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In this paper we introduce the Higher Order Lipschitz Greedy Recombination Interpolation Method (HOLGRIM) for finding sparse approximations of Lip$(\gamma)$ functions, in the sense of Stein, given as a linear combination of a (large) number of simpler Lip$(\gamma)$ functions. HOLGRIM is developed as a refinement of the Greedy Recombination Interpolation Method (GRIM) in the setting of Lip$(\gamma)$ functions. HOLGRIM combines dynamic growth-based interpolation techniques with thinning-based reduction techniques in a data-driven fashion. The dynamic growth is driven by a greedy selection algorithm in which multiple new points may be selected at each step. The thinning reduction is carried out by recombination, the linear algebra technique utilised by GRIM. We establish that the number of non-zero weights for the approximation returned by HOLGRIM is controlled by a particular packing number of the data. The level of data concentration required to guarantee that HOLGRIM returns a good sparse approximation is decreasing with respect to the regularity parameter $\gamma > 0$. Further, we establish complexity cost estimates verifying that implementing HOLGRIM is feasible.

We investigate the consequence of two Lip$(\gamma)$ functions, in the sense of Stein, being close throughout a subset of their domain. A particular consequence of our results is the following. Given $K_0 > \varepsilon > 0$ and $\gamma > \eta > 0$ there is a constant $\delta = \delta(\gamma,\eta,\varepsilon,K_0) > 0$ for which the following is true. Let $\Sigma \subset \mathbb{R}^d$ be closed and $f , h : \Sigma \to \mathbb{R}$ be Lip$(\gamma)$ functions whose Lip$(\gamma)$ norms are both bounded above by $K_0$. Suppose $B \subset \Sigma$ is closed and that $f$ and $h$ coincide throughout $B$. Then over the set of points in $\Sigma$ whose distance to $B$ is at most $\delta$ we have that the Lip$(\eta)$ norm of the difference $f-h$ is bounded above by $\varepsilon$. More generally, we establish that this phenomenon remains valid in a less restrictive Banach space setting under the weaker hypothesis that the two Lip$(\gamma)$ functions $f$ and $h$ are only close in a pointwise sense throughout the closed subset $B$. We require only that the subset $\Sigma$ be closed; in particular, the case that $\Sigma$ is finite is covered by our results. The restriction that $\eta < \gamma$ is sharp in the sense that our result is false for $\eta := \gamma$., Comment: Minor refinement of main result statements. arXiv admin note: substantial text overlap with arXiv:2205.07495

Signature kernels are at the core of several machine learning algorithms for analysing multivariate time series. The kernel of two bounded variation paths (such as piecewise linear interpolations of time series data) is typically computed by solving a Goursat problem for a hyperbolic partial differential equation (PDE) in two independent time variables. However, this approach becomes considerably less practical 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. To mitigate this issue, we first show that the signature kernel of a broader class of paths, known as \emph{smooth rough paths}, also satisfies a PDE, albeit in the form of a system of coupled equations. We then use this result to introduce new algorithms for the numerical approximation of signature kernels. As bounded variation paths (and more generally geometric $p$-rough paths) can be approximated by piecewise smooth rough paths, one can replace the PDE with rapidly varying coefficients in the original Goursat problem by an explicit system of coupled equations with piecewise constant coefficients derived from the first few iterated integrals of the original input paths. While this approach requires solving more equations, they do not require looking back at the complex and fine structure of the initial paths, which significantly reduces the computational complexity associated with the analysis of highly oscillatory time series.

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., Comment: 5 figures, 2 tables

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.

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., Comment: 23 pages, 5 figures, International Conference on Machine Learning 2024

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).

We introduce Optimistix: a nonlinear optimisation library built in JAX and Equinox. Optimistix introduces a novel, modular approach for its minimisers and least-squares solvers. This modularity relies on new practical abstractions for optimisation which we call search and descent, and which generalise classical notions of line search, trust-region, and learning-rate algorithms. It provides high-level APIs and solvers for minimisation, nonlinear least-squares, root-finding, and fixed-point iteration. Optimistix is available at https://github.com/patrick-kidger/optimistix., Comment: 8 pages, 4 figures, 2 tables

Quantum state discrimination plays an essential role in quantum technology, crucial for quantum error correction, metrology, and sensing. While conventional methods rely on integrating readout signals or classifying raw signals, we developed a method to extract information about state transitions during readout, based on the path signature method, a tool for analyzing stochastic time series. The hardware experiments demonstrate an improvement in transmon qutrit state readout fidelity from 85.9 $\pm$ 1.0% to 91.0 $\pm$ 0.5%, without the need for additional hardware. This method has the potential to become a foundational tool for quantum technology.

The signature of a path, as a fundamental object in Rough path theory, serves as a generating function for non-commutative monomials on path space. It transforms the path into a grouplike element in the tensor algebra space, summarising the path faithfully up to a generalised form of re-parameterisation (a negligible equivalence class in this context). Our paper concerns stochastic processes and studies the characteristic function of the path signature of the stochastic process. In contrast to the expected signature, it determines the law on the random signatures without any regularity condition. The computation of the characteristic function of the random signature offers potential applications in stochastic analysis and machine learning, where the expected signature plays an important role. In this paper, we focus on a time-homogeneous It\^o diffusion process, and adopt a PDE approach to derive the characteristic function of its signature defined at any fixed time horizon. A key ingredient of our approach is the introduction of the generalised-signature process. This lifting enables us to establish the Feynman-Kac-type theorem for the characteristic function of the generalised-signature process by following the martingale approach. Moreover, as an application of our results, we present a novel derivation of the joint characteristic function of Brownian motion coupled with the L\'evy area, leveraging the structure theorem of anti-symmetric matrices.

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., Comment: To appear in EACL 2024: System Demonstrations

In 2015, Bailleul presented a mechanism to solve rough differential equations by constructing flows, using the log-ODE method. We extend this notion in two ways: On the one hand, we localize Bailleul's notion of an almost-flow to solve RDEs on manifolds. On the other hand, we extend his results to non-geometric rough paths, living in any connected, cocommutative, graded Hopf algebra. This requires a new concept, which we call a pseudo bialgebra map. We further connect our results to Curry et al (2020), who solved planarly branched RDEs on homogeneous spaces.

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., Comment: Set to be published in the PLoS ONE Journal

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.

We present an adaptive algorithm for effectively solving rough differential equations (RDEs) using the log-ODE method. The algorithm is based on an error representation formula that accurately describes the contribution of local errors to the global error. By incorporating a cost model, our algorithm efficiently determines whether to refine the time grid or increase the order of the log-ODE method. In addition, we provide several examples that demonstrate the effectiveness of our adaptive algorithm in solving RDEs.

The signature of a rectifiable path is a tensor series in the tensor algebra whose coefficients are definite iterated integrals of the path. The signature characterises the path up to a generalised form of reparametrisation. It is a classical result of K. T. Chen that the log-signature (the logarithm of the signature) is a Lie series. A Lie series is polynomial if it has finite degree. We show that the log-signature is polynomial if and only if the path is a straight line up to reparametrisation. Consequently, the log-signature of a rectifiable path either has degree one or infinite support. Though our result pertains to rectifiable paths, the proof uses results from rough path theory, in particular that the signature characterises a rough path up to reparametrisation., Comment: 11 pages

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., Comment: 5 pages, 3 figures

The signature is a representation of a path as an infinite sequence of its iterated integrals. Under certain assumptions, the signature characterizes the path, up to translation and reparameterization. Therefore, a crucial question of interest is the development of efficient algorithms to invert the signature, i.e., to reconstruct the path from the information of its (truncated) signature. In this article, we study the insertion procedure, originally introduced by Chang and Lyons (2019), from both a theoretical and a practical point of view. After describing our version of the method, we give its rate of convergence for piecewise linear paths, accompanied by an implementation in Pytorch. The algorithm is parallelized, meaning that it is very efficient at inverting a batch of signatures simultaneously. Its performance is illustrated with both real-world and simulated examples.

Distribution Regression on path-space refers to the task of learning functions mapping the law of a stochastic process to a scalar target. The learning procedure based on the notion of path-signature, i.e. a classical transform from rough path theory, was widely used to approximate weakly continuous functionals, such as the pricing functionals of path--dependent options' payoffs. However, this approach fails for Optimal Stopping Problems arising from mathematical finance, such as the pricing of American options, because the corresponding value functions are in general discontinuous with respect to the weak topology. In this paper we develop a rigorous mathematical framework to resolve this issue by recasting an Optimal Stopping Problem as a higher order kernel mean embedding regression based on the notions of higher rank signatures of measure--valued paths and adapted topologies. The core computational component of our algorithm consists in solving a family of two--dimensional hyperbolic PDEs., Comment: 33 pages

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 process of physical dynamic models in deep neural networks, we propose Graph Neural Controlled Differential Equation (GN-CDE) model, a generic differential model for dynamic graphs that characterise the continuously dynamic evolution of node embedding trajectories with a neural network parameterised vector field and the derivatives of interactions w.r.t. time. Our framework exhibits several desirable characteristics, including the ability to express dynamics on evolving graphs without integration by segments, 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 superiority of our proposed approach compared to the baselines., Comment: 13 pages, 3 figures

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.

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., Comment: 23 pages, 8 figures

We analyze the Nystr\"om approximation of a positive definite kernel associated with a probability measure. We first prove an improved error bound for the conventional Nystr\"om 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 Nystr\"om 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., Comment: 22 pages, ICML 2023 camera-ready version. Typos fixed

We identify the free half shuffle algebra of Sch\"utzenberger (1958) with an algebra of real-valued functionals on paths, where the half shuffle emulates integration of a functional against another. We then provide two, to our knowledge, new identities in arity 3 involving its commutator (area), and show that these are sufficient to recover the Zinbiel and Tortkara identities of Dzhumadil'daev (2007). We use these identities to prove that any element of the free half shuffle algebra can be expressed as a polynomial over iterated areas. Moreover, we consider minimal sets of iterated integrals defined through the recursive application of the half shuffle on Hall trees. Leveraging the duality between this set of Hall integrals and classical Hall bases of the free Lie algebra, we prove using combinatorial arguments that any element of the free half shuffle algebra can be written uniquely as a polynomial over Hall integrals. We interpret this result as a structure theorem for streamed information, loosely analogous to the unique prime factorisation of integers, allowing to split any real valued function on streamed data into two parts: a first that extracts and packages the streamed information into recursively defined atomic objects (Hall integrals), and a second that evaluates a polynomial function in these objects without further reference to the original stream. The question of whether a similar result holds if Hall integrals are replaced by Hall areas is left as an open conjecture. Finally, we construct a canonical, but to our knowledge, new decomposition of the free half shuffle algebra as shuffle power series in the greatest letter of the original alphabet with coefficients in a sub-algebra freely generated by a new alphabet with an infinite number of letters. We use this construction to provide a second proof of our structure theorem.

Given a probability measure $\mu$ on a set $\mathcal{X}$ and a vector-valued function $\varphi$, a common problem is to construct a discrete probability measure on $\mathcal{X}$ such that the push-forward of these two probability measures under $\varphi$ is the same. This construction is at the heart of numerical integration methods that run under various names such as quadrature, cubature, or recombination. A natural approach is to sample points from $\mu$ until their convex hull of their image under $\varphi$ includes the mean of $\varphi$. Here we analyze the computational complexity of this approach when $\varphi$ exhibits a graded structure by using so-called hypercontractivity. The resulting theorem not only covers the classical cubature case of multivariate polynomials, but also integration on pathspace, as well as kernel quadrature for product measures., Comment: 20 pages

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.

In this paper we develop the Greedy Recombination Interpolation Method (GRIM) for finding sparse approximations of functions initially given as linear combinations of some (large) number of simpler functions. In a similar spirit to the CoSaMP algorithm, GRIM combines dynamic growth-based interpolation techniques and thinning-based reduction techniques. The dynamic growth-based aspect is a modification of the greedy growth utilised in the Generalised Empirical Interpolation Method (GEIM). A consequence of the modification is that our growth is not restricted to being one-per-step as it is in GEIM. The thinning-based aspect is carried out by recombination, which is the crucial component of the recent ground-breaking convex kernel quadrature method. GRIM provides the first use of recombination outside the setting of reducing the support of a measure. The sparsity of the approximation found by GRIM is controlled by the geometric concentration of the data in a sense that is related to a particular packing number of the data. We apply GRIM to a kernel quadrature task for the radial basis function kernel, and verify that its performance matches that of other contemporary kernel quadrature techniques.

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.

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., Comment: This paper is accepted by British Machine Vision Conference 2021

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., Comment: Published at NeurIPS 2021

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., Comment: 29 pages, NeurIPS 2022 camera-ready version

Suppose that $\gamma$ and $\sigma$ are two continuous bounded variation paths which take values in a finite-dimensional inner product space $V$. Recent papers have introduced the truncated and the untruncated signature kernel of $\gamma$ and $\sigma$, and showed how these concepts can be used in classification and prediction tasks involving multivariate time series. In this paper, we introduce general signature kernels and show how they can be interpreted, in many examples, as an average of PDE solutions, and hence how they can be computed using suitable quadrature rules. We extend this analysis to derive closed-form formulae for expressions involving the expected (Stratonovich) signature of Brownian motion. In doing so, we articulate a novel connection between signature kernels and the hyperbolic development map, the latter of which has been a broadly useful tool in the analysis of the signature. As an application we evaluate the use of different general signature kernels as the basis for non-parametric goodness-of-fit tests to Wiener measure on path space.

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.

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., Comment: Accepted at NeurIPS 2021

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., Comment: Published at ICML 2021

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.

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%., Comment: Published at IEEE-CSR (International Conference on Cybersecurity and Resilience) 2021

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., Comment: Published at ICML 2021

For a $d$-dimensional random vector $X$, let $p_{n, X}(\theta)$ be the probability that the convex hull of $n$ independent copies of $X$ contains a given point $\theta$. We provide several sharp inequalities regarding $p_{n, X}(\theta)$ and $N_X(\theta)$ denoting the smallest $n$ for which $p_{n, X}(\theta)\ge1/2$. As a main result, we derive the totally general inequality $1/2 \le \alpha_X(\theta)N_X(\theta)\le 3d + 1$, where $\alpha_X(\theta)$ (a.k.a. the Tukey depth) is the minimum probability that $X$ is in a fixed closed halfspace containing the point $\theta$. We also show several applications of our general results: one is a moment-based bound on $N_X(\mathbb{E}[X])$, which is an important quantity in randomized approaches to cubature construction or measure reduction problem. Another application is the determination of the canonical convex body included in a random convex polytope given by independent copies of $X$, where our combinatorial approach allows us to generalize existing results in random matrix community significantly., Comment: 34 pages

In this paper, we consider the underdamped Langevin diffusion (ULD) and propose a numerical approximation using its associated ordinary differential equation (ODE). When used as a Markov Chain Monte Carlo (MCMC) algorithm, we show that the ODE approximation achieves a $2$-Wasserstein error of $\varepsilon$ in $\mathcal{O}\big(d^{\frac{1}{3}}/\varepsilon^{\frac{2}{3}}\big)$ steps under the standard smoothness and strong convexity assumptions on the target distribution. This matches the complexity of the randomized midpoint method proposed by Shen and Lee [NeurIPS 2019] which was shown to be order optimal by Cao, Lu and Wang. However, the main feature of the proposed numerical method is that it can utilize additional smoothness of the target log-density $f$. More concretely, we show that the ODE approximation achieves a $2$-Wasserstein error of $\varepsilon$ in $\mathcal{O}\big(d^{\frac{2}{5}}/\varepsilon^{\frac{2}{5}}\big)$ and $\mathcal{O}\big(\sqrt{d}/\varepsilon^{\frac{1}{3}}\big)$ steps when Lipschitz continuity is assumed for the Hessian and third derivative of $f$. By discretizing this ODE using a third order Runge-Kutta method, we can obtain a practical MCMC method that uses just two additional gradient evaluations per step. In our experiment, where the target comes from a logistic regression, this method shows faster convergence compared to other unadjusted Langevin MCMC algorithms.

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.

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., Comment: Published at ICML 2021

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., Comment: Published at ICML 2021

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.

The availability of mobile technologies has enabled the efficient collection prospective longitudinal, ecologically valid self-reported mood data from psychiatric patients. These data streams have potential for improving the efficiency and accuracy of psychiatric diagnosis as well predicting future mood states enabling earlier intervention. However, missing responses are common in such datasets and there is little consensus as to how this should be dealt with in practice. A signature-based method was used to capture different elements of self-reported mood alongside missing data to both classify diagnostic group and predict future mood in patients with bipolar disorder, borderline personality disorder and healthy controls. The missing-response-incorporated signature-based method achieves roughly 66\% correct diagnosis, with f1 scores for three different clinic groups 59\% (bipolar disorder), 75\% (healthy control) and 61\% (borderline personality disorder) respectively. This was significantly more efficient than the naive model which excluded missing data. Accuracies of predicting subsequent mood states and scores were also improved by inclusion of missing responses. The signature method provided an effective approach to the analysis of prospectively collected mood data where missing data was common and should be considered as an approach in other similar datasets.

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.

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., Comment: 27 pages, 9 figures

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., Comment: Published at AISTATS 2021