Ridge Functions and Applications in Neural Networks

Ridge Functions and Applications in Neural Networks
Author: Vugar E. Ismailov
Publisher: American Mathematical Society
Total Pages: 186
Release: 2021-12-17
Genre: Mathematics
ISBN: 1470467658

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.

Ridge Functions

Ridge Functions
Author: Allan Pinkus
Publisher: Cambridge University Press
Total Pages: 218
Release: 2015-08-07
Genre: Computers
ISBN: 1107124395

Presents the state of the art in the theory of ridge functions, providing a solid theoretical foundation.

Machine Learning for the Physical Sciences

Machine Learning for the Physical Sciences
Author: Carlo Requião da Cunha
Publisher: CRC Press
Total Pages: 289
Release: 2023-12-05
Genre: Computers
ISBN: 1003821146

Machine learning is an exciting topic with a myriad of applications. However, most textbooks are targeted towards computer science students. This, however, creates a complication for scientists across the physical sciences that also want to understand the main concepts of machine learning and look ahead to applica- tions and advancements in their fields. This textbook bridges this gap, providing an introduction to the mathematical foundations for the main algorithms used in machine learning for those from the physical sciences, without a formal background in computer science. It demon- strates how machine learning can be used to solve problems in physics and engineering, targeting senior undergraduate and graduate students in physics and electrical engineering, alongside advanced researchers. Key Features: Includes detailed algorithms Supplemented by codes in Julia: a high-performing language and one that is easy to read for those in the natural sciences All algorithms are presented with a good mathematical background

Recent Advances in Harmonic Analysis and Applications

Recent Advances in Harmonic Analysis and Applications
Author: Dmitriy Bilyk
Publisher: Springer Science & Business Media
Total Pages: 400
Release: 2012-10-16
Genre: Mathematics
ISBN: 1461445647

Recent Advances in Harmonic Analysis and Applications features selected contributions from the AMS conference which took place at Georgia Southern University, Statesboro in 2011 in honor of Professor Konstantin Oskolkov's 65th birthday. The contributions are based on two special sessions, namely "Harmonic Analysis and Applications" and "Sparse Data Representations and Applications." Topics covered range from Banach space geometry to classical harmonic analysis and partial differential equations. Survey and expository articles by leading experts in their corresponding fields are included, and the volume also features selected high quality papers exploring new results and trends in Muckenhoupt-Sawyer theory, orthogonal polynomials, trigonometric series, approximation theory, Bellman functions and applications in differential equations. Graduate students and researchers in analysis will be particularly interested in the articles which emphasize remarkable connections between analysis and analytic number theory. The readers will learn about recent mathematical developments and directions for future work in the unexpected and surprising interaction between abstract problems in additive number theory and experimentally discovered optical phenomena in physics. This book will be useful for number theorists, harmonic analysts, algorithmists in multi-dimensional signal processing and experts in physics and partial differential equations.

Radial Basis Function Neural Networks with Sequential Learning

Radial Basis Function Neural Networks with Sequential Learning
Author: N. Sundararajan
Publisher: World Scientific
Total Pages: 236
Release: 1999
Genre: Science
ISBN: 9789810237714

A review of radial basis founction (RBF) neural networks. A novel sequential learning algorithm for minimal resource allocation neural networks (MRAN). MRAN for function approximation & pattern classification problems; MRAN for nonlinear dynamic systems; MRAN for communication channel equalization; Concluding remarks; A outline source code for MRAN in MATLAB; Bibliography; Index.

Theory of Ridge Regression Estimation with Applications

Theory of Ridge Regression Estimation with Applications
Author: A. K. Md. Ehsanes Saleh
Publisher: John Wiley & Sons
Total Pages: 384
Release: 2019-02-12
Genre: Mathematics
ISBN: 1118644611

A guide to the systematic analytical results for ridge, LASSO, preliminary test, and Stein-type estimators with applications Theory of Ridge Regression Estimation with Applications offers a comprehensive guide to the theory and methods of estimation. Ridge regression and LASSO are at the center of all penalty estimators in a range of standard models that are used in many applied statistical analyses. Written by noted experts in the field, the book contains a thorough introduction to penalty and shrinkage estimation and explores the role that ridge, LASSO, and logistic regression play in the computer intensive area of neural network and big data analysis. Designed to be accessible, the book presents detailed coverage of the basic terminology related to various models such as the location and simple linear models, normal and rank theory-based ridge, LASSO, preliminary test and Stein-type estimators. The authors also include problem sets to enhance learning. This book is a volume in the Wiley Series in Probability and Statistics series that provides essential and invaluable reading for all statisticians. This important resource: Offers theoretical coverage and computer-intensive applications of the procedures presented Contains solutions and alternate methods for prediction accuracy and selecting model procedures Presents the first book to focus on ridge regression and unifies past research with current methodology Uses R throughout the text and includes a companion website containing convenient data sets Written for graduate students, practitioners, and researchers in various fields of science, Theory of Ridge Regression Estimation with Applications is an authoritative guide to the theory and methodology of statistical estimation.

Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic $K$-Theory

Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic $K$-Theory
Author: Donald Yau
Publisher: American Mathematical Society
Total Pages: 555
Release: 2024-10-08
Genre: Mathematics
ISBN: 1470478099

Bimonoidal categories are categorical analogues of rings without additive inverses. They have been actively studied in category theory, homotopy theory, and algebraic $K$-theory since around 1970. There is an abundance of new applications and questions of bimonoidal categories in mathematics and other sciences. The three books published by the AMS in the Mathematical Surveys and Monographs series under the general title Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic $K$-Theory (Volume I: Symmetric Bimonoidal Categories and Monoidal Bicategories?this book, Volume II: Braided Bimonoidal Categories with Applications, and Volume III: From Categories to Structured Ring Spectra) provide a unified treatment of bimonoidal and higher ring-like categories, their connection with algebraic $K$-theory and homotopy theory, and applications to quantum groups and topological quantum computation. With ample background material, extensive coverage, detailed presentation of both well-known and new theorems, and a list of open questions, this work is a user-friendly resource for beginners and experts alike. Part 1 of this book proves in detail Laplaza's two coherence theorems and May's strictification theorem of symmetric bimonoidal categories, as well as their bimonoidal analogues. This part includes detailed corrections to several inaccurate statements and proofs found in the literature. Part 2 proves Baez's Conjecture on the existence of a bi-initial object in a 2-category of symmetric bimonoidal categories. The next main theorem states that a matrix construction, involving the matrix product and the matrix tensor product, sends a symmetric bimonoidal category with invertible distributivity morphisms to a symmetric monoidal bicategory, with no strict structure morphisms in general.

Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic $K$-Theory

Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic $K$-Theory
Author: Niles Johnson
Publisher: American Mathematical Society
Total Pages: 633
Release: 2024-10-23
Genre: Mathematics
ISBN: 1470478110

Bimonoidal categories are categorical analogues of rings without additive inverses. They have been actively studied in category theory, homotopy theory, and algebraic $K$-theory since around 1970. There is an abundance of new applications and questions of bimonoidal categories in mathematics and other sciences. The three books published by the AMS in the Mathematical Surveys and Monographs series under the title Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic $K$-Theory (Volume I: Symmetric Bimonoidal Categories and Monoidal Bicategories, Volume II: Braided Bimonoidal Categories with Applications, and Volume III: From Categories to Structured Ring Spectra?this book) provide a unified treatment of bimonoidal and higher ring-like categories, their connection with algebraic $K$-theory and homotopy theory, and applications to quantum groups and topological quantum computation. With ample background material, extensive coverage, detailed presentation of both well-known and new theorems, and a list of open questions, this work is a user-friendly resource for beginners and experts alike. Part 1 of this book is a detailed study of enriched monoidal categories, pointed diagram categories, and enriched multicategories. Using this machinery, Part 2 discusses the rich interconnection between the higher ring-like categories, homotopy theory, and algebraic $K$-theory. Starting with a chapter on homotopy theory background, the first half of Part 2 constructs the Segal $K$-theory functor and the Elmendorf-Mandell $K$-theory multifunctor from permutative categories to symmetric spectra. For the latter, the detailed treatment here includes identification and correction of some subtle errors concerning its extended domain. The second half applies the $K$-theory multifunctor to small ring, bipermutative, braided ring, and $E_n$-monoidal categories to obtain, respectively, strict ring, $E_{infty}$-, $E_2$-, and $E_n$-symmetric spectra.

Multivariate Approximation and Splines

Multivariate Approximation and Splines
Author: Günther Nürnberger
Publisher: Birkhäuser
Total Pages: 329
Release: 2012-12-06
Genre: Mathematics
ISBN: 3034888716

This book contains the refereed papers which were presented at the interna tional conference on "Multivariate Approximation and Splines" held in Mannheim, Germany, on September 7-10,1996. Fifty experts from Bulgaria, England, France, Israel, Netherlands, Norway, Poland, Switzerland, Ukraine, USA and Germany participated in the symposium. It was the aim of the conference to give an overview of recent developments in multivariate approximation with special emphasis on spline methods. The field is characterized by rapidly developing branches such as approximation, data fit ting, interpolation, splines, radial basis functions, neural networks, computer aided design methods, subdivision algorithms and wavelets. The research has applications in areas like industrial production, visualization, pattern recognition, image and signal processing, cognitive systems and modeling in geology, physics, biology and medicine. In the following, we briefly describe the contents of the papers. Exact inequalities of Kolmogorov type which estimate the derivatives of mul the paper of BABENKO, KOFANovand tivariate periodic functions are derived in PICHUGOV. These inequalities are applied to the approximation of classes of mul tivariate periodic functions and to the approximation by quasi-polynomials. BAINOV, DISHLIEV and HRISTOVA investigate initial value problems for non linear impulse differential-difference equations which have many applications in simulating real processes. By applying iterative techniques, sequences of lower and upper solutions are constructed which converge to a solution of the initial value problem.

Trees of Hyperbolic Spaces

Trees of Hyperbolic Spaces
Author: Michael Kapovich
Publisher: American Mathematical Society
Total Pages: 295
Release: 2024-08-15
Genre: Mathematics
ISBN: 1470474255

This book offers an alternative proof of the Bestvina?Feighn combination theorem for trees of hyperbolic spaces and describes uniform quasigeodesics in such spaces. As one of the applications of their description of uniform quasigeodesics, the authors prove the existence of Cannon?Thurston maps for inclusion maps of total spaces of subtrees of hyperbolic spaces and of relatively hyperbolic spaces. They also analyze the structure of Cannon?Thurston laminations in this setting. Furthermore, some group-theoretic applications of these results are discussed. This book also contains background material on coarse geometry and geometric group theory.