Recent Advances in Rough Path and Signature Theory

Date: September 15 to 19

Venue: Room 408, South Building, School of Creativity & Arts, ShanghaiTech University

This conference will focus on mathematical issues and aspects of general rough path theory and their applications in various areas such as machine learning, mathematical finance, numerical analysis and statistics.

Invited speakers: 

Christian Bayer (WIAS Berlin)

Xin Chen (Shanghai Jiao Tong University)

Samuel Crew (National Tsing Hua University)

Joscha Diehl (Greifswald University)

Xi Geng (Melbourne University)

Paul Hager (Vienna University)

Anna Kwossek (Vienna University)

Darrick Lee (Edinburgh University)

Terry Lyons (Oxford University)

Qi Meng (Chinese Academy of Sciences)

Zhongmin Qian (Oxford University)

Nikolas Tapia (WIAS Berlin)

Yue Wu (Strathclyde University)

Danyu Yang (Chongqing University)

Lingyi Yang (Oxford University)

Huilin Zhang (Shandong University)

Organizing committee: 

Siran Li (Shanghai Jiao Tong University)

Chong Liu (ShanghaiTech University)

Hao Ni (University College London)

Shi Wang (ShanghaiTech University)

Contact: Liu Chong (liuchong@shanghaitech.edu.cn)   

Program:

Global and local regression: a signature approach with applications

Christian Bayer (WIAS Berlin)

Abstract: The path signature is a powerful tool for solving regression problems on path space, i.e., for computing conditional expectations E[Y|X]\mathbb{E}[Y | X] when the random variable XX is a stochastic process -- or a time-series. We provide new theoretical convergence guarantees for two different, complementary approaches to regression using signature methods. In the context of global regression, we show that linear functionals of the robust signature are universal in the LpL^p sense in a wide class of examples. In addition, we present a local regression method based on signature semi-metrics, and show universality as well as rates of convergence. Based on joint works with Davit Gogolashvili, Luca Pelizzari, and John Schoenmakers.

Title: tbc

Xin Chen (Shanghai Jiao Tong University)

Quantum path signatures

Samuel Crew (National Tsing Hua University)

Abstract: I discuss recent work on quantum path signatures that places path signatures and associated kernels in a physical gauge-theoretic context. Specifically, I will discuss random unitary developments of smooth paths and derive governing integro-differential that generalise loop equations from random matrix theory. I will discuss a quantum circuit construction and a sparse GUE ensemble that give rise to an efficient quantum algorithm in the one clean qubit model to compute the development.

CosAttention: some non-linearity for linear-cost attention alternatives

Joscha Diehl (Greifswald University)

Abstract: Based on a trignonometric identity, we introduce CosAttention, a nonlinear, attention-like procedure with linear runtime. We show some preliminary success in the long range arena in using it to improve the accuracy of linear-cost attention alternatives (linear attention, linear SSMs). Work in progress

with Richard Krieg.

Cartan's path development, the logarithmic signature and a conjecture of Lyons-Sidorova

Xi Geng (Melbourne University)

Abstract: It is well known that the signature coefficients of a rough path decay factorially fast, hence possessing an infinite radius of convergence (R.O.C.). On the other hand, it is a highly non-trivial fact that the logarithmic signature coefficients only possess geometric decay. This was confirmed for two special classes of paths in the work of Lyons-Sidorova 2006 and they conjectured that the only BV paths whose logarithmic signature can have infinite R.O.C. are straight lines.

In this talk, we show that if the logarithmic signature of a path has infinite R.O.C., its signature coefficients must satisfy a rigid system of algebraic relations which impose strong geometric constraints on the path, and in some special situations, confirms the Lyons-Sidorova conjecture. As an application of these algebraic identities, we prove a weak version of the conjecture, which asserts that if the logarithmic signature of a BV path has infinite R.O.C. over all time intervals [s,t], the path must live on a straight line.

This is based on a recent joint work with Horatio Boedihardjo (Warwick) and my PhD student Sheng Wang.

Expected Signature Kernels of Lévy Processes

Paul Hager (Vienna University)

Abstract: The expected signature kernel arises in statistical learning tasks as a similarity measure of probability measures on path space. Computing this kernel for known classes of stochastic processes is an important problem that, in particular, can help reduce computational costs. Building on the representation of the expected signature of inhomogeneous Lévy processes as the development of a smooth path in the extended tensor algebra [F.-H.-Tapia, Forum of Mathematics: Sigma (2022), ”Unified signature cumulants and generalized Magnus expansions”], we extend the arguments developed for smooth rough paths in [Lemercier-Lyons (2024), ”A high-order solver for signature kernels”] to derive a PDE system for the expected signature of inhomogeneous Lévy processes. As a specific example, we demonstrate that the expected signature kernel of Gaussian martingales satisfies a Goursat PDE.

A rough path approach to pathwise stochastic integration à la Föllmer

Anna Kwossek (Vienna University)

Abstract: In this talk, we present a general framework for pathwise stochastic integration that extends Föllmer integration and provides pathwise analogues of Itô and Stratonovich integration. Using the concepts of quadratic variation and Lévy area of a continuous path along a sequence of partitions, we define pathwise stochastic integrals as limits of general Riemann sums and show that these coincide with suitable rough path integrals. Furthermore, we state necessary and sufficient conditions for the quadratic variation and Lévy area of a continuous path to be invariant with respect to the choice of the partition sequence. This talk is based on joint work with Purba Das and David Prömel.

The Signature of Piecewise Linear Surfaces

Darrick Lee (Edinburgh University)

Abstract: In this talk, we introduce the surface signature for piecewise linear surfaces, which satisfies a 2D Chen’s identity: it preserves horizontal and vertical concatenation of surfaces. Furthermore, we discuss the injectivity of the surface signature up to thin homotopy (analogous to tree-like equivalence of paths). Based on joint work with Francis Bischoff.

Streamed multimodal data is everywhere: applications of rough path theory

Terry Lyons (Oxford University)

Abstract: When data arrives in different channels, and the order of arrival matters, (the person arrives before or after the bus) then classical time series become a very ineffective way to record this data. Sampling needs to be frequent to capture order, but then the dimension of the feature set becomes extraordinarily high requiring huge data or ad hoc assumptions to learn patterns.

Rough path theory breaks this contradiction using group elements to represent paths over moderate intervals.

We will survey many real world applications of rough path theory to tackle practical data driven challenges.

AI for Solving SPDE

Qi Meng (Chinese Academy of Sciences)

Abstract: Stochastic Partial Differential Equations (SPDEs) driven by random noise play a central role in modelling physical processes whose spatio-temporal dynamics can be rough, such as turbulence flows, superconductors, and quantum dynamics. To efficiently model these processes and make predictions, machine learning (ML)-based surrogate models are proposed, with their network architectures incorporating the spatio-temporal roughness in their design.In this talk, I will introduce a ML-based surrogate model named DLR-Net, which is specially designed for solving SPDEs, and then, I will introduce SPDEBench, which includes typical datasets, benchmark ML models and unified evaluation for solving regular and singular SPDEs.

Conditional law duality and Feynman-Kac formula

Zhongmin Qian (Oxford University)

Abstract: There are families of probability measures on path spaces of continuous paths, the diffusion measures and their conditional laws, associated with a time-dependent elliptic operator. These infinite dimensional measures enhance the function of fundamental solutions and are very useful in the study of parabolic equations. I shall report a result on the duality of conditional laws associated with an elliptic operator of second-order, and derive new kind of Feynman-Kac formulas representing implicitly solutions of parabolic systems via diffusion measures.

Title: tbc

Nikolas Tapia (WIAS Berlin)

Randomised quadrature, stochastic sewing lemma and beyond

Yue Wu (Strathclyde University)

Abstract: The numerical error of a method used to approximate the solution of a differential equation can often be traced back to the underlying quadrature error. In settings where the driving signal is irregular in time—such as time-irregular ordinary differential equations—or where the coefficients themselves are irregular—as in stochastic differential equations—the role of quadrature becomes subtler. In these cases, randomised quadrature rules provide a natural framework, since they connect closely with the stochastic sewing lemma, which allows one to rigorously control approximations under irregularity. This framework extends beyond classical ODE and SDE solvers: the same principle can be generalized to finite element methods, where numerical errors similarly reduce to localized quadrature-type approximations, and randomised  variants can be exploited to handle irregular data or coefficients.

The Lipschitz continuity of the solution to branched rough differential equations

Danyu Yang (Chongqing University)

Abstract: Based on an isomorphism between the Grossman–Larson Hopf algebra and the tensor Hopf algebra, we apply a sub-Riemannian geometry technique to branched rough differential equations. This allows us to establish the explicit Lipschitz continuity of the solution with respect to the initial value, the vector field, and the driving rough path.

Structured Linear CDEs: trade-off between expressivity and parallelism for sequence models

Lingyi Yang (Oxford University)

Abstract: When designing the architecture of deep sequence models, we want state-transition matrices that are expressive enough to capture complex patterns while maintaining the ability to be trained at scale. In this talk, I will introduce Structured Linear Controlled Differential Equations (SLiCEs), a unifying framework that brings together existing structured approaches and introduces new ones motivated by this issue. SLiCEs show how block-diagonal, sparse, and Walsh-Hadamard transition structures can retain the full expressivity of dense models while being cheaper to compute. On benchmarks, SLiCEs solve the A5 state-tracking task with a single layer, achieve best-in-class generalisation on regular language tasks, and match state-of-the-art performance on time-series classification while cutting per-step training time by a factor of twenty.

Rough Stochastic Filtering

Huilin Zhang (Shandong University)

Abstract: In this talk, the theory of rough stochastic differential equations [Friz-Hocquet-Le '21] is applied to revisit classical problems in stochastic filtering.

We provide rough counterparts to the Kallianpur-Striebel formula and the Zakai and Kushner--Stratonovich equations, seen to coincide with classical objects upon randomization. We follow [Crisan-Pardoux '24] in doing so in a correlated diffusion setting, where classical Ito-based duality arguments break down.

Well-posedness of the (rough) filtering equation is seen to hold under dimension-independent regularity assumption, in contrast to the stochastic case.