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- W4367000465 abstract "Motivated by the developing mathematics of deep learning, we build universal functions approximators of continuous maps between arbitrary Polish metric spaces $mathcal{X}$ and $mathcal{Y}$ using elementary functions between Euclidean spaces as building blocks. Earlier results assume that the target space $mathcal{Y}$ is a topological vector space. We overcome this limitation by ``randomization'': our approximators output discrete probability measures over $mathcal{Y}$. When $mathcal{X}$ and $mathcal{Y}$ are Polish without additional structure, we prove very general qualitative guarantees; when they have suitable combinatorial structure, we prove quantitative guarantees for H{o}lder-like maps, including maps between finite graphs, solution operators to rough differential equations between certain Carnot groups, and continuous non-linear operators between Banach spaces arising in inverse problems. In particular, we show that the required number of Dirac measures is determined by the combinatorial structure of $mathcal{X}$ and $mathcal{Y}$. For barycentric $mathcal{Y}$, including Banach spaces, $mathbb{R}$-trees, Hadamard manifolds, or Wasserstein spaces on Polish metric spaces, our approximators reduce to $mathcal{Y}$-valued functions. When the Euclidean approximators are neural networks, our constructions generalize transformer networks, providing a new probabilistic viewpoint of geometric deep learning." @default.
- W4367000465 created "2023-04-27" @default.
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- W4367000465 date "2023-04-24" @default.
- W4367000465 modified "2023-10-18" @default.
- W4367000465 title "An Approximation Theory for Metric Space-Valued Functions With A View Towards Deep Learning" @default.
- W4367000465 doi "https://doi.org/10.48550/arxiv.2304.12231" @default.
- W4367000465 hasPublicationYear "2023" @default.
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