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Found 7 papers, 2 papers with code
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.
A unified framework for learning with nonlinear model classes from arbitrary linear samples
In summary, our work not only introduces a unified way to study learning unknown objects from general types of data, but also establishes a series of general theoretical guarantees which consolidate and improve various known results.
CS4ML: A general framework for active learning with arbitrary data based on Christoffel functions
Our framework extends the standard setup by allowing for general types of data, rather than merely pointwise samples of the target function.
CAS4DL: Christoffel Adaptive Sampling for function approximation via Deep Learning
In this work, we propose an adaptive sampling strategy, CAS4DL (Christoffel Adaptive Sampling for Deep Learning) to increase the sample efficiency of DL for multivariate function approximation.
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.
The gap between theory and practice in function approximation with deep neural networks
Our main conclusion from these experiments is that there is a crucial gap between the approximation theory of DNNs and their practical performance, with trained DNNs performing relatively poorly on functions for which there are strong approximation results (e. g. smooth functions), yet performing well in comparison to best-in-class methods for other functions.
