Publisher Summary This chapter explains basic examples of wavelet methods in numerical
analysis. It introduces the approximations and shows show the way they are related to …

Deep learning has shown high performances in various types of tasks from visual
recognition to natural language processing, which indicates superior flexibility and adaptivity …

We derive fundamental lower bounds on the connectivity and the memory requirements of
deep neural networks guaranteeing uniform approximation rates for arbitrary function …

Weak Convergence Page 1 1.3 Weak Convergence In this section IDl and IE are metric spaces
with metrics d and e, respectively. The set of all continuous, bounded functions f: IDl 1--+ IR is …

Compressive sensing (CS) is an alternative to Shannon/Nyquist sampling for the acquisition
of sparse or compressible signals that can be well approximated by just K¿ N elements from …

Inspired by the recent work of Bertalmio et al. on digital inpaintings [SIGGRAPH 2000], we
develop general mathematical models for local inpaintings of nontexture images. On smooth …

A method for concise, faithful approximation of complex 3D datasets is key to reducing the
computational cost of graphics applications. Despite numerous applications ranging from …

This paper is a selective review of the regularization methods scattered in statistics literature.
We introduce a general conceptual approach to regularization and fit most existing methods …

We consider the quality of learning a response function by a nonparametric Bayesian
approach using a Gaussian process (GP) prior on the response function. We upper bound …

Cartoon-like images, ie, C2 functions which are smooth apart from a C2 discontinuity curve,
have by now become a standard model for measuring sparse (nonlinear) approximation …