Modelling High-Dimensional Data with Likelihood-Based Generative Models

Deep generative models are a class of techniques that use neural networks to form a probabilistic model of training data so that new samples from the data distribution can be generated. This type of learning method necessitates greater level of understanding of the underlying data than common supervised learning approaches. As such, generative models have proven to be useful for much more than simply generation, with applications including, but far from limited to, segmentation, semantic correspondence, classification, image translation, representation learning, solving inverse and ill-posed problems, disentanglement, and out of distribution detection. This thesis explores how to efficiently scale deep generative models to very high dimensional data in order to aid application to cases where fine details are crucial to capture, as well as data such as 3D models and video. A comprehensive review is carried out, from fundamentals to recent state-of-the-art approaches, to understand the properties of different classes of approaches, diagnose what makes scaling difficult, and explore how techniques can be combined to trade off speed, quality, and diversity. Following this, three new generative modelling approaches are introduced, each with a different angle to enable greater scaling. The first approach enables greater scaling by representing samples as continuous functions thereby allowing arbitrary resolution data to be modelled; this is made possible by using latent conditional neural networks to directly map coordinates to content. The second approach compresses data to a highly informative discrete space then models this space with a powerful unconstrained discrete diffusion process, thereby improving sample quality over the first method while allowing faster sampling and better scaling than comparable discrete methods. The final approach extends diffusion models to infinite dimensional spaces, combining the advantages of the first two approaches to allow diverse, high quality samples, at arbitrary resolutions. Based on these findings and current research, the thesis closes by discussing the state of generative models, future research directions, and ethical considerations of the field.