A Generalised Signature Method for Multivariate Time Series Feature Extraction

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.Comment: 25 page

Sequential path signature networks for personalised longitudinal language modeling

Longitudinal user modeling can provide a strong signal for various downstream tasks. Despite the rapid progress in representation learning, dynamic aspects of modelling individuals’ language have only been sparsely addressed. We present a novel extension of neural sequential models using the notion of path signatures from rough path theory, which constitute graduated summaries of continuous paths and have the ability to capture non-linearities in trajectories. By combining path signatures of users’ history with contextual neural representations and recursive neural networks we can produce compact time-sensitive user representations. Given the magnitude of mental health conditions with symptoms manifesting in language, we show the applicability of our approach on the task of identifying changes in individuals’ mood by analysing their online textual content. By directly integrating signature transforms of users’ history in the model architecture we jointly address the two most important aspects of the task, namely sequentiality and temporality. Our approach achieves state-of-the-art performance on macro-average F1 score on the two available datasets for the task, outperforming or performing on-par with state-of-the-art models utilising only historical posts and even outperforming prior models which also have access to future posts of users

Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU

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.Comment: Published at ICLR 202

ImageSig: A signature transform for ultra-lightweight image recognition

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

Topologies on unparameterised path space

The signature of a path, introduced by K.T. Chen [10] in 1954, has been extensively studied in recent years. The fundamental 2010 paper [20] of Hambly and Lyons showed that the signature is an injective function on the space of continuous, finite-variation paths up to a general notion of reparameterisation called tree-like equivalence. This result has been extended to geometric rough paths by Boedihardjo et al. [5]. More recently, the approximation theory of the signature has been widely used in the literature in applications. The archetypal instance of these results, see e.g. [24], guarantees uniform approximation, on compact sets, of a continuous function by a linear functional on the (extended) tensor algebra acting on the signature. In this paper we study in detail, and for the first time, the properties of three natural candidate topologies on the set of unparameterised paths, i.e. the tree-like equivalence classes. These are obtained by privileging different properties of the signature and are: (1) the product topology, obtained by equipping the range of the signature with the (subspace topology of the) product topology in the extended tensor algebra and then requiring S to be an embedding, (2) the quotient topology derived from the 1-variation topology on the underlyind path space, and (3) the metric topology associated to ([γ], [σ]) := ||γ* - σ*||₁ using the (constant-speed) tree-reduced representatives γ* and σ* of the respective equivalence classes. We evaluate these spaces from the point of view of their suitability when it comes to studying (probability) measures on them. We prove that the respective collections of open sets are ordered by strict inclusion, (1) being the weakest and (3) the strongest. Our other conclusions can be summarised as follows. All three topological spaces are separable and Hausdorff, (1) being both metrisable and σ-compact, but not a Baire space and hence being neither Polish nor locally compact. The completion of (1), in any metric inducing the product topology, is the subspace G* of group-like elements. The quotient topology (2) is not metrisable and the metric d is not complete. We also discuss some open problems related to these spaces. We consider finally the implications of the selection of the topology for uniform approximation results involving the signature. A stereotypical model for a continuous function on (unparameterised) path space is the solution of a controlled differential equation. We thus prove, for a broad class of these equations, well-definedness and measurability of the (fixed-time) solution map with respect to the Borel sigma-algebra of each topology. Under stronger regularity assumptions, we further show continuity of this same map on explicit compact subsets of the product topology (1). We relate these results to the expected signature model of [24]

Advanced Techniques for Ground Penetrating Radar Imaging

Ground penetrating radar (GPR) has become one of the key technologies in subsurface sensing and, in general, in non-destructive testing (NDT), since it is able to detect both metallic and nonmetallic targets. GPR for NDT has been successfully introduced in a wide range of sectors, such as mining and geology, glaciology, civil engineering and civil works, archaeology, and security and defense. In recent decades, improvements in georeferencing and positioning systems have enabled the introduction of synthetic aperture radar (SAR) techniques in GPR systems, yielding GPR–SAR systems capable of providing high-resolution microwave images. In parallel, the radiofrequency front-end of GPR systems has been optimized in terms of compactness (e.g., smaller Tx/Rx antennas) and cost. These advances, combined with improvements in autonomous platforms, such as unmanned terrestrial and aerial vehicles, have fostered new fields of application for GPR, where fast and reliable detection capabilities are demanded. In addition, processing techniques have been improved, taking advantage of the research conducted in related fields like inverse scattering and imaging. As a result, novel and robust algorithms have been developed for clutter reduction, automatic target recognition, and efficient processing of large sets of measurements to enable real-time imaging, among others. This Special Issue provides an overview of the state of the art in GPR imaging, focusing on the latest advances from both hardware and software perspectives

Rotation-free online handwritten character recognition using dyadic path signature features, hanging normalization, and deep neural network

The path signature feature (PSF) which was initially introduced in rough paths theory as a branch of stochastic analysis, has recently been successfully applied to the field of pattern recognition for extracting sufficient quantity of information contained in a finite trajectory, but with potentially high dimension. In this paper, we propose a variation of path signature representation, namely the dyadic path signature feature (D-PSF), to fully characterize the trajectory using a hierarchical structure to solve the rotation-free online handwritten character recognition (OLHCR) problem. We adopt the deep neural network (DNN) as classifier, and investigate three hanging normalization methods to improve the robustness of the DNN to rotational distortions. Extensive experiments on digits, English letters, and Chinese radicals demonstrated that the proposed D-PSF, jointly with hanging normalization and DNN, achieved very promising results for rotated OLHCR, significantly outperforming previous methods

Rotation-free online handwritten character recognition using dyadic path signature features, hanging normalization, and deep neural network

The path signature feature (PSF) which was initially introduced in rough paths theory as a branch of stochastic analysis, has recently been successfully applied to the field of pattern recognition for extracting sufficient quantity of information contained in a finite trajectory, but with potentially high dimension. In this paper, we propose a variation of path signature representation, namely the dyadic path signature feature (D-PSF), to fully characterize the trajectory using a hierarchical structure to solve the rotation-free online handwritten character recognition (OLHCR) problem. We adopt the deep neural network (DNN) as classifier, and investigate three hanging normalization methods to improve the robustness of the DNN to rotational distortions. Extensive experiments on digits, English letters, and Chinese radicals demonstrated that the proposed D-PSF, jointly with hanging normalization and DNN, achieved very promising results for rotated OLHCR, significantly outperforming previous methods