{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/374898"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/374898","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"Probabilistic Modelling in Function Space","abstract":"Gaussian processes have established themselves as powerful tools for inferring functions from data. They provide a flexible framework for defining distributions over functions, enabling closed form solutions and principled handling of uncertainty. However, their application is often hindered by the difficulty of formulating suitable priors through analytical covariance functions and ensuring computational scalability. In contrast, deep generative models, while not offering closed form solutions, excel at learning data-driven priors and inherently scale to large datasets using neural networks. 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We demonstrate the potential of this approach as a viable alternative to Gaussian processes for complex data, especially when ample data is available to learn suitable priors. In the final chapter, we introduce geometric structure into these functional diffusion models. We show that this is crucial for tackling real-world scientific problems, particularly those dealing with non-Euclidean spaces and symmetries, which we exemplify by scenarios such as tracking cyclones on the Earth's surface.","abstract_html":"Gaussian processes have established themselves as powerful tools for inferring functions from data. They provide a flexible framework for defining distributions over functions, enabling closed form solutions and principled handling of uncertainty. However, their application is often hindered by the difficulty of formulating suitable priors through analytical covariance functions and ensuring computational scalability. In contrast, deep generative models, while not offering closed form solutions, excel at learning data-driven priors and inherently scale to large datasets using neural networks. This thesis aims to bridge the gap between the principled but computationally expensive Gaussian process approaches and the more flexible, yet less transparent, deep generative model approaches. The first contribution is an in-depth analysis of Gaussian processes on the hypersphere, an idea inspired by infinite-width neural networks. This leads to a class of kernels, known as zonal kernels, and a linear-time Gaussian process method that address the scalability concerns. Building on this, the second contribution introduces a compositional Gaussian process model that mirrors the structure of deep neural networks. This endows Gaussian processes with the ability to learn features from data, akin to neural networks, while retaining its principled uncertainty estimates. 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