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Organizing Committee
Imperial College London
University of Oxford
University College London and Alan Turing Institute
University of Oxford
University of Cambridge
Abstract
Rough path theory emerged as a branch of stochastic analysis to give an improved approach to dealing with the interactions of complex random systems. In that context, it continues to resolve important questions, but its broader theoretical footprint has been substantial. Most notable is its contribution to Hairer’s Fields-Medal-winning work on regularity structures. At the core of rough path theory is the so-called signature transform which, while being simple to define, has rich mathematical properties bringing in aspects of analysis, geometry, and algebra. Hambly and Lyons (Annals of Math, 2010) built upon earlier work of Chen, showing how the signature represents the path uniquely up to generalized reparameterizations. This turns out to have practical implications allowing one to summarise the space of functions on unparameterized paths and data streams in a very economical way.
Over the past five years, a significant strand of applied work has been undertaken to exploit the mathematical richness of this object in diverse data science challenges from healthcare, to computer vision to gesture recognition. The log signature is becoming a powerful way to summarise the fine structure of a data stream in a neural net. The emergence of neural differential equations as an important tool in data science further deepens the connections with rough paths.
This four-day workshop will bring together key expertise across disciplines to advance understanding of some of the most pressing and exciting challenges. The week will start with structured, tutorial-style lectures on the foundational aspects of signatures their use in data science, and topics of broad appeal. These will include:
There will also be an extended interactive practical session on computing with signatures. This session will be directed by leaders with participants working in groups with others of a similar level of previous experience. The aim of this session will be for participants to develop skills using the latest packages to implement data-focused tasks involving signatures.
The rest of the workshop will consist of technical talks and contributed talks from participants. There will be both organized discussions as well as opportunities for informal interaction.
To be considered for the practical portions of the workshop please apply by June 20, 2021.
Over the past five years, a significant strand of applied work has been undertaken to exploit the mathematical richness of this object in diverse data science challenges from healthcare, to computer vision to gesture recognition. The log signature is becoming a powerful way to summarise the fine structure of a data stream in a neural net. The emergence of neural differential equations as an important tool in data science further deepens the connections with rough paths.
This four-day workshop will bring together key expertise across disciplines to advance understanding of some of the most pressing and exciting challenges. The week will start with structured, tutorial-style lectures on the foundational aspects of signatures their use in data science, and topics of broad appeal. These will include:
- Mathematical foundations
- Neural rough differential equations
- Signature-based kernel methods
- Expected signatures
- Applications of signatures to action recognition and healthcare
There will also be an extended interactive practical session on computing with signatures. This session will be directed by leaders with participants working in groups with others of a similar level of previous experience. The aim of this session will be for participants to develop skills using the latest packages to implement data-focused tasks involving signatures.
The rest of the workshop will consist of technical talks and contributed talks from participants. There will be both organized discussions as well as opportunities for informal interaction.
To be considered for the practical portions of the workshop please apply by June 20, 2021.
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Workshop Schedule
Tuesday, July 06, 2021
8:50 - 9:00 AM EDT
Welcome
Virtual
Kavita Ramanan, Brown University
9:00 - 10:00 AM EDT
Tutorial: Mathematical foundations of the signature
Virtual
Terry Lyons, University of Oxford
Harald Oberhauser, University of Oxford
10:05 - 10:30 AM EDT
Tutorial: Signatures in the Wild
Virtual
Peter Foster, Alan Turing Institute
10:35 - 11:00 AM EDT
ML Without Unnecessary Harm: Blind Pareto Fairness and Subgroup Robustness
Virtual
Guillermo Sapiro, Duke University
Abstract
With the wide adoption of machine learning algorithms across various application domains, there is a growing interest in the fairness properties of such algorithms. The vast majority of the activity in the field of group fairness addresses disparities between predefined groups based on protected features such as gender, age, and race, which need to be available at train, and often also at test, time. These approaches are static and retrospective, since algorithms designed to protect groups identified a priori cannot anticipate and protect the needs of different at-risk groups in the future. In this work we analyze the space of solutions for worst-case fairness beyond demographics, and propose Blind Pareto Fairness (BPF), a method that leverages no-regret dynamics to recover a fair minimax classifier that reduces worst-case risk of any potential subgroup of sufficient size, and guarantees that the remaining population receives the best possible level of service. BPF addresses fairness beyond demographics, that is, it does not rely on predefined notions of at-risk groups, neither at train nor at test time. Our experimental results show that the proposed framework improves worst-case risk in multiple standard datasets, while simultaneously providing better levels of service for the remaining population, in comparison to competing methods.
11:05 - 11:30 AM EDT
Coffee Break
Virtual
11:30 AM - 12:00 PM EDT
Tutorial: Signatures in the Wild
Virtual
James Morrill, University of Oxford
12:00 - 12:30 PM EDT
Coffee Break
Virtual
12:30 - 1:00 PM EDT
Infancy Longitudinal Structural MRI Data Analysis with Path Signature Features for the Cognitive Scores Prediction
Virtual
Xin (Cindy) Zhang, South China University of Technology
Abstract
Path signature has unique advantages on extracting high order differential features of sequential data. Our team has been studying the path signature theory and actively applied it to various applications, including infant cognitive score prediction, human motion recognition, hand-written character recognition, hand-written text line recognition and writer identification etc. In this talk, I will share our most recent works on infant cognitive score prediction using learnable path signature features and simple deep learning models. The cognitive score can reveal individual’s abilities on intelligence, motion, language abilities. Recent research discovered that the cognitive ability is closely related with individual’s cortical structure and its development. We have proposed two frameworks to predict the cognitive score with different path signature features. For the first framework, we construct the temporal path signature along the age growth and extract signature features from longitudinal structural MRI data. By incorporating the cortical temporal path signature into the multi-stream deep learning model, the individual cognitive score can be predicted, even with missing data issues. For the second framework, we propose the learnable path signature algorithm to compute the developmental feature. Further, we obtain the brain region-wise development graph for the first two-year infant. Then we have employed the graph convolutional network for the score prediction. These two frameworks have been tested on two in-house cognitive data sets and reached state-of-the-art results.
1:00 - 2:30 PM EDT
Lunch/Free Time
Virtual
2:30 - 3:30 PM EDT
Practical Session 1 : Computing some examples
Virtual
Peter Foster, Alan Turing Institute
Sam Morley, University of Oxford
Wednesday, July 07, 2021
9:00 - 9:30 AM EDT
Tutorial: Log-signatures and Neural Rough Differential Equations
Virtual
James Foster, University of Oxford
9:35 - 10:05 AM EDT
Neural Stochastic Differential Equations
Virtual
Patrick Kidger, University of Oxford
10:10 - 10:40 AM EDT
Tutorial: Kernels and Signatures
Virtual
Cristopher Salvi, University of Oxford
10:40 - 11:10 AM EDT
Coffee Break
Virtual
11:10 - 11:35 AM EDT
Distribution Regression for Sequential Data
Virtual
Maud Lemercier, University of Warwick
Abstract
Distribution regression on sequential data describes the task of learning a function from a group of time series to a single scalar target. I will present a generic framework, based on the expected signature, which enables to compactly summarise a cloud of time series and make decisions on it. I will then demonstrate empirically how this framework achieves state-of-the-art performance on both synthetic and real-world examples from thermodynamics, mathematical finance and agricultural science.
11:40 AM - 3:30 PM EDT
Lunch/Free Time
Virtual
3:30 - 5:00 PM EDT
Practical Session: Computing some examples
Virtual
Peter Foster, Alan Turing Institute
Sam Morley, University of Oxford
Thursday, July 08, 2021
9:00 - 9:30 AM EDT
Tutorial: Generative models and Signature-Based Machine Learning Models
Virtual
Hao Ni, University College London and Alan Turing Institute
9:35 - 10:05 AM EDT
Tutorial: Path recovery from signature feature representation
Virtual
Weixin Yang, The Alan Turing Institute
Abstract
To recover the underlying paths from a given signature representation can not only give confidence in using signature as features in machine learning tasks but also be useful for data augmentation or key-frame extraction. The "Signatory" python package allows the signature transformation to act as a layer in a trainable neural network. Inspired by it, we tried to recover different types of underlying paths from a given signature-based representation. By visualizing the results, we discussed the effects of different hyper-parameters for our signature feature set.
10:10 - 10:40 AM EDT
Tutorial: Gaussian Processes
Virtual
Csaba Toth, University of Oxford
10:45 - 11:15 AM EDT
Coffee Break
Virtual
11:15 - 11:45 AM EDT
Framing RNN as a kernel method: A neural ODE approach
Virtual
Adeline Fermanian, LPSM
Abstract
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 a specific feature set of the input sequence, known as 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. Our results are illustrated on simulated datasets.
11:45 AM - 12:10 PM EDT
Signature kernels and expected signatures
Virtual
Thomas Cass, Imperial College London
12:10 - 1:30 PM EDT
Lunch/Free Time
Virtual
1:40 - 2:05 PM EDT
Machine Learning for PDEs
Virtual
Michael Brenner, Harvard University
Abstract
I will discuss methods for using machine learning to speed up solutions of nonlinear partial differential equations, focusing on learning discretizations for coarse graining the numerical solutions of PDEs. I will start with examples in 1d, and then move on to the Navier Stokes equation.
2:05 - 2:30 PM EDT
Data -Driven Market Simulators some simple applications of signature kernel methods in mathematical finance
Virtual
Blanka Horvath, King's College London
Abstract
Techniques that address sequential data have been a central theme in machine learning research in the past years. More recently, such considerations have entered the field of finance-related ML applications in several areas where we face inherently path dependent problems: from (deep) pricing and hedging (of path-dependent options) to generative modeling of synthetic market data, which we refer to as market generation.
We revisit Deep Hedging from the perspective of the role of the data streams used for training and highlight how this perspective motivates the use of highly accurate generative models for synthetic data generation. From this, we draw conclusions regarding the implications for risk management and model governance of these applications, in contrast torisk-management in classical quantitative finance approaches.
Indeed, financial ML applications and their risk-management heavily rely on a solid means of measuring and efficiently computing (smilarity-)metrics between datasets consisting of sample paths of stochastic processes. Stochastic processes are at their core random variables with values on path space. However, while the distance between two (finite dimensional) distributions was historically well understood, the extension of this notion to the level of stochastic processes remained a challenge until recently. We discuss the effect of different choices of such metrics while revisiting some topics that are central to ML-augmented quantitative finance applications (such as the synthetic generation and the evaluation of similarity of data streams) from a regulatory (and model governance) perpective. Finally, we discuss the effect of considering refined metrics which respect and preserve the information structure (the filtration) of the marketand the implications and relevance of such metrics on financial results.
We revisit Deep Hedging from the perspective of the role of the data streams used for training and highlight how this perspective motivates the use of highly accurate generative models for synthetic data generation. From this, we draw conclusions regarding the implications for risk management and model governance of these applications, in contrast torisk-management in classical quantitative finance approaches.
Indeed, financial ML applications and their risk-management heavily rely on a solid means of measuring and efficiently computing (smilarity-)metrics between datasets consisting of sample paths of stochastic processes. Stochastic processes are at their core random variables with values on path space. However, while the distance between two (finite dimensional) distributions was historically well understood, the extension of this notion to the level of stochastic processes remained a challenge until recently. We discuss the effect of different choices of such metrics while revisiting some topics that are central to ML-augmented quantitative finance applications (such as the synthetic generation and the evaluation of similarity of data streams) from a regulatory (and model governance) perpective. Finally, we discuss the effect of considering refined metrics which respect and preserve the information structure (the filtration) of the marketand the implications and relevance of such metrics on financial results.
3:35 - 4:15 PM EDT
Coffee Break
Virtual
4:15 - 4:45 PM EDT
Tutorial: Action recognition from landmark data
Virtual
Kevin Schlegel, The Alan Turing Institute
4:50 - 5:20 PM EDT
Tutorial: Two transforms for signature features
Virtual
Yue Wu, University of Oxford
Abstract
In this tutorial, I will introduce two transforms that work with signature transforms, one is designed to handle missing data, and the other one is designed to embed the effect of the absolute position of the data stream into signature features in a unified and efficient way.
Friday, July 09, 2021
9:00 - 9:25 AM EDT
Signatures, tensor decompositions, and nonlinear algebra
Virtual
Anna Seigal, University of Oxford
Abstract
I will begin by discussing tensor rank, and the equations that arise when decomposing tensors into rank one terms. I will then consider decompositions of signature tensors, and their systems of equations. Along the way, I will mention joint work with Max Pfeffer and Bernd Sturmfels, and with Terry Lyons and Cris Salvi.
9:30 - 9:55 AM EDT
espilon-Strong Simulation of Stochastic Differential Equations Driven by Levy Processes
Virtual
Jing Dong, Columbia Business School
Abstract
Consider a stochastic differential equation dY(t)=f(X(t))dX(t), where X(t) is a pure jump Levy process with finite p-variation, 1<= p < 2, and f is alpha-Lipschitz for some alpha>p. Following the geometric solution construction of Levy-driven stochastic differential equations, we develop a class of epsilon-strong simulation algorithms that allows us to construct a probability space, supporting both Y and a fully simulatable process Y_epsilon, such that Y_epsilon is within epsilon distance from Y under the Skorokhod J1 topology on compact time intervals with probability 1. Moreover, the user can adaptively refine the accuracy levels. This tolerance-enforcement feature allows us to easily combine our algorithm with multilevel Monte Carlo method for efficient estimation of expectations, and adding as a benefit a straightforward analysis of the rate of convergence.
10:05 - 10:30 AM EDT
Stochastic gradient descent for noise with ML-type scaling
Virtual
Stephan Wojtowytsch, Princeton University
Abstract
There are two types of convergence results for stochastic gradient descent: (1) SGD finds minimizers of convex objective functions and (2) SGD finds critical points of smooth objective functions. We show that, if the objective landscape and noise possess certain properties which are reminiscent of deep learning problems, then we can obtain global convergence guarantees of first type under second type assumptions for a fixed (small, but positive) learning rate. The convergence is exponential, but with a large random coefficient. If the learning rate exceeds a certain threshold, we discuss minimum selection by studying the invariant distribution of a continuous time SGD model. We show that at a critical threshold, SGD prefers minimizers where the objective function is 'flat' in a precise sense.
10:35 AM - 12:30 PM EDT
Lunch/Free Time
Virtual
12:30 - 1:15 PM EDT
Discussion of Challenges
Virtual
1:15 - 1:25 PM EDT
Coffee Break
Virtual
1:25 - 1:40 PM EDT
Open Session
Virtual
1:40 - 2:35 PM EDT
Coffee Break
Virtual
2:35 - 3:00 PM EDT
Controlled Rough Paths Revisited
Virtual
Bruce Driver, University of California, San Diego
Abstract
In this talk, I will discuss some of the details needed for the theory of controlled rough paths in the infinite dimensional Banach space setting. A key point will be that one may use truncated signatures of smooth curves as ``generating functions'' of the algebraic identities needed to make the theory work. This is a report on work in progress.
3:00 - 3:25 PM EDT
Exact Sampling of Stochastic Differential Equations
Virtual
Jose Blanchet, Columbia University and Stanford University
Abstract
We consider the problem of generating exact samples at finitely many locations from the solution of a generic multidimensional stochastic differential equation (SDE) driven by Brownian motion at a given time. If the SDE can be transformed into one with a constant diffusion coefficient and gradient drift exact samples can be obtained by sequential acceptance / rejection. In general, in this talk, we will explain how to use the theory of rough paths to obtain such exact samples. This is the first generic algorithm for exact samples of generic multivariate diffusions.
3:30 - 4:00 PM EDT
Closing Remarks
Virtual