resampling particles in a spatially balanced manner, we introduce a novel interacting particle algorithm for Bayesian inference.
[01377] Iterated integrals of Gaussian fields and ill-posedness of heat equations
Author(s) :
Ilya Chevyrev (University of Edinburgh)
Abstract : In this talk, I will present a probabilistic method to show norm inflation, and thus local ill-posedness, for non-linear heat equations above scaling criticality. One of the motivations is a proof that the DeTurck-Yang-Mills heat flow is ill-posed on any Banach space that carries the 3D Gaussian free field, which complements recent well-posedness results. The method is inspired by work of Lyons, 1991, on iterated integrals of Brownian paths. Based on arXiv:2205.14350.
