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Learning smooth functions in high dimensions: from sparse polynomials to deep neural networks
For the latter, there is currently a significant gap between the approximation theory of DNNs and the practical performance of deep learning.
On efficient algorithms for computing near-best polynomial approximations to high-dimensional, Hilbert-valued functions from limited samples
On the one hand, there is a well-developed theory of best $s$-term polynomial approximation, which asserts exponential or algebraic rates of convergence for holomorphic functions.
Deep Neural Networks Are Effective At Learning High-Dimensional Hilbert-Valued Functions From Limited Data
Such problems are challenging: 1) pointwise samples are expensive to acquire, 2) the function domain is high dimensional, and 3) the range lies in a Hilbert space.
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