It is widely believed that engineering a model to be invariant/equivariant improves generalisation.Despite the growing popularity of this approach, a precise characterisation of the generalisation benefit is lacking. By considering the simplest case of linear models, this paper provides the first provably non-zero improvement in generalisation for invariant/equivariant models when the target distribution is invariant/equivariant with respect to a compact group. Moreover, our work reveals an interesting relationship between generalisation, the number of training examples and properties of the group action.Our results rest on an observation of the structure of function spaces under averaging operators which, along with its consequences for feature averaging, may be of independent interest.

Many important problems involving molecular property prediction from 3D structures have limited data, posing a generalization challenge for neural networks. In this paper, we describe a pre-training technique that utilizes large datasets of 3D molecular structures at equilibrium to learn meaningful representations for downstream tasks. Inspired by recent advances in noise regularization, our pre-training objective is based on denoising. Relying on the well-known link between denoising autoencoders and score-matching, we also show that the objective corresponds to learning a molecular force field -arising from approximating the physical state distribution with a mixture of Gaussians -directly from equilibrium structures. Our experiments demonstrate that using this pre-training objective significantly improves performance on multiple benchmarks, achieving a new state-of-the-art on the majority of targets in the widely used QM9 dataset. Our analysis then provides practical insights into the effects of different factors -dataset sizes, model size and architecture, and the choice of upstream and downstream datasets -on pre-training.

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information about population flows to model potential transmissions across local areas. A simple approach to posterior simulation quickly becomes computationally infeasible, which is problematic if the system is required to provide timely predictions. We describe how to make posterior simulation for the model efficient, so that we are able to provide daily updates on epidemic developments.The results can be found at our web site https://localcovid.info, which is updated daily to display estimated instantaneous reproduction numbers and predicted case counts for the next weeks, across local authorities in Great Britain. The codebase updating the web site can be found at https://github.com/oxcsml/Rmap. We hope that our methodology and web site will be of interest to researchers, policy-makers and the public alike, to help identify upcoming local outbreaks and to aid in the containment of Covid-19 through both public health measures and personal decisions taken by the general public. DATAOur model is applied to publicly available daily counts of positive test results reported under the combined Pillars 1 (NHS and PHE) and 2 (commercial partners) of the UK's Covid-19 testing strategy. 1 The data are available for 312 lower-tier local authorities (LTLAs) in England, 14 NHS Health Boards in Scotland (each covering multiple local authorities) and 22 unitary local authorities in Wales, for a total of n = 348 local areas. The data are daily counts of lab-confirmed (PCR swab) cases presented by specimen date, starting from 30 January 2020. The original data are from the respective national public health authorities of England 2 , Scotland 3 and Wales 4 and we access them through the DELVE Global Covid-19 Dataset 5 (Bhoopchand et al., 2020). Due to delays in processing tests, we ignore the last 7 days of case counts. METHODOur method is based on an approach to infectious disease modelling using discrete renewal processes. These have a long history, and have served as the basis for a number of recent studies estimating instantaneous reproduction numbers, (Cori et al., 2013;Flaxman et al., 2020;Fraser, 2007; Wallinga & Teunis, 2004). See Bhatt et al. ( 2020) and references therein for historical and mathematical background, as well as Gostic et al. ( 2020) for important practical considerations.Following Flaxman et al. (2020), we model latent time series of incidence rates via renewal processes, and separate observations of reported cases using negative binomial distributions, to account for uncertainties in case reporting, outliers in case counts, and delays between infection and testing. We introduce a number of extensions and differences addressing issues that arise for applications to modelling epidemics at local authority level rather than regional 1 https://www.gov.uk/government/publications/coronavirus-covid-19-scaling-up-testing-programmes 2 https://coronavirus.data.gov.uk 3 https://publichealthscotland.scot/our-areas-of-work/sharing-our-data-and-intelligence/coronavirus-covid-19-dataand-guidance/ 4 https://phw.nhs.wal...

We use an individual-level transmission and contact simulation model to explore the effectiveness and resource requirements of various test-trace-isolate (TTI) strategies for reducing the spread of SARS-CoV-2 in the UK, in the context of different scenarios with varying levels of stringency of non-pharmaceutical interventions. Based on modelling results, we show that self-isolation of symptomatic individuals and quarantine of their household contacts has a substantial impact on the number of new infections generated by each primary case. We further show that adding contact tracing of non-household contacts of confirmed cases to this broader package of interventions reduces the number of new infections otherwise generated by 5–15%. We also explore impact of key factors, such as tracing application adoption and testing delay, on overall effectiveness of TTI.