Terry Lyons

Terry Lyons, Andrew D. McLeod · oxford

Signature-based techniques give mathematical insight into the interactions between complex streams of evolving data. These insights can be quite naturally translated into numerical approaches to understanding streamed data, and perhaps because of their mathematical precision, have proved useful in analysing streamed data in situations where the data is irregular, and not stationary, and the dimension of the data and the sample sizes are both moderate. Understanding streamed multi-modal data is exponential: a word in $n$ letters from an alphabet of size $d$ can be any one of $d^n$ messages. Signatures remove the exponential amount of noise that arises from sampling irregularity, but an exponential amount of information still remain. This survey aims to stay in the domain where that exponential scaling can be managed directly. Scalability issues are an important challenge in many problems but would require another survey article and further ideas. This survey describes a range of contexts where the data sets are small enough to remove the possibility of massive machine learning, and the existence of small sets of context free and principled features can be used effectively. The mathematical nature of the tools can make their use intimidating to non-mathematicians. The examples presented in this article are intended to bridge this communication gap and provide tractable working examples drawn from the machine learning context. Notebooks are available online for several of these examples. This survey builds on the earlier paper of Ilya Chevryev and Andrey Kormilitzin which had broadly similar aims at an earlier point in the development of this machinery. This article illustrates how the theoretical insights offered by signatures are simply realised in the analysis of application data in a way that is largely agnostic to the data type.

Adeline Fermanian, Terry Lyons, James Morrill et al. · oxford

This article provides a concise overview of some of the recent advances in the application of rough path theory to machine learning. Controlled differential equations (CDEs) are discussed as the key mathematical model to describe the interaction of a stream with a physical control system. A collection of iterated integrals known as the signature naturally arises in the description of the response produced by such interactions. The signature comes equipped with a variety of powerful properties rendering it an ideal feature map for streamed data. We summarise recent advances in the symbiosis between deep learning and CDEs, studying the link with RNNs and culminating with the Neural CDE model. We concluded with a discussion on signature kernel methods.

Satoshi Hayakawa, Harald Oberhauser, Terry Lyons · oxford

We analyze the Nyström approximation of a positive definite kernel associated with a probability measure. We first prove an improved error bound for the conventional Nyström approximation with i.i.d. sampling and singular-value decomposition in the continuous regime; the proof techniques are borrowed from statistical learning theory. We further introduce a refined selection of subspaces in Nyström approximation with theoretical guarantees that is applicable to non-i.i.d. landmark points. Finally, we discuss their application to convex kernel quadrature and give novel theoretical guarantees as well as numerical observations.

Mohamed R. Ibrahim, Terry Lyons · oxford

This paper introduces a new lightweight method for image recognition. ImageSig is based on computing signatures and does not require a convolutional structure or an attention-based encoder. It is striking to the authors that it achieves: a) an accuracy for 64 X 64 RGB images that exceeds many of the state-of-the-art methods and simultaneously b) requires orders of magnitude less FLOPS, power and memory footprint. The pretrained model can be as small as 44.2 KB in size. ImageSig shows unprecedented performance on hardware such as Raspberry Pi and Jetson-nano. ImageSig treats images as streams with multiple channels. These streams are parameterized by spatial directions. We contribute to the functionality of signature and rough path theory to stream-like data and vision tasks on static images beyond temporal streams. With very few parameters and small size models, the key advantage is that one could have many of these "detectors" assembled on the same chip; moreover, the feature acquisition can be performed once and shared between different models of different tasks - further accelerating the process. This contributes to energy efficiency and the advancements of embedded AI at the edge.

Tiexin Qin, Benjamin Walker, Terry Lyons et al. · oxford

This paper focuses on representation learning for dynamic graphs with temporal interactions. A fundamental issue is that both the graph structure and the nodes own their own dynamics, and their blending induces intractable complexity in the temporal evolution over graphs. Drawing inspiration from the recent progress of physical dynamic models in deep neural networks, we propose Graph Neural Controlled Differential Equations (GN-CDEs), a continuous-time framework that jointly models node embeddings and structural dynamics by incorporating a graph enhanced neural network vector field with a time-varying graph path as the control signal. Our framework exhibits several desirable characteristics, including the ability to express dynamics on evolving graphs without piecewise integration, the capability to calibrate trajectories with subsequent data, and robustness to missing observations. Empirical evaluation on a range of dynamic graph representation learning tasks demonstrates the effectiveness of our proposed approach in capturing the complex dynamics of dynamic graphs.

Mohamed R. Ibrahim, Terry Lyons · oxford

Through our respiratory system, many viruses and diseases frequently spread and pass from one person to another. Covid-19 served as an example of how crucial it is to track down and cut back on contacts to stop its spread. There is a clear gap in finding automatic methods that can detect hand-to-face contact in complex urban scenes or indoors. In this paper, we introduce a computer vision framework, called FaceTouch, based on deep learning. It comprises deep sub-models to detect humans and analyse their actions. FaceTouch seeks to detect hand-to-face touches in the wild, such as through video chats, bus footage, or CCTV feeds. Despite partial occlusion of faces, the introduced system learns to detect face touches from the RGB representation of a given scene by utilising the representation of the body gestures such as arm movement. This has been demonstrated to be useful in complex urban scenarios beyond simply identifying hand movement and its closeness to faces. Relying on Supervised Contrastive Learning, the introduced model is trained on our collected dataset, given the absence of other benchmark datasets. The framework shows a strong validation in unseen datasets which opens the door for potential deployment.

Elena Gal, Shaun Singh, Aldo Pacchiano et al. · oxford

In many real world settings binary classification decisions are made based on limited data in near real-time, e.g. when assessing a loan application. We focus on a class of these problems that share a common feature: the true label is only observed when a data point is assigned a positive label by the principal, e.g. we only find out whether an applicant defaults if we accepted their loan application. As a consequence, the false rejections become self-reinforcing and cause the labelled training set, that is being continuously updated by the model decisions, to accumulate bias. Prior work mitigates this effect by injecting optimism into the model, however this comes at the cost of increased false acceptance rate. We introduce adversarial optimism (AdOpt) to directly address bias in the training set using adversarial domain adaptation. The goal of AdOpt is to learn an unbiased but informative representation of past data, by reducing the distributional shift between the set of accepted data points and all data points seen thus far. AdOpt significantly exceeds state-of-the-art performance on a set of challenging benchmark problems. Our experiments also provide initial evidence that the introduction of adversarial domain adaptation improves fairness in this setting.

Zehong Zhang, Fei Lu, Esther Xu Fei et al. · oxford

Statistical optimality benchmarking is crucial for analyzing and designing time series classification (TSC) algorithms. This study proposes to benchmark the optimality of TSC algorithms in distinguishing diffusion processes by the likelihood ratio test (LRT). The LRT is an optimal classifier by the Neyman-Pearson lemma. The LRT benchmarks are computationally efficient because the LRT does not need training, and the diffusion processes can be efficiently simulated and are flexible to reflect the specific features of real-world applications. We demonstrate the benchmarking with three widely-used TSC algorithms: random forest, ResNet, and ROCKET. These algorithms can achieve the LRT optimality for univariate time series and multivariate Gaussian processes. However, these model-agnostic algorithms are suboptimal in classifying high-dimensional nonlinear multivariate time series. Additionally, the LRT benchmark provides tools to analyze the dependence of classification accuracy on the time length, dimension, temporal sampling frequency, and randomness of the time series.

Benjamin Walker, Alexandre Bloch, Lingyi Yang et al.

Continuous-time models are a natural choice for irregular and asynchronous data. A central design choice is how to embed discrete observations into continuous time. Interpolation- and imputation-based embeddings reconstruct a continuous observation path, making the model sensitive to the choice of reconstruction. We show that this reconstruction step is unnecessary; under mild conditions, compact-set universality on the model input space transfers to the data space whenever the embedding from data to input is continuous and injective. Guided by this result, and building on the rectilinear control path for Neural Controlled Differential Equations (NCDEs), we introduce a continuous and injective embedding for Log-NCDEs, a universal class of continuous-time models. Our approach records observations as increments and composes them over arbitrary query intervals to directly form log-signatures. This provides interval-level summaries without first interpolating the observed variables, while supporting online computation. Experiments on synthetic controlled dynamics and real-world time-series datasets show that the representation is accurate, efficient, and robust to irregular, asynchronous, and sparse observations.

Benjamin Walker, Terry Lyons

World models require state tracking, which is the ability to maintain a correct latent state across action sequences. Existing benchmarks are often synthetic or language-based, limiting their value as tests of structured state updates in realistic domains. We introduce Chess-World-Model, a large-scale state-tracking benchmark built from 10 million real chess games, where models predict the exact board state reached after a sequence of legal moves. Alongside a held-out real-game split, we include an out-of-distribution split from uniformly random legal play, which tests whether models learn the transition rules rather than shortcuts from common human positions. Prior theoretical and empirical work has shown that Transformers struggle to state-track, while input-dependent linear RNNs require expressive state-transition matrices to do so. We therefore benchmark a causal Transformer, block-diagonal SLiCE, Mamba-3, and Gated DeltaNet with negative eigenvalues under a matched interface and training protocol. The recurrent models strongly outperform the Transformer at 3 and 8 million parameters. Real-game performance saturates above 18 million parameters, but the random-uniform split remains discriminative up to 40 million, exposing failures otherwise hidden by scale. Additionally, ablations show that less expressive state-transition mechanisms reduce performance on the out-of-distribution split for all three recurrent models. Together, these results establish Chess-World-Model as a practical large-scale benchmark for state tracking that exposes failures model scale would otherwise conceal.

Ioannis Gasteratos, Antoine Jacquier, Maud Lemercier et al.

We frame novelty detection on path space as a hypothesis testing problem with signature-based test statistics. Using transportation-cost inequalities of Gasteratos and Jacquier (2023), we obtain tail bounds for false positive rates that extend beyond Gaussian measures to laws of RDE solutions with smooth bounded vector fields, yielding estimates of quantiles and p-values. Exploiting the shuffle product, we derive exact formulae for smooth surrogates of conditional value-at-risk (CVaR) in terms of expected signatures, leading to new one-class SVM algorithms optimising smooth CVaR objectives. We then establish lower bounds on type-$\mathrm{II}$ error for alternatives with finite first moment, giving general power bounds when the reference measure and the alternative are absolutely continuous with respect to each other. Finally, we evaluate numerically the type-$\mathrm{I}$ error and statistical power of signature-based test statistic, using synthetic anomalous diffusion data and real-world molecular biology data.

Talia Tseriotou, Ryan Sze-Yin Chan, Adam Tsakalidis et al.

We present an open-source, pip installable toolkit, Sig-Networks, the first of its kind for longitudinal language modelling. A central focus is the incorporation of Signature-based Neural Network models, which have recently shown success in temporal tasks. We apply and extend published research providing a full suite of signature-based models. Their components can be used as PyTorch building blocks in future architectures. Sig-Networks enables task-agnostic dataset plug-in, seamless pre-processing for sequential data, parameter flexibility, automated tuning across a range of models. We examine signature networks under three different NLP tasks of varying temporal granularity: counselling conversations, rumour stance switch and mood changes in social media threads, showing SOTA performance in all three, and provide guidance for future tasks. We release the Toolkit as a PyTorch package with an introductory video, Git repositories for preprocessing and modelling including sample notebooks on the modeled NLP tasks.

Alexandre Bloch, Samuel N. Cohen, Terry Lyons et al.

The signature is a canonical representation of a multidimensional path over an interval. However, it treats all historical information uniformly, offering no intrinsic mechanism for contextualising the relevance of the past. To address this, we introduce the Exponentially Weighted Signature (EWS), generalising the Exponentially Fading Memory (EFM) signature from diagonal to general bounded linear operators. These operators enable cross-channel coupling at the level of temporal weighting together with richer memory dynamics including oscillatory, growth, and regime-dependent behaviour, while preserving the algebraic strengths of the classical signature. We show that the EWS is the unique solution to a linear controlled differential equation on the tensor algebra, and that it generalises both state-space models and the Laplace and Fourier transforms of the path. The group-like structure of the EWS enables efficient computation and makes the framework amenable to gradient-based learning, with the full semigroup action parametrised by and learned through its generator. We use this framework to empirically demonstrate the expressivity gap between the EWS and both the signature and EFM on two SDE-based regression tasks.

Raeid Saqur, Christoph Bergmeir, Blanka Horvath et al.

We argue that the current practice of evaluating AI/ML time-series forecasting models, predominantly on benchmarks characterized by strong, persistent periodicities and seasonalities, obscures real progress by overlooking the performance of efficient classical methods. We demonstrate that these "standard" datasets often exhibit dominant autocorrelation patterns and seasonal cycles that can be effectively captured by simpler linear or statistical models, rendering complex deep learning architectures frequently no more performant than their classical counterparts for these specific data characteristics, and raising questions as to whether any marginal improvements justify the significant increase in computational overhead and model complexity. We call on the community to (I) retire or substantially augment current benchmarks with datasets exhibiting a wider spectrum of non-stationarities, such as structural breaks, time-varying volatility, and concept drift, and less predictable dynamics drawn from diverse real-world domains, and (II) require every deep learning submission to include robust classical and simple baselines, appropriately chosen for the specific characteristics of the downstream tasks' time series. By doing so, we will help ensure that reported gains reflect genuine scientific methodological advances rather than artifacts of benchmark selection favoring models adept at learning repetitive patterns.

Shujian Liao, Terry Lyons, Weixin Yang et al.

This paper contributes to the challenge of skeleton-based human action recognition in videos. The key step is to develop a generic network architecture to extract discriminative features for the spatio-temporal skeleton data. In this paper, we propose a novel module, namely Logsig-RNN, which is the combination of the log-signature layer and recurrent type neural networks (RNNs). The former one comes from the mathematically principled technology of signatures and log-signatures as representations for streamed data, which can manage high sample rate streams, non-uniform sampling and time series of variable length. It serves as an enhancement of the recurrent layer, which can be conveniently plugged into neural networks. Besides we propose two path transformation layers to significantly reduce path dimension while retaining the essential information fed into the Logsig-RNN module. Finally, numerical results demonstrate that replacing the RNN module by the Logsig-RNN module in SOTA networks consistently improves the performance on both Chalearn gesture data and NTU RGB+D 120 action data in terms of accuracy and robustness. In particular, we achieve the state-of-the-art accuracy on Chalearn2013 gesture data by combining simple path transformation layers with the Logsig-RNN. Codes are available at https://github.com/steveliao93/GCN_LogsigRNN.

Cristopher Salvi, Thomas Cass, James Foster et al.

Recently, there has been an increased interest in the development of kernel methods for learning with sequential data. The signature kernel is a learning tool with potential to handle irregularly sampled, multivariate time series. In "Kernels for sequentially ordered data" the authors introduced a kernel trick for the truncated version of this kernel avoiding the exponential complexity that would have been involved in a direct computation. Here we show that for continuously differentiable paths, the signature kernel solves a hyperbolic PDE and recognize the connection with a well known class of differential equations known in the literature as Goursat problems. This Goursat PDE only depends on the increments of the input sequences, does not require the explicit computation of signatures and can be solved efficiently using state-of-the-arthyperbolic PDE numerical solvers, giving a kernel trick for the untruncated signature kernel, with the same raw complexity as the method from "Kernels for sequentially ordered data", but with the advantage that the PDE numerical scheme is well suited for GPU parallelization, which effectively reduces the complexity by a full order of magnitude in the length of the input sequences. In addition, we extend the previous analysis to the space of geometric rough paths and establish, using classical results from rough path theory, that the rough version of the signature kernel solves a rough integral equation analogous to the aforementioned Goursat PDE. Finally, we empirically demonstrate the effectiveness of our PDE kernel as a machine learning tool in various machine learning applications dealing with sequential data. We release the library sigkernel publicly available at https://github.com/crispitagorico/sigkernel.

Zhen Shao, Ryan Sze-Yin Chan, Thomas Cochrane et al.

In this paper, we propose a dimensionless anomaly detection method for multivariate streams. Our method is independent of the unit of measurement for the different stream channels, therefore dimensionless. We first propose the variance norm, a generalisation of Mahalanobis distance to handle infinite-dimensional feature space and singular empirical covariance matrix rigorously. We then combine the variance norm with the path signature, an infinite collection of iterated integrals that provide global features of streams, to propose SigMahaKNN, a method for anomaly detection on (multivariate) streams. We show that SigMahaKNN is invariant to stream reparametrisation, stream concatenation and has a graded discrimination power depending on the truncation level of the path signature. We implement SigMahaKNN as an open-source software, and perform extensive numerical experiments, showing significantly improved anomaly detection on streams compared to isolation forest and local outlier factors in applications ranging from language analysis, hand-writing analysis, ship movement paths analysis and univariate time-series analysis.

James Morrill, Adeline Fermanian, Patrick Kidger et al.

The 'signature method' refers to a collection of feature extraction techniques for multivariate time series, derived from the theory of controlled differential equations. There is a great deal of flexibility as to how this method can be applied. On the one hand, this flexibility allows the method to be tailored to specific problems, but on the other hand, can make precise application challenging. This paper makes two contributions. First, the variations on the signature method are unified into a general approach, the \emph{generalised signature method}, of which previous variations are special cases. A primary aim of this unifying framework is to make the signature method more accessible to any machine learning practitioner, whereas it is now mostly used by specialists. Second, and within this framework, we derive a canonical collection of choices that provide a domain-agnostic starting point. We derive these choices as a result of an extensive empirical study on 26 datasets and go on to show competitive performance against current benchmarks for multivariate time series classification. Finally, to ease practical application, we make our techniques available as part of the open-source [redacted] project.

Patrick Kidger, Terry Lyons

Signatory is a library for calculating and performing functionality related to the signature and logsignature transforms. The focus is on machine learning, and as such includes features such as CPU parallelism, GPU support, and backpropagation. To our knowledge it is the first GPU-capable library for these operations. Signatory implements new features not available in previous libraries, such as efficient precomputation strategies. Furthermore, several novel algorithmic improvements are introduced, producing substantial real-world speedups even on the CPU without parallelism. The library operates as a Python wrapper around C++, and is compatible with the PyTorch ecosystem. It may be installed directly via \texttt{pip}. Source code, documentation, examples, benchmarks and tests may be found at \texttt{\url{https://github.com/patrick-kidger/signatory}}. The license is Apache-2.0.

Patric Bonnier, Patrick Kidger, Imanol Perez Arribas et al.

The signature is an infinite graded sequence of statistics known to characterise a stream of data up to a negligible equivalence class. It is a transform which has previously been treated as a fixed feature transformation, on top of which a model may be built. We propose a novel approach which combines the advantages of the signature transform with modern deep learning frameworks. By learning an augmentation of the stream prior to the signature transform, the terms of the signature may be selected in a data-dependent way. More generally, we describe how the signature transform may be used as a layer anywhere within a neural network. In this context it may be interpreted as a pooling operation. We present the results of empirical experiments to back up the theoretical justification. Code available at https://github.com/patrick-kidger/Deep-Signature-Transforms.

Nicola Muca Cirone, Antonio Orvieto, Benjamin Walker et al. · oxford

Structured state-space models (SSMs) such as S4, stemming from the seminal work of Gu et al., are gaining popularity as effective approaches for modeling sequential data. Deep SSMs demonstrate outstanding performance across a diverse set of domains, at a reduced training and inference cost compared to attention-based transformers. Recent developments show that if the linear recurrence powering SSMs allows for multiplicative interactions between inputs and hidden states (e.g. GateLoop, Mamba, GLA), then the resulting architecture can surpass in both in accuracy and efficiency attention-powered foundation models trained on text, at scales of billion parameters. In this paper, we give theoretical grounding to this recent finding using tools from Rough Path Theory: we show that when random linear recurrences are equipped with simple input-controlled transitions (selectivity mechanism), then the hidden state is provably a low-dimensional projection of a powerful mathematical object called the signature of the input -- capturing non-linear interactions between tokens at distinct timescales. Our theory not only motivates the success of modern selective state-space models such as Mamba but also provides a solid framework to understand the expressive power of future SSM variants.

Benjamin Walker, Andrew D. McLeod, Tiexin Qin et al. · oxford

The vector field of a controlled differential equation (CDE) describes the relationship between a control path and the evolution of a solution path. Neural CDEs (NCDEs) treat time series data as observations from a control path, parameterise a CDE's vector field using a neural network, and use the solution path as a continuously evolving hidden state. As their formulation makes them robust to irregular sampling rates, NCDEs are a powerful approach for modelling real-world data. Building on neural rough differential equations (NRDEs), we introduce Log-NCDEs, a novel, effective, and efficient method for training NCDEs. The core component of Log-NCDEs is the Log-ODE method, a tool from the study of rough paths for approximating a CDE's solution. Log-NCDEs are shown to outperform NCDEs, NRDEs, the linear recurrent unit, S5, and MAMBA on a range of multivariate time series datasets with up to $50{,}000$ observations.

Paola Arrubarrena, Maud Lemercier, Bojan Nikolic et al. · oxford

We introduce SigNova, a new semi-supervised framework for detecting anomalies in streamed data. While our initial examples focus on detecting radio-frequency interference (RFI) in digitized signals within the field of radio astronomy, it is important to note that SigNova's applicability extends to any type of streamed data. The framework comprises three primary components. Firstly, we use the signature transform to extract a canonical collection of summary statistics from observational sequences. This allows us to represent variable-length visibility samples as finite-dimensional feature vectors. Secondly, each feature vector is assigned a novelty score, calculated as the Mahalanobis distance to its nearest neighbor in an RFI-free training set. By thresholding these scores we identify observation ranges that deviate from the expected behavior of RFI-free visibility samples without relying on stringent distributional assumptions. Thirdly, we integrate this anomaly detector with Pysegments, a segmentation algorithm, to localize consecutive observations contaminated with RFI, if any. This approach provides a compelling alternative to classical windowing techniques commonly used for RFI detection. Importantly, the complexity of our algorithm depends on the RFI pattern rather than on the size of the observation window. We demonstrate how SigNova improves the detection of various types of RFI (e.g., broadband and narrowband) in time-frequency visibility data. We validate our framework on the Murchison Widefield Array (MWA) telescope and simulated data and the Hydrogen Epoch of Reionization Array (HERA).

Maud Lemercier, Terry Lyons, Cristopher Salvi · oxford

Signature kernels, inner products of path signatures, underpin several machine learning algorithms for multivariate time series analysis. For bounded variation paths, signature kernels were recently shown to solve a Goursat PDE. However, existing PDE solvers only use increments as input data, leading to first order approximation errors. These approaches become computationally intractable 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. In this paper, we extend the analysis to rough paths, and show, leveraging the framework of smooth rough paths, that the resulting rough signature kernels can be approximated by a novel system of PDEs whose coefficients involve higher order iterated integrals of the input rough paths. We show that this system of PDEs admits a unique solution and establish quantitative error bounds yielding a higher order approximation to rough signature kernels.

Benjamin Walker, Lingyi Yang, Nicola Muca Cirone et al.

This work introduces Structured Linear Controlled Differential Equations (SLiCEs), a unifying framework for sequence models with structured, input-dependent state-transition matrices that retain the maximal expressivity of dense matrices whilst being cheaper to compute. The framework encompasses existing architectures, such as input-dependent block-diagonal linear recurrent neural networks and DeltaNet's diagonal-plus-low-rank structure, as well as two novel variants based on sparsity and the Walsh-Hadamard transform. We prove that, unlike the diagonal state-transition matrices of S4D and Mamba, SLiCEs employing block-diagonal, sparse, or Walsh-Hadamard matrices match the maximal expressivity of dense matrices. Empirically, SLiCEs solve the $A_5$ state-tracking benchmark with a single layer, achieve best-in-class length generalisation on regular language tasks among parallel-in-time models, and match the performance of log neural controlled differential equations on six multivariate time-series classification datasets while cutting the average time per training step by a factor of twenty.

Felix Krones, Ben Walker, Terry Lyons et al.

This work presents our team's (SignalSavants) winning contribution to the 2024 George B. Moody PhysioNet Challenge. The Challenge had two goals: reconstruct ECG signals from printouts and classify them for cardiac diseases. Our focus was the first task. Despite many ECGs being digitally recorded today, paper ECGs remain common throughout the world. Digitising them could help build more diverse datasets and enable automated analyses. However, the presence of varying recording standards and poor image quality requires a data-centric approach for developing robust models that can generalise effectively. Our approach combines the creation of a diverse training set, Hough transform to rotate images, a U-Net based segmentation model to identify individual signals, and mask vectorisation to reconstruct the signals. We assessed the performance of our models using the 10-fold stratified cross-validation (CV) split of 21,799 recordings proposed by the PTB-XL dataset. On the digitisation task, our model achieved an average CV signal-to-noise ratio of 17.02 and an official Challenge score of 12.15 on the hidden set, securing first place in the competition. Our study shows the challenges of building robust, generalisable, digitisation approaches. Such models require large amounts of resources (data, time, and computational power) but have great potential in diversifying the data available.

Felix H. Krones, Ben Walker, Guy Parsons et al.

This work showcases our team's (The BEEGees) contributions to the 2023 George B. Moody PhysioNet Challenge. The aim was to predict neurological recovery from coma following cardiac arrest using clinical data and time-series such as multi-channel EEG and ECG signals. Our modelling approach is multimodal, based on two-dimensional spectrogram representations derived from numerous EEG channels, alongside the integration of clinical data and features extracted directly from EEG recordings. Our submitted model achieved a Challenge score of $0.53$ on the hidden test set for predictions made $72$ hours after return of spontaneous circulation. Our study shows the efficacy and limitations of employing transfer learning in medical classification. With regard to prospective implementation, our analysis reveals that the performance of the model is strongly linked to the selection of a decision threshold and exhibits strong variability across data splits.

Mohamed R. Ibrahim, Terry Lyons

Air pollution in cities, especially NO\textsubscript{2}, is linked to numerous health problems, ranging from mortality to mental health challenges and attention deficits in children. While cities globally have initiated policies to curtail emissions, real-time monitoring remains challenging due to limited environmental sensors and their inconsistent distribution. This gap hinders the creation of adaptive urban policies that respond to the sequence of events and daily activities affecting pollution in cities. Here, we demonstrate how city CCTV cameras can act as a pseudo-NO\textsubscript{2} sensors. Using a predictive graph deep model, we utilised traffic flow from London's cameras in addition to environmental and spatial factors, generating NO\textsubscript{2} predictions from over 133 million frames. Our analysis of London's mobility patterns unveiled critical spatiotemporal connections, showing how specific traffic patterns affect NO\textsubscript{2} levels, sometimes with temporal lags of up to 6 hours. For instance, if trucks only drive at night, their effects on NO\textsubscript{2} levels are most likely to be seen in the morning when people commute. These findings cast doubt on the efficacy of some of the urban policies currently being implemented to reduce pollution. By leveraging existing camera infrastructure and our introduced methods, city planners and policymakers could cost-effectively monitor and mitigate the impact of NO\textsubscript{2} and other pollutants.

Benjamin Walker, Felix Krones, Ivan Kiskin et al.

This study presents our team PathToMyHeart's contribution to the George B. Moody PhysioNet Challenge 2022. Two models are implemented. The first model is a Dual Bayesian ResNet (DBRes), where each patient's recording is segmented into overlapping log mel spectrograms. These undergo two binary classifications: present versus unknown or absent, and unknown versus present or absent. The classifications are aggregated to give a patient's final classification. The second model is the output of DBRes integrated with demographic data and signal features using XGBoost.DBRes achieved our best weighted accuracy of $0.771$ on the hidden test set for murmur classification, which placed us fourth for the murmur task. (On the clinical outcome task, which we neglected, we scored 17th with costs of $12637$.) On our held-out subset of the training set, integrating the demographic data and signal features improved DBRes's accuracy from $0.762$ to $0.820$. However, this decreased DBRes's weighted accuracy from $0.780$ to $0.749$. Our results demonstrate that log mel spectrograms are an effective representation of heart sound recordings, Bayesian networks provide strong supervised classification performance, and treating the ternary classification as two binary classifications increases performance on the weighted accuracy.

Cristopher Salvi, Maud Lemercier, Chong Liu et al.

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 and captures 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 pick up 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.

Satoshi Hayakawa, Harald Oberhauser, Terry Lyons

We study kernel quadrature rules with convex weights. Our approach combines the spectral properties of the kernel with recombination results about point measures. This results in effective algorithms that construct convex quadrature rules using only access to i.i.d. samples from the underlying measure and evaluation of the kernel and that result in a small worst-case error. In addition to our theoretical results and the benefits resulting from convex weights, our experiments indicate that this construction can compete with the optimal bounds in well-known examples.

James Morrill, Patrick Kidger, Lingyi Yang et al.

Neural controlled differential equations (Neural CDEs) are a continuous-time extension of recurrent neural networks (RNNs), achieving state-of-the-art (SOTA) performance at modelling functions of irregular time series. In order to interpret discrete data in continuous time, current implementations rely on non-causal interpolations of the data. This is fine when the whole time series is observed in advance, but means that Neural CDEs are not suitable for use in \textit{online prediction tasks}, where predictions need to be made in real-time: a major use case for recurrent networks. Here, we show how this limitation may be rectified. First, we identify several theoretical conditions that interpolation schemes for Neural CDEs should satisfy, such as boundedness and uniqueness. Second, we use these to motivate the introduction of new schemes that address these conditions, offering in particular measurability (for online prediction), and smoothness (for speed). Third, we empirically benchmark our online Neural CDE model on three continuous monitoring tasks from the MIMIC-IV medical database: we demonstrate improved performance on all tasks against ODE benchmarks, and on two of the three tasks against SOTA non-ODE benchmarks.

Patrick Kidger, James Foster, Xuechen Li et al.

Neural SDEs combine many of the best qualities of both RNNs and SDEs: memory efficient training, high-capacity function approximation, and strong priors on model space. This makes them a natural choice for modelling many types of temporal dynamics. Training a Neural SDE (either as a VAE or as a GAN) requires backpropagating through an SDE solve. This may be done by solving a backwards-in-time SDE whose solution is the desired parameter gradients. However, this has previously suffered from severe speed and accuracy issues, due to high computational cost and numerical truncation errors. Here, we overcome these issues through several technical innovations. First, we introduce the \textit{reversible Heun method}. This is a new SDE solver that is \textit{algebraically reversible}: eliminating numerical gradient errors, and the first such solver of which we are aware. Moreover it requires half as many function evaluations as comparable solvers, giving up to a $1.98\times$ speedup. Second, we introduce the \textit{Brownian Interval}: a new, fast, memory efficient, and exact way of sampling \textit{and reconstructing} Brownian motion. With this we obtain up to a $10.6\times$ speed improvement over previous techniques, which in contrast are both approximate and relatively slow. Third, when specifically training Neural SDEs as GANs (Kidger et al. 2021), we demonstrate how SDE-GANs may be trained through careful weight clipping and choice of activation function. This reduces computational cost (giving up to a $1.87\times$ speedup) and removes the numerical truncation errors associated with gradient penalty. Altogether, we outperform the state-of-the-art by substantial margins, with respect to training speed, and with respect to classification, prediction, and MMD test metrics. We have contributed implementations of all of our techniques to the torchsde library to help facilitate their adoption.

Maud Lemercier, Cristopher Salvi, Thomas Cass et al.

Making predictions and quantifying their uncertainty when the input data is sequential is a fundamental learning challenge, recently attracting increasing attention. We develop SigGPDE, a new scalable sparse variational inference framework for Gaussian Processes (GPs) on sequential data. Our contribution is twofold. First, we construct inducing variables underpinning the sparse approximation so that the resulting evidence lower bound (ELBO) does not require any matrix inversion. Second, we show that the gradients of the GP signature kernel are solutions of a hyperbolic partial differential equation (PDE). This theoretical insight allows us to build an efficient back-propagation algorithm to optimize the ELBO. We showcase the significant computational gains of SigGPDE compared to existing methods, while achieving state-of-the-art performance for classification tasks on large datasets of up to 1 million multivariate time series.

Bo Wang, Yue Wu, Nemanja Vaci et al.

Bipolar disorder (BD) and borderline personality disorder (BPD) are two chronic mental health conditions that clinicians find challenging to distinguish based on clinical interviews, due to their overlapping symptoms. In this work, we investigate the automatic detection of these two conditions by modelling both verbal and non-verbal cues in a set of interviews. We propose a new approach of modelling short-term features with visibility-signature transform, and compare it with widely used high-level statistical functions. We demonstrate the superior performance of our proposed signature-based model. Furthermore, we show the role of different sets of features in characterising BD and BPD.

Thomas Cochrane, Peter Foster, Varun Chhabra et al.

The development of machine learning algorithms in the cyber security domain has been impeded by the complex, hierarchical, sequential and multimodal nature of the data involved. In this paper we introduce the notion of a streaming tree as a generic data structure encompassing a large portion of real-world cyber security data. Starting from host-based event logs we represent computer processes as streaming trees that evolve in continuous time. Leveraging the properties of the signature kernel, a machine learning tool that recently emerged as a leading technology for learning with complex sequences of data, we develop the SK-Tree algorithm. SK-Tree is a supervised learning method for systematic malware detection on streaming trees that is robust to irregular sampling and high dimensionality of the underlying streams. We demonstrate the effectiveness of SK-Tree to detect malicious events on a portion of the publicly available DARPA OpTC dataset, achieving an AUROC score of 98%.

Patrick Kidger, James Foster, Xuechen Li et al.

Stochastic differential equations (SDEs) are a staple of mathematical modelling of temporal dynamics. However, a fundamental limitation has been that such models have typically been relatively inflexible, which recent work introducing Neural SDEs has sought to solve. Here, we show that the current classical approach to fitting SDEs may be approached as a special case of (Wasserstein) GANs, and in doing so the neural and classical regimes may be brought together. The input noise is Brownian motion, the output samples are time-evolving paths produced by a numerical solver, and by parameterising a discriminator as a Neural Controlled Differential Equation (CDE), we obtain Neural SDEs as (in modern machine learning parlance) continuous-time generative time series models. Unlike previous work on this problem, this is a direct extension of the classical approach without reference to either prespecified statistics or density functions. Arbitrary drift and diffusions are admissible, so as the Wasserstein loss has a unique global minima, in the infinite data limit any SDE may be learnt. Example code has been made available as part of the \texttt{torchsde} repository.

John Pougue Biyong, Bo Wang, Terry Lyons et al.

Relying on large pretrained language models such as Bidirectional Encoder Representations from Transformers (BERT) for encoding and adding a simple prediction layer has led to impressive performance in many clinical natural language processing (NLP) tasks. In this work, we present a novel extension to the Transformer architecture, by incorporating signature transform with the self-attention model. This architecture is added between embedding and prediction layers. Experiments on a new Swedish prescription data show the proposed architecture to be superior in two of the three information extraction tasks, comparing to baseline models. Finally, we evaluate two different embedding approaches between applying Multilingual BERT and translating the Swedish text to English then encode with a BERT model pretrained on clinical notes.

Patrick Kidger, Ricky T. Q. Chen, Terry Lyons

Neural differential equations may be trained by backpropagating gradients via the adjoint method, which is another differential equation typically solved using an adaptive-step-size numerical differential equation solver. A proposed step is accepted if its error, \emph{relative to some norm}, is sufficiently small; else it is rejected, the step is shrunk, and the process is repeated. Here, we demonstrate that the particular structure of the adjoint equations makes the usual choices of norm (such as $L^2$) unnecessarily stringent. By replacing it with a more appropriate (semi)norm, fewer steps are unnecessarily rejected and the backpropagation is made faster. This requires only minor code modifications. Experiments on a wide range of tasks -- including time series, generative modeling, and physical control -- demonstrate a median improvement of 40% fewer function evaluations. On some problems we see as much as 62% fewer function evaluations, so that the overall training time is roughly halved.

James Morrill, Cristopher Salvi, Patrick Kidger et al.

Neural controlled differential equations (CDEs) are the continuous-time analogue of recurrent neural networks, as Neural ODEs are to residual networks, and offer a memory-efficient continuous-time way to model functions of potentially irregular time series. Existing methods for computing the forward pass of a Neural CDE involve embedding the incoming time series into path space, often via interpolation, and using evaluations of this path to drive the hidden state. Here, we use rough path theory to extend this formulation. Instead of directly embedding into path space, we instead represent the input signal over small time intervals through its \textit{log-signature}, which are statistics describing how the signal drives a CDE. This is the approach for solving \textit{rough differential equations} (RDEs), and correspondingly we describe our main contribution as the introduction of Neural RDEs. This extension has a purpose: by generalising the Neural CDE approach to a broader class of driving signals, we demonstrate particular advantages for tackling long time series. In this regime, we demonstrate efficacy on problems of length up to 17k observations and observe significant training speed-ups, improvements in model performance, and reduced memory requirements compared to existing approaches.

Bo Wang, Yue Wu, Niall Taylor et al.

Bipolar disorder (BD) and borderline personality disorder (BPD) are both chronic psychiatric disorders. However, their overlapping symptoms and common comorbidity make it challenging for the clinicians to distinguish the two conditions on the basis of a clinical interview. In this work, we first present a new multi-modal dataset containing interviews involving individuals with BD or BPD being interviewed about a non-clinical topic . We investigate the automatic detection of the two conditions, and demonstrate a good linear classifier that can be learnt using a down-selected set of features from the different aspects of the interviews and a novel approach of summarising these features. Finally, we find that different sets of features characterise BD and BPD, thus providing insights into the difference between the automatic screening of the two conditions.

Hans Bühler, Blanka Horvath, Terry Lyons et al.

Neural network based data-driven market simulation unveils a new and flexible way of modelling financial time series without imposing assumptions on the underlying stochastic dynamics. Though in this sense generative market simulation is model-free, the concrete modelling choices are nevertheless decisive for the features of the simulated paths. We give a brief overview of currently used generative modelling approaches and performance evaluation metrics for financial time series, and address some of the challenges to achieve good results in the latter. We also contrast some classical approaches of market simulation with simulation based on generative modelling and highlight some advantages and pitfalls of the new approach. While most generative models tend to rely on large amounts of training data, we present here a generative model that works reliably in environments where the amount of available training data is notoriously small. Furthermore, we show how a rough paths perspective combined with a parsimonious Variational Autoencoder framework provides a powerful way for encoding and evaluating financial time series in such environments where available training data is scarce. Finally, we also propose a suitable performance evaluation metric for financial time series and discuss some connections of our Market Generator to deep hedging.

Maud Lemercier, Cristopher Salvi, Theodoros Damoulas et al.

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.

Patrick Kidger, James Morrill, Terry Lyons

The shapelet transform is a form of feature extraction for time series, in which a time series is described by its similarity to each of a collection of `shapelets'. However it has previously suffered from a number of limitations, such as being limited to regularly-spaced fully-observed time series, and having to choose between efficient training and interpretability. Here, we extend the method to continuous time, and in doing so handle the general case of irregularly-sampled partially-observed multivariate time series. Furthermore, we show that a simple regularisation penalty may be used to train efficiently without sacrificing interpretability. The continuous-time formulation additionally allows for learning the length of each shapelet (previously a discrete object) in a differentiable manner. Finally, we demonstrate that the measure of similarity between time series may be generalised to a learnt pseudometric. We validate our method by demonstrating its performance and interpretability on several datasets; for example we discover (purely from data) that the digits 5 and 6 may be distinguished by the chirality of their bottom loop, and that a kind of spectral gap exists in spoken audio classification.

Patrick Kidger, James Morrill, James Foster et al.

Neural ordinary differential equations are an attractive option for modelling temporal dynamics. However, a fundamental issue is that the solution to an ordinary differential equation is determined by its initial condition, and there is no mechanism for adjusting the trajectory based on subsequent observations. Here, we demonstrate how this may be resolved through the well-understood mathematics of \emph{controlled differential equations}. The resulting \emph{neural controlled differential equation} model is directly applicable to the general setting of partially-observed irregularly-sampled multivariate time series, and (unlike previous work on this problem) it may utilise memory-efficient adjoint-based backpropagation even across observations. We demonstrate that our model achieves state-of-the-art performance against similar (ODE or RNN based) models in empirical studies on a range of datasets. Finally we provide theoretical results demonstrating universal approximation, and that our model subsumes alternative ODE models.

Shujian Liao, Terry Lyons, Weixin Yang et al.

This paper contributes to the challenge of learning a function on streamed multimodal data through evaluation. The core of the result of our paper is the combination of two quite different approaches to this problem. One comes from the mathematically principled technology of signatures and log-signatures as representations for streamed data, while the other draws on the techniques of recurrent neural networks (RNN). The ability of the former to manage high sample rate streams and the latter to manage large scale nonlinear interactions allows hybrid algorithms that are easy to code, quicker to train, and of lower complexity for a given accuracy. We illustrate the approach by approximating the unknown functional as a controlled differential equation. Linear functionals on solutions of controlled differential equations are the natural universal class of functions on data streams. Following this approach, we propose a hybrid Logsig-RNN algorithm that learns functionals on streamed data. By testing on various datasets, i.e. synthetic data, NTU RGB+D 120 skeletal action data, and Chalearn2013 gesture data, our algorithm achieves the outstanding accuracy with superior efficiency and robustness.

Patrick Kidger, Terry Lyons

The classical Universal Approximation Theorem holds for neural networks of arbitrary width and bounded depth. Here we consider the natural `dual' scenario for networks of bounded width and arbitrary depth. Precisely, let $n$ be the number of inputs neurons, $m$ be the number of output neurons, and let $ρ$ be any nonaffine continuous function, with a continuous nonzero derivative at some point. Then we show that the class of neural networks of arbitrary depth, width $n + m + 2$, and activation function $ρ$, is dense in $C(K; \mathbb{R}^m)$ for $K \subseteq \mathbb{R}^n$ with $K$ compact. This covers every activation function possible to use in practice, and also includes polynomial activation functions, which is unlike the classical version of the theorem, and provides a qualitative difference between deep narrow networks and shallow wide networks. We then consider several extensions of this result. In particular we consider nowhere differentiable activation functions, density in noncompact domains with respect to the $L^p$-norm, and how the width may be reduced to just $n + m + 1$ for `most' activation functions.

Terry Lyons, Imanol Perez Arribas

Unravelling hidden patterns in datasets is a classical problem with many potential applications. In this paper, we present a challenge whose objective is to discover nonlinear relationships in noisy cloud of points. If a set of point satisfies a nonlinear relationship that is unlikely to be due to randomness, we will label the set with this relationship. Since points can satisfy one, many or no such nonlinear relationships, cloud of points will typically have one, multiple or no labels at all. This introduces the labelling problem that will be studied in this paper. The objective of this paper is to develop a framework for the labelling problem. We introduce a precise notion of a label, and we propose an algorithm to discover such labels in a given dataset, which is then tested in synthetic datasets. We also analyse, using tools from random matrix theory, the problem of discovering false labels in the dataset.

Terry Lyons, Harald Oberhauser

We introduce features for massive data streams. These stream features can be thought of as "ordered moments" and generalize stream sketches from "moments of order one" to "ordered moments of arbitrary order". In analogy to classic moments, they have theoretical guarantees such as universality that are important for learning algorithms.

Andrey Kormilitzin, Kate E. A. Saunders, Paul J. Harrison et al.

Recurrent major mood episodes and subsyndromal mood instability cause substantial disability in patients with bipolar disorder. Early identification of mood episodes enabling timely mood stabilisation is an important clinical goal. Recent technological advances allow the prospective reporting of mood in real time enabling more accurate, efficient data capture. The complex nature of these data streams in combination with challenge of deriving meaning from missing data mean pose a significant analytic challenge. The signature method is derived from stochastic analysis and has the ability to capture important properties of complex ordered time series data. To explore whether the onset of episodes of mania and depression can be identified using self-reported mood data.