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We use a rough path-based approach to investigate the degeneracy problem in the context of pathwise control. We extend the framework developed in arXiv:1902.05434 to treat admissible controls from a suitable class of H\"older continuous paths and simultaneously to handle a broader class of noise terms. Our approach uses fractional calculus to augment the original control equation, resulting in a system with added fractional dynamics. We adapt the existing analysis of fractional systems from the work of Gomoyunov arXiv:1908.01747, arXiv:2111.14400v1 , arXiv:2109.02451 to this new setting, providing a notion of a rough fractional viscosity solution for fractional systems that involve a noise term of arbitrarily low regularity. In this framework, following the method outlined in arXiv:1902.05434, we derive sufficient conditions to ensure that the control problem remains non-degenerate.

The signature of a $p$-weakly geometric rough path summarises a path up to a generalised notion of reparameterisation. The quotient space of equivalence classes on which the signature is constant yields unparameterised path space. The study of topologies on unparameterised path space, initiated in [CT24b] for paths of bounded variation, has practical bearing on the use of signature based methods in a variety applications. This note extends the majority of results from [CT24b] to unparameterised weakly geometric rough path space. We study three classes of topologies: metrisable topologies for which the quotient map is continuous; the quotient topology derived from the underlying path space; and an explicit metric between the tree-reduced representatives of each equivalence class. We prove that topologies of the first type (under an additional assumption) are separable and Lusin, but not locally compact or completely metrisable. The quotient topology is Hausdorff but not metrisable, while the metric generating the third topology is not complete and its topology is not locally compact. We also show that the third topology is Polish when $p=1$.

We present a unified theory for Mahalanobis-type anomaly detection on Banach spaces, using ideas from Cameron-Martin theory applied to non-Gaussian measures. This approach leads to a basis-free, data-driven notion of anomaly distance through the so-called variance norm of a probability measure, which can be consistently estimated using empirical measures. Our framework generalizes the classical $\mathbb{R}^d$, functional $(L^2[0,1])^d$, and kernelized settings, including the general case of non-injective covariance operator. We prove that the variance norm depends solely on the inner product in a given Hilbert space, and hence that the kernelized Mahalanobis distance can naturally be recovered by working on reproducing kernel Hilbert spaces. Using the variance norm, we introduce the notion of a kernelized nearest-neighbour Mahalanobis distance for semi-supervised anomaly detection. In an empirical study on 12 real-world datasets, we demonstrate that the kernelized nearest-neighbour Mahalanobis distance outperforms the traditional kernelized Mahalanobis distance for multivariate time series anomaly detection, using state-of-the-art time series kernels such as the signature, global alignment, and Volterra reservoir kernels. Moreover, we provide an initial theoretical justification of nearest-neighbour Mahalanobis distances by developing concentration inequalities in the finite-dimensional Gaussian case.

In August 2023, IT experts and scientists came together for a workshop to discuss the possibilities of building a computer cluster fully on renewable energies, as a test-case at Havana University in Cuba. The discussion covered the scientific needs for a computer cluster for particle physics at the InSTEC institute at Havana University, the possibilities to use solar energy, new developments in computing technologies, and computer cluster operation as well as operational needs for computing in particle physics. This computer cluster on renewable energies at the InSTEC institute is seen as a prototype for a large-scale computer cluster on renewable energies for scientific computing in the Caribbean, hosted in Cuba. The project is called "Humboldt Highway", to remember Alexander von Humboldt's achievements in bringing cultures of the American and European continents closer together by exchange and travel. In this spirit, we propose a project that enables and intensifies the scientific exchange between research laboratories and universities in Europe and the Caribbean, in particular Cuba.

These notes expound the recent use of the signature transform and rough path theory in data science and machine learning. We develop the core theory of the signature from first principles and then survey some recent popular applications of this approach, including signature-based kernel methods and neural rough differential equations. The notes are based on a course given by the two authors at Imperial College London.

We construct a refinement of Gaitsgory's central functor for integral motivic sheaves, and show it preserves stratified Tate motives. Towards this end, we develop a reformulation of unipotent motivic nearby cycles, which also works over higher-dimensional bases. We moreover introduce Wakimoto motives and use them to show that our motivic central functor is t-exact. A decategorification of these functors yields a new approach to generic Hecke algebras for general parahorics., Comment: 56 pages, comments welcome

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).

Random developments of a path into a matrix Lie group $G_N$ have recently been used to construct signature-based kernels on path space. Two examples include developments into GL$(N;\mathbb{R})$ and $U(N;\mathbb{C})$, the general linear and unitary groups of dimension $N$. For the former, [MLS23] showed that the signature kernel is obtained via a scaling limit of developments with Gaussian vector fields. The second instance was used in [LLN23] to construct a metric between probability measures on path space. We present a unified treatment to obtaining large $N$ limits by leveraging the tools of free probability theory. An important conclusion is that the limiting kernels, while dependent on the choice of Lie group, are nonetheless universal limits with respect to how the development map is randomised. For unitary developments, the limiting kernel is given by the contraction of a signature against the monomials of freely independent semicircular random variables. Using the Schwinger-Dyson equations, we show that this kernel can be obtained by solving a novel quadratic functional equation. We provide a convergent numerical scheme for this equation, together with rates, which does not require computation of signatures themselves.

This report presents the takeaways of the inaugural Workshop on Generative AI and Law (GenLaw), held in July 2023. A cross-disciplinary group of practitioners and scholars from computer science and law convened to discuss the technical, doctrinal, and policy challenges presented by law for Generative AI, and by Generative AI for law, with an emphasis on U.S. law in particular. We begin the report with a high-level statement about why Generative AI is both immensely significant and immensely challenging for law. To meet these challenges, we conclude that there is an essential need for 1) a shared knowledge base that provides a common conceptual language for experts across disciplines; 2) clarification of the distinctive technical capabilities of generative-AI systems, as compared and contrasted to other computer and AI systems; 3) a logical taxonomy of the legal issues these systems raise; and, 4) a concrete research agenda to promote collaboration and knowledge-sharing on emerging issues at the intersection of Generative AI and law. In this report, we synthesize the key takeaways from the GenLaw workshop that begin to address these needs. All of the listed authors contributed to the workshop upon which this report is based, but they and their organizations do not necessarily endorse all of the specific claims in this report.

It is well known that, when numerically simulating solutions to SDEs, achieving a strong convergence rate better than O(\sqrt{h}) (where h is the step size) requires the use of certain iterated integrals of Brownian motion, commonly referred to as its "L\'{e}vy areas". However, these stochastic integrals are difficult to simulate due to their non-Gaussian nature and for a d-dimensional Brownian motion with d > 2, no fast almost-exact sampling algorithm is known. In this paper, we propose L\'{e}vyGAN, a deep-learning-based model for generating approximate samples of L\'{e}vy area conditional on a Brownian increment. Due to our "Bridge-flipping" operation, the output samples match all joint and conditional odd moments exactly. Our generator employs a tailored GNN-inspired architecture, which enforces the correct dependency structure between the output distribution and the conditioning variable. Furthermore, we incorporate a mathematically principled characteristic-function based discriminator. Lastly, we introduce a novel training mechanism termed "Chen-training", which circumvents the need for expensive-to-generate training data-sets. This new training procedure is underpinned by our two main theoretical results. For 4-dimensional Brownian motion, we show that L\'{e}vyGAN exhibits state-of-the-art performance across several metrics which measure both the joint and marginal distributions. We conclude with a numerical experiment on the log-Heston model, a popular SDE in mathematical finance, demonstrating that high-quality synthetic L\'{e}vy area can lead to high order weak convergence and variance reduction when using multilevel Monte Carlo (MLMC).

Fake news emerged as an apparent global problem during the 2016 U.S. Presidential election. Addressing it requires a multidisciplinary effort to define the nature and extent of the problem, detect fake news in real time, and mitigate its potentially harmful effects. This will require a better understanding of how the Internet spreads content, how people process news, and how the two interact. We review the state of knowledge in these areas and discuss two broad potential mitigation strategies: better enabling individuals to identify fake news, and intervention within the platforms to reduce the attention given to fake news. The cooperation of Internet platforms (especially Facebook, Google, and Twitter) with researchers will be critical to understanding the scale of the issue and the effectiveness of possible interventions., Comment: This is the accepted version of the article, posted to comply with public access mandates. The final published version is available at https://doi.org/10.1126/science.aao2998

Advanced AI models hold the promise of tremendous benefits for humanity, but society needs to proactively manage the accompanying risks. In this paper, we focus on what we term "frontier AI" models: highly capable foundation models that could possess dangerous capabilities sufficient to pose severe risks to public safety. Frontier AI models pose a distinct regulatory challenge: dangerous capabilities can arise unexpectedly; it is difficult to robustly prevent a deployed model from being misused; and, it is difficult to stop a model's capabilities from proliferating broadly. To address these challenges, at least three building blocks for the regulation of frontier models are needed: (1) standard-setting processes to identify appropriate requirements for frontier AI developers, (2) registration and reporting requirements to provide regulators with visibility into frontier AI development processes, and (3) mechanisms to ensure compliance with safety standards for the development and deployment of frontier AI models. Industry self-regulation is an important first step. However, wider societal discussions and government intervention will be needed to create standards and to ensure compliance with them. We consider several options to this end, including granting enforcement powers to supervisory authorities and licensure regimes for frontier AI models. Finally, we propose an initial set of safety standards. These include conducting pre-deployment risk assessments; external scrutiny of model behavior; using risk assessments to inform deployment decisions; and monitoring and responding to new information about model capabilities and uses post-deployment. We hope this discussion contributes to the broader conversation on how to balance public safety risks and innovation benefits from advances at the frontier of AI development., Comment: Update July 11th: - Added missing footnote back in. - Adjusted author order (mistakenly non-alphabetical among the first 6 authors) and adjusted affiliations (Jess Whittlestone's affiliation was mistagged and Gillian Hadfield had SRI added to her affiliations) Updated September 4th: Various typos

We show that the flagellar beat of bull spermatozoa and Chlamydomonas Reinhardtii can be modelled by a minimal, geometrically nonlinear reaction-diffusion system. Model solutions are spatio-temporally animated patterns describing flagellar bending waves, connecting beating patterns of cilia and flagella with chemical patterns from classical reaction-diffusion systems. Instead of chemical species freely reacting and diffusing in space, our system describes the tug-of-war reaction-kinetics of molecular motors that are anchored in the flagellar structure, but the shear deformation that they generate can diffuse away via the bending elasticity of the flagellum. Synchronization of the reaction-kinetics in neighbouring elements occurs via a sliding-control mechanism. We derive from first principles the reaction-diffusion basis of animated patterns as a direct consequence of the high internal energy dissipation by the flagellum relative to the external dissipation by the surrounding fluid. By fitting nonlinear, large-amplitude solutions of a specific motor cross-bridge reaction-kinetics, we show that reaction-diffusion successfully accounts for beating patterns of both bull sperm and Chlamydomonas (wild-type and mbo2-mutant). Our results suggest that the flagellar beat occurs far from equilibrium, in the strongly nonlinear regime, and that a unified mechanism may exist for dynein molecular motor control that is regulated by axonemal sliding, without requiring curvature-sensing or the fine-tuning of basal compliance, and only weakly influenced by hydrodynamic dissipation and the cell body boundary condition. High internal dissipation allows the emergence of base-to-tip autonomous travelling waves, independently of the external fluid viscosity. This enables progressive swimming in low viscosity environments, and may be critical for external fertilizers and aquatic microorganisms., Comment: 18 pages, 7 figures

The ARCHER2 service, a CPU based HPE Cray EX system with 750,080 cores (5,860 nodes), has been deployed throughout 2020 and 2021, going into full service in December of 2021. A key part of the work during this deployment was the integration of ARCHER2 into our local monitoring systems. As ARCHER2 was one of the very first large-scale EX deployments, this involved close collaboration and development work with the HPE team through a global pandemic situation where collaboration and co-working was significantly more challenging than usual. The deployment included the creation of automated checks and visual representations of system status which needed to be made available to external parties for diagnosis and interpretation. We will describe how these checks have been deployed and how data gathered played a key role in the deployment of ARCHER2, the commissioning of the plant infrastructure, the conduct of HPL runs for submission to the Top500 and contractual monitoring of the availability of the ARCHER2 service during its commissioning and early life., Comment: 7 pages