Denoising Diffusion Probabilistic Models (DDPMs) have accomplished much in the realm of generative AI. Despite their high performance, there is room for improvement, especially in terms of sample fidelity by utilizing statistical properties that impose structural integrity, such as isotropy. Minimizing the mean squared error between the additive and predicted noise alone does not impose constraints on the predicted noise to be isotropic. Thus, we were motivated to utilize the isotropy of the additive noise as a constraint on the objective function to enhance the fidelity of DDPMs. Our approach is simple and can be applied to any DDPM variant. We validate our approach by presenting experiments conducted on four synthetic 2D datasets as well as on unconditional image generation. As demonstrated by the results, the incorporation of this constraint improves the fidelity metrics, Precision and Density for the 2D datasets as well as for the unconditional image generation.

COVID-19 continues to cause a significant impact on public health. To minimize this impact, policy makers undertake containment measures that however, when carried out disproportionately to the actual threat, as a result if errorneous threat assessment, cause undesirable long-term socio-economic complications. In addition, macro-level or national level decision making fails to consider the localized sensitivities in small regions. Hence, the need arises for region-wise threat assessments that provide insights on the behaviour of COVID-19 through time, enabled through accurate forecasts. In this study, a forecasting solution is proposed, to predict daily new cases of COVID-19 in regions small enough where containment measures could be locally implemented, by targeting three main shortcomings that exist in literature; the unreliability of existing data caused by inconsistent testing patterns in smaller regions, weak deploy-ability of forecasting models towards predicting cases in previously unseen regions, and model training biases caused by the imbalanced nature of data in COVID-19 epi-curves. Hence, the contributions of this study are three-fold; an optimized smoothing technique to smoothen less deterministic epi-curves based on epidemiological dynamics of that region, a Long-Short-Term-Memory (LSTM) based forecasting model trained using data from select regions to create a representative and diverse training set that maximizes deploy-ability in regions with lack of historical data, and an adaptive loss function whilst training to mitigate the data imbalances seen in epi-curves. The proposed smoothing technique, the generalized training strategy and the adaptive loss function largely increased the overall accuracy of the forecast, which enables efficient containment measures at a more localized micro-level.