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A Multi-Resolution Framework for U-Nets with Applications to Hierarchical VAEs
Fabian Falck · Christopher Williams · Dominic Danks · George Deligiannidis · Christopher Yau · Chris C Holmes · Arnaud Doucet · Matthew Willetts

Wed Nov 30 02:00 PM -- 04:00 PM (PST) @ Hall J #508

U-Net architectures are ubiquitous in state-of-the-art deep learning, however their regularisation properties and relationship to wavelets are understudied. In this paper, we formulate a multi-resolution framework which identifies U-Nets as finite-dimensional truncations of models on an infinite-dimensional function space. We provide theoretical results which prove that average pooling corresponds to projection within the space of square-integrable functions and show that U-Nets with average pooling implicitly learn a Haar wavelet basis representation of the data. We then leverage our framework to identify state-of-the-art hierarchical VAEs (HVAEs), which have a U-Net architecture, as forward Euler discretisations of multi-resolution diffusion processes which flow from a point mass, introducing sampling instabilities. We also demonstrate that HVAEs learn a representation of time which allows for improved parameter efficiency through weight-sharing. We use this observation to achieve state-of-the-art HVAE performance with half the number of parameters of existing models, exploiting the properties of our continuous-time formulation.

Author Information

Fabian Falck (University of Oxford)
Christopher Williams (University of Oxford)
Dominic Danks (University of Birmingham + Alan Turing Institute)
George Deligiannidis (Oxford)
Christopher Yau (University of Oxford)
Chris C Holmes (University of Oxford)
Arnaud Doucet (Oxford)
Matthew Willetts (University College London)

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