Jitse NiesenI recently read two interesting survey articles by my academic brother Ben Adcock at Simon Fraser University about theoretical aspect of sampling: how to approximate a function π given random point samples π(π₯α΅’) with noise. This is a fundamental problem in Machine Learning.
The first paper, "Learning smooth functions in high dimensions: from sparse polynomials to deep neural networks" (by Ben and co-authors), is about how fast the approximation error may decrease as you take more samples. We can overcome the curse of dimensionality if the function gets increasingly smooth in higher dimensions. URL: https://arxiv.org/abs/2404.03761
The second paper, "Optimal sampling for least-squares approximation", is about choosing where to sample in order to get as close to the unknown function (in least-square sense) as possible. https://arxiv.org/abs/2409.02342
Learning smooth functions in high dimensions: from sparse polynomials to deep neural networks
Learning approximations to smooth target functions of many variables from finite sets of pointwise samples is an important task in scientific computing and its many applications in computational science and engineering. Despite well over half a century of research on high-dimensional approximation, this remains a challenging problem. Yet, significant advances have been made in the last decade towards efficient methods for doing this, commencing with so-called sparse polynomial approximation methods and continuing most recently with methods based on Deep Neural Networks (DNNs). In tandem, there have been substantial advances in the relevant approximation theory and analysis of these techniques. In this work, we survey this recent progress. We describe the contemporary motivations for this problem, which stem from parametric models and computational uncertainty quantification; the relevant function classes, namely, classes of infinite-dimensional, Banach-valued, holomorphic functions; fundamental limits of learnability from finite data for these classes; and finally, sparse polynomial and DNN methods for efficiently learning such functions from finite data. For the latter, there is currently a significant gap between the approximation theory of DNNs and the practical performance of deep learning. Aiming to narrow this gap, we develop the topic of practical existence theory, which asserts the existence of dimension-independent DNN architectures and training strategies that achieve provably near-optimal generalization errors in terms of the amount of training data.
