An Approximation Theory for Metric Space-Valued Functions With A View Towards Deep Learning

• URL: http://arxiv.org/abs/2304.12231v2
• Date: Mon, 24 Jul 2023 16:00:37 GMT
• Title: An Approximation Theory for Metric Space-Valued Functions With A View Towards Deep Learning
• Authors: Anastasis Kratsios, Chong Liu, Matti Lassas, Maarten V. de Hoop, Ivan Dokmani\'c
• Abstract summary: We build universal functions approximators of continuous maps between arbitrary Polish metric spaces $mathcalX$ and $mathcalY$. In particular, we show that the required number of Dirac measures is determined by the structure of $mathcalX$ and $mathcalY$.
• Score: 25.25903127886586
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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