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Incorporating group symmetry directly into the learning process has proved to be an effective guideline for model design. By producing features that are guaranteed to transform covariantly to the group actions on the inputs, group-equivariant convolutional neural net- works (G-CNNs) achieve significantly improved generalization performance in learning tasks with intrinsic symmetry. General theory and practical implementation of G-CNNs have been studied for planar images under either rotation or scaling transformation, but only individu- ally. We present, in this paper, a roto-scale-translation equivariant CNN (RST-CNN), that is guaranteed to achieve equivariance jointly over these three groups via coupled group con- volutions. Moreover, as symmetry transformations in reality are rarely perfect and typically subject to input deformation, we provide a stability analysis of the equivariance of representation to input distortion, which motivates the truncated expansion of the convolutional filters under (pre-fixed) low-frequency spatial modes. The resulting model provably achieves deformation-robust RS T equivariance, i.e., the RST symmetry is still “approximately” preserved when the transformation is “contaminated” by a nuisance data deformation, a property that is especially important for out-of-distribution generalization. Numerical experiments on MNIST, Fashion-MNIST, and STL-10 demonstrate that the proposed model yields remarkable gains over prior arts, especially in the small data regime where both rotation and scaling variations are present within the data. 

Incorporating group symmetry directly into the learning process has proved to be an effective guideline for model design. By producing features that are guaranteed to transform covariantly to the group actions on the inputs, group-equivariant convolutional neural networks (G-CNNs) achieve significantly improved generalization performance in learning tasks with intrinsic symmetry. General theory and practical implementation of G-CNNs have been studied for planar images under either rotation or scaling transformation, but only individually. We present, in this paper, a roto-scale-translation equivariant CNN (RST-CNN), that is guaranteed to achieve equivariance jointly over these three groups via coupled group convolutions. Moreover, as symmetry transformations in reality are rarely perfect and typically subject to input deformation, we provide a stability analysis of the equivariance of representation to input distortion, which motivates the truncated expansion of the convolutional filters under (pre-fixed) low-frequency spatial modes. The resulting model provably achieves deformation-robust RST-equivariance, i.e., the RST-symmetry is still "approximately” preserved when the transformation is "contaminated” by a nuisance data deformation, a property that is especially important for out-of-distribution generalization. Numerical experiments on MNIST, Fashion-MNIST, and STL-10 demonstrate that the proposed model yields remarkable gains over prior arts, especially in the small data regime where both rotation and scaling variations are present within the data. 

Background: COVID-19 vaccines have been approved and made available. While questions of vaccine allocation strategies have received significant attention, important questions remain regarding the potential impact of the vaccine given uncertainties regarding efficacy against transmission, availability, timing, and durability. Methods: We adapted a susceptible-exposed-infectious-recovered (SEIR) model to examine the potential impact on hospitalization and mortality assuming increasing rates of vaccine efficacy, coverage, and administration. We also evaluated the uncertainty of the vaccine to prevent infectiousness as well as the impact on outcomes based on the timing of distribution and the potential effects of waning immunity. Findings: Increased vaccine efficacy against disease reduces hospitalizations and deaths from COVID-19; however, the relative benefit of transmission blocking varied depending on the timing of vaccine distribution. Early in an outbreak, a vaccine that reduces transmission will be relatively more effective than one introduced later in the outbreak. In addition, earlier and accelerated implementation of a less effective vaccine is more impactful than later implementation of a more effective vaccine. These findings are magnified when considering the durability of the vaccine. Vaccination in the spring will be less impactful when immunity is less durable. Interpretation: Policy choices regarding non-pharmaceutical interventions, such as social distancing and face mask use, will need to remain in place longer if the vaccine is less effective at reducing transmission or distributed slower. In addition, the stage of the local outbreak greatly impacts the overall effectiveness of the vaccine in a region and should be considered when allocating vaccines.