A two-day CRiSM-Heilbronn-EPSRC funded workshop “New Impacts of Rough Analysis” will be held at MB2.24 in the Mathematical Sciences Building, University of Warwick on 25 and 26 July 2024.

The workshop is kindly supported by the CRISM Centre at the University of Warwick, the Heilbronn Institute Small Grant and UKRI EPSRC.

Invited Speakers

Attendance and Registration

For non-speakers, please register at the link below so that we have an idea of numbers for catering.

REGISTRATION PAGE

Registration closes on Thursday 18th July 2024

Book of abstracts

Zulit Paola Arrubarrena Tame (Imperial)

Title: Novelty Detection on Radio Astronomy Data using Signatures

Abstract: An anomaly detection methodology is presented that identifies if a given observation is unusual by deviating from a corpus of non-contaminated observations. The signature transform is applied to the streamed data as a vectorization to obtain a faithful representation in a fixed-dimensional feature space. This talk is applied to radio astronomy data to identify very faint radio frequency interference (RFI) contaminating the rest of the data.

Youness Boutaib (Liverpool)

Title: High order expansions of RDE solutions: old and new

Abstract: We revisit the (high order) Euler expansion of solutions to (geometric) RDEs (in possibly infinite dimensional spaces) as presented in the papers by Friz and Victoir (2008) then B., Gyurko, Lyons and Yang (2014). We will highlight some of its analytical and algebraic aspects, then, based on the example of the signature differential equation, inquire about the existence of alternative expansions that may have faster rates of convergence.

Emilio Ferrucci (Oxford)

Title: Branched Itô formula and natural Itô-Stratonovich isomorphism

James Foster (Bath)

Title: On the convergence of adaptive approximations for SDEs

Abstract: When using ordinary differential equations (ODEs) in practice, numerical solutions are often approximated and propagated in time via discrete step sizes. For a large variety of ODE problems, performance can be improved by making these step sizes “adaptive” – that is, adaptively changed based on the state of system. However, for stochastic differential equations (SDEs), adaptive numerical methods can be difficult to study and even fail to converge due to the rough nature of Brownian motion. In this talk, we will show that convergence does indeed occur, provided the underlying Brownian motion is discretized in an adaptive but “martingale-like” fashion. Whilst this prevents adaptive steps from skipping over time points (which is typically the case for ODEs), we believe our convergence theory is the first that is applicable to standard SDE solvers. We will discuss the key ingredients in this analysis – including martingale convergence, rough path theory and the approximation of Brownian motion by polynomials. To conclude the talk, we shall present an experiment and give some ideas for closing the gap between adaptive ODE and SDE numerics. (joint work with Andraž Jelinčič)

Ioannis Gasteratos (Imperial)

Title: Transportation-cost inequalities for nonlinear Gaussian functionals

Abstract: In this talk, we study concentration properties for laws of non-linear Gaussian functionals on metric spaces. Our focus lies on measures with non-Gaussian tail behaviour which are beyond the reach of Talagrand’s classical Transportation Cost Inequalities (TCIs). Motivated by solutions of Rough Differential Equations and relying on a suitable contraction principle, we prove generalised TCIs for functionals that arise in the theory of regularity structures and, in particular, in the cases of rough volatility and the two-dimensional Parabolic Anderson Model. Our work also extends existing results on TCIs for diffusions driven by Gaussian processes

Rasheed Ibraheem (Edinburgh)

Title: Path Signature-Based Life Prognostics of Li-ion Battery Using Pulse Test Data.

Abstract: Common models predicting the End of Life (EOL) and Remaining Useful Life (RUL) of Li-ion cells use long cycling data samples. This is a bottleneck when predictions are needed for decision-making but no historical data is available. A machine learning model to predict the EOL and RUL of Li-ion cells using only data contained in a single Hybrid Pulse Power Characterization (HPPC) test is proposed. The model ignores the cell's prior cycling usage and is validated across nine different datasets each with its cathode chemistry. A model able to classify cells by whether they have passed EOL given an HPPC test has also been developed. The underpinning data-centric modeling concept for feature generation is the notion of 'path signature' which is combined with an explainable tree-based machine learning model and an in-depth study of the models is provided. Model validation across different SOC ranges shows that data collected from the HPPC test across a 20% SOC window suffices for effective prediction. The EOL and RUL models achieve 85 and 91 cycles MAE respectively while the classification model has an accuracy of 94% on the test data. Code for data processing and modelling is publicly available athttps://github.com/Rasheed19/pulse-projectLink opens in a new window.

Darrick Lee (Oxford)

Title: The Surface Signature

Abstract: Path development is a representation of paths which preserves the underlying concatenation structure of paths, and the path signature is the universal path development map. Surfaces and images are equipped with two distinct (horizontal and vertical) concatenation structures, and the notion of surface development is a representation of surfaces which preserves this complex concatenation structure. Finally, building upon work of Kapranov, we define the surface signature as the universal surface development map.

Lei Jiang (UCL)

Title: GCN-DevLSTM: Path Development for Skeleton-Based Action Recognition

Abstract: Skeleton-based action recognition (SAR) in videos is an important but challenging task in computer vision. The recent state-of-the-art (SOTA) models for SAR are primarily based on graph convolutional neural networks (GCNs), which are powerful in extracting the spatial information of skeleton data. However, it is yet clear that such GCN-based models can effectively capture the temporal dynamics of human action sequences. To this end, we propose the G-Dev layer, which exploits the path development– a principled and parsimonious representation for sequential data by leveraging the Lie group structure. By integrating the G-Dev layer, the hybrid G-DevLSTM module enhances the traditional LSTM to reduce the time dimension while retaining high-frequency information. It can be conveniently applied to any temporal graph data, complementing existing advanced GCN-based models. Our empirical studies on the NTU60, NTU120 and Chalearn2013 datasets demonstrate that our proposed GCN-DevLSTM network consistently improves the strong GCN baseline models and achieves SOTA results with superior robustness in SAR tasks.

Maud Lemercier (Oxford)

Title: A High Order Solver for Signature Kernels

Abstract: Signature kernels are at the core of several machine learning algorithms for analysing multivariate time series. The kernels of bounded variation paths, such as piecewise linear interpolations of time series data, are typically computed by solving a linear hyperbolic second-order PDE. However, this approach becomes considerably less practical for highly oscillatory inputs, due to significant time and memory complexities.

To mitigate this issue, I will introduce a high order method which involves replacing the original PDE, which has rapidly varying coefficients, with a system of coupled equations with piecewise constant coefficients. These coefficients are derived from the first few terms of the log-signatures of the input paths and can be computed efficiently using existing Python libraries.

Sam Morley (Oxford)

Title: RoughPy: bridging the gap between mathematics and data science

Abstract: RoughPy is a Python library for working with rough paths in data science. It provides an abstraction around streams of data that allows us to work with these data in a way that is much closer to the mathematics. Our central object is a Stream, which can be queried over an interval to obtain a signature or log-signature that describes the evolution over the interval, which can then be used for analysis. RoughPy provides free tensors, shuffle tensors, and Lie objects that makes working with the signatures more natural, along with all the associated operations. This makes RoughPy an effective tool for bridging the mathematics of rough paths with data science of streamed data.

Cris Salvi (Imperial)

Title: Graph Expansions of Deep Neural Networks and their Universal Scaling Limits

Abstract: I will present a novel approach to obtain scaling limits of neural networks using the genus expansion technique from random matrix theory. This approach begins with an expansion of neural networks reminiscent of Butcher series for ODEs, where the role of monomials is played by random multilinear maps indexed by directed graphs whose edges correspond to random matrices. This expansion linearises the effect of the activation functions, allowing for the direct application of Wick's principle to compute the expectation of each of its terms. The leading contribution to each term can be determined by embedding the corresponding graphs onto surfaces, and computing their Euler characteristic. By developing a calculus bridging analytic and graphical operations, I will explain how to obtain similar graph expansions for the neural tangent kernel as well as the input-output Jacobian of the original neural network, and derive their infinite-width limits with relative ease.

Will Turner(Imperial)

Danyu Yang (Chongqing)

Title: The Lipschitz continuity of the solution to branched rough differential equations

Lingyi Yang (Oxford)

Location

All talks take place in Zeeman Building room MB.2.24 at the University of Warwick.

Information on travelling to the university can be found here. In particular, the best train station is Coventry, to which there are direct trains from Birmingham (around 20 minutes) and London Euston (around 1 hour). The closest airport is Birmingham International, though London airports have more availability for longer flights.

Schedule

Zoom meeting ID: 931 0497 4122

Passcode: 215959

Thursday 25th July 

10:30-11:00 Registration and Welcome

11:00-11:30 Zulit Paola Arrubarrena Tame (Imperial College)

11:30-12:00 Maud Lemercier (Oxford University)

12:00-13:30 Lunch in Radcliffe, by invitation only

13:30-14:00 Danyu Yang (Chongqing University)
14:00-14:30 William Turner (Imperial College)
14:30-15:00 Ioannis Gasteratos (Imperial College)

15:00-15:30 Coffee and Tea

15:30-16:00 Rasheed Ibraheem (Edinburgh University)
16:00-16:30 Cristopher Salvi (Imperial College)
16:30-17:00 James Foster (Bath University)

17:30 Workshop Dinner in Radcliffe (by invitation only)

Friday 26th July 

09:30-10:00 Sam Morley (Oxford University)

10:00-10:30 Khoa Le (Leeds University)
10:30-11:00 Coffee and Tea

11:00-11:30 Youness Boutaib (Liverpool University)
11:30-12:00 Lei Jiang (University College London)
12:00-13:30 Lunch in Radcliffe, by invitation only

13:30-14:00 Darrick Lee (Oxford University)

14:00-14:30 Emilio Rossi-Ferrucci (Oxford University)
14:30-15:00 Lingyi Yang (Oxford University)
15:00-15:30 Coffee and Tea and Departure