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Autoencoders in function space in JMLR
The article “Autoencoders in function space” by Justin Bunker, Mark Girolami, Hefin Lambley, Andrew Stuart and myself has just appeared in its final form in the Journal of Machine Learning Research. This article continues one of the main themes of my and collaborators' work, namely that powerful discretisation-invariant learning methods can be obtained by examining the problem in an infinite-dimensional function space instead of on a fixed grid.
Abstract. Autoencoders have found widespread application in both their original deterministic form and in their variational formulation (VAEs). In scientific applications and in image processing it is often of interest to consider data that are viewed as functions; while discretisation (of differential equations arising in the sciences) or pixellation (of images) renders problems finite dimensional in practice, conceiving first of algorithms that operate on functions, and only then discretising or pixellating, leads to better algorithms that smoothly operate between resolutions. In this paper function-space versions of the autoencoder (FAE) and variational autoencoder (FVAE) are introduced, analysed, and deployed. Well-definedness of the objective governing VAEs is a subtle issue, particularly in function space, limiting applicability. For the FVAE objective to be well defined requires compatibility of the data distribution with the chosen generative model; this can be achieved, for example, when the data arise from a stochastic differential equation, but is generally restrictive. The FAE objective, on the other hand, is well defined in many situations where FVAE fails to be. Pairing the FVAE and FAE objectives with neural operator architectures that can be evaluated on any mesh enables new applications of autoencoders to inpainting, superresolution, and generative modelling of scientific data.
Published on Sunday 7 September 2025 at 12:00 UTC #publication #jmlr #bunker #girolami #lambley #stuart #autoencoders
Autoencoders in function space
Justin Bunker, Mark Girolami, Hefin Lambley, Andrew Stuart and I have just uploaded a preprint of our paper “Autoencoders in function space” to the arXiv.
Abstract. Autoencoders have found widespread application, in both their original deterministic form and in their variational formulation (VAEs). In scientific applications it is often of interest to consider data that are comprised of functions; the same perspective is useful in image processing. In practice, discretisation (of differential equations arising in the sciences) or pixellation (of images) renders problems finite dimensional, but conceiving first of algorithms that operate on functions, and only then discretising or pixellating, leads to better algorithms that smoothly operate between different levels of discretisation or pixellation. In this paper function-space versions of the autoencoder (FAE) and variational autoencoder (FVAE) are introduced, analysed, and deployed. Well-definedness of the objective function governing VAEs is a subtle issue, even in finite dimension, and more so on function space. The FVAE objective is well defined whenever the data distribution is compatible with the chosen generative model; this happens, for example, when the data arise from a stochastic differential equation. The FAE objective is valid much more broadly, and can be straightforwardly applied to data governed by differential equations. Pairing these objectives with neural operator architectures, which can thus be evaluated on any mesh, enables new applications of autoencoders to inpainting, superresolution, and generative modelling of scientific data.
Published on Monday 5 August 2024 at 12:00 UTC #preprint #bunker #girolami #lambley #stuart #autoencoders