Recent years have witnessed a growth of interest in the special functions called ridge functions. These functions appear in various fields and under various guises. They appear in partial differential equations (where they are called plane waves), in computerized tomography, and in statistics. Ridge functions are also the underpinnings of many central models in neural network theory. In this book various approximation theoretic properties of ridge functions are described. This book also describes properties of generalized ridge functions, and their relation to linear superpositions and Kolmogorov's famous superposition theorem. In the final part of the book, a single and two hidden layer neural networks are discussed. The results obtained in this part are based on properties of ordinary and generalized ridge functions. Novel aspects of the universal approximation property of feedforward neural networks are revealed. This book will be of interest to advanced graduate students and researchers working in functional analysis, approximation theory, and the theory of real functions, and will be of particular interest to those wishing to learn more about neural network theory and applications and other areas where ridge functions are used.
Author(s): Vugar E. Ismailov
Series: Mathematical Surveys and Monographs, 263
Publisher: American Mathematical Society
Year: 2021
Language: English
Pages: 197
City: Providence
Cover
Title page
Preface
Introduction
Chapter 1. Properties of linear combinations of ridge functions
1.1. A brief excursion into the approximation theory of ridge functions
1.2. Representation of multivariate functions by linear combinations of ridge functions
1.3. Characterization of an extremal sum of ridge functions
1.4. Sums of continuous ridge functions
1.5. On the proximinality of ridge functions
1.6. On the approximation by weighted ridge functions
Chapter 2. The smoothness problem in ridge function representation
2.1. A solution to the problem up to a multivariate polynomial
2.2. A solution to the smoothness problem under certain conditions
2.3. A constructive analysis of the smoothness problem
Chapter 3. Approximation of multivariate functions by sums of univariate functions
3.1. Characterization of some bivariate function classes by formulas for the error of approximation
3.2. Approximation by sums of univariate functions on certain domains
3.3. On the theorem of M. Golomb
Chapter 4. Generalized ridge functions and linear superpositions
4.1. Representation theorems
4.2. Uniqueness theorems
4.3. An approximation algorithm
Chapter 5. Applications to neural networks
5.1. Universal approximation property of neural networks
5.2. Single hidden layer neural networks with restricted weights
5.3. Two hidden layer neural networks
5.4. Construction of a universal sigmoidal function
Bibliography
Index
Back Cover
