High-dimensional Approximation: From Sparse Polynomials To Wavelets
Prof.
Francesca Pelosi
University of Siena - Dipartimento di Ingegneria dell'Informazione e Scienze Matematiche Giuseppe Alessio D'Inverno
SISSA , Trieste
Course Type
Type B
Calendar
Sept 8 h. 9.00-13.00 aula 456 Sept 9 h.9.00-11 aula 456 h. 11.00-13.00 lab. inf. 124 Sept 10 h. 9.00-13.00 aula 456 Sept 11 h. 9.00-13.00 aula 456 Sept 12 h. 9.00-13.00 lab. inf. 124
Room
Program
Abstract
Approximating functions of many variables from limited data is a crucial task in modern
scientific computing, machine learning, and main signal processing tasks such as signal compression or super-resolution. The goal of this course is to introduce recent approximation techniques, based on sparse polynomials and wavelets reconstruction.
In the first part of this course we will begin by explaining the fundamentals of sparse polynomial
approximation theory. Subsequently, we will introduce numerical methods for computing sparse
polynomial approximations from limited Monte Carlo samples. Our focus will be on techniques
based on least squares and compressed sensing, and we will discuss the circumstances under which they are provably able to mitigate the so-called curse of dimensionality. We will demonstrate how these ideas can be applied to deep learning theory and showcase what are known as practical existence theorems for deep neural networks. Additionally, we will present open problems and current research directions in the field.
In the second part of the course we will introduce the idea of wavelets, that decompose a
signal into approximations and details at different scales, focusing on applications such as data
compression, detecting features and removing noise from signals. We will explain some of the theory behind continuous, discrete and stationary wavelet transforms and we will develop multiscale algorithms for compression and signal/image processing. In addition we will give some insight on the wavelet scattering network that enables to derive, with minimal configuration, low-variance features from real-valued time series and image data for use in machine learning and deep learning applications.
