A Multi-Resolution Framework for U-Nets with Applications to Hierarchical VAEs

• URL: http://arxiv.org/abs/2301.08187v1
• Date: Thu, 19 Jan 2023 17:33:48 GMT
• Title: A Multi-Resolution Framework for U-Nets with Applications to Hierarchical VAEs
• Authors: Fabian Falck, Christopher Williams, Dominic Danks, George Deligiannidis, Christopher Yau, Chris Holmes, Arnaud Doucet, Matthew Willetts
• Abstract summary: We formulate a multi-resolution framework which identifies U-Nets as finite-dimensional truncations of models on an infinite-dimensional function space. We then leverage our framework to identify state-of-the-art hierarchical VAEs (HVAEs) which have a U-Net architecture.
• Score: 29.995904718691204
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 a type of two-step forward Euler discretisation 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.

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