This book describes how neural networks operate from the mathematical point of view. As a result, neural networks can be interpreted both as function universal approximators and information processors. The book bridges the gap between ideas and concepts of neural networks, which are used nowadays at an intuitive level, and the precise modern mathematical language, presenting the best practices of the former and enjoying the robustness and elegance of the latter. This book can be used in a graduate course in deep learning, with the first few parts being accessible to senior undergraduates. In addition, the book will be of wide interest to machine learning researchers who are interested in a theoretical understanding of the subject.
This book covers topics of Informational Geometry, a field which deals with the differential geometric study of the manifold probability density functions. This is a field that is increasingly attracting the interest of researchers from many different areas of science, including mathematics, statistics, geometry, computer science, signal processing, physics and neuroscience. It is the authors' hope that the present book will be a valuable reference for researchers and graduate students in one of the aforementioned fields. This textbook is a unified presentation of differential geometry and probability theory, and constitutes a text for a course directed at graduate or advanced undergraduate students interested in applications of differential geometry in probability and statistics. The book contains over 100 proposed exercises meant to help students deepen their understanding, and it is accompanied by software that is able to provide numerical computations of several information geometric objects. The reader will understand a flourishing field of mathematics in which very few books have been written so far. Inhaltsverzeichnis Part I: The Geometry of Statistical Models.- Statistical Models.- Explicit Examples.- Entropy on Statistical Models.- Kullback Leibler Relative Entropy.- Informational Energy.- Maximum Entropy Distributions.- Part II: Statistical Manifolds.- An Introduction to Manifolds.- Dualistic Structure.- Dual Volume Elements.- Dual Laplacians.- Contrast Functions Geometry.- Contrast Functions on Statistical Models.- Statistical Submanifolds.- Appendix A: Information Geometry Calculator.
The book delves into the connections between Stochastic Analysis, Geometry, and Partial Differential Equations (PDEs), emphasizing how geometric structures like Riemannian and sub-Riemannian geometries influence diffusion processes. It explores the implications for solving PDEs and applications in mathematical finance, while unifying these disciplines through the bracket-generating condition, also known as Hörmander's condition. The primary objective is to highlight the shared conditions on vector fields that link PDEs, nonholonomic geometry, and stochastic processes.
Focusing on sub-Riemannian and Heisenberg manifolds, this comprehensive text presents a novel variational approach that blends theory with practical applications. It serves as both a detailed reference and an educational resource, emphasizing key concepts and methodologies in the field. The book is designed for researchers and graduate students, providing valuable insights into advanced mathematical frameworks and their implications in geometry and analysis.
Focusing on an introductory approach, this book presents Stochastic Calculus in a manner that mirrors elementary deterministic Calculus, making it accessible for students already familiar with basic calculus concepts. The author emphasizes clarity and understanding over intricate mathematical detail, aiming to ease the transition to more complex topics in Stochastic Calculus by retaining familiar properties from deterministic methods.
The book uniquely combines probabilistic methods and partial differential equations to effectively price derivatives under both constant and stochastic volatility models. This approach allows readers to explicitly compute a wide range of prices for European, American, and Asian derivatives, offering a comprehensive understanding of mathematical finance.
Focusing on various methodologies, this monograph provides an in-depth examination of theories for deriving explicit formulas for heat kernels associated with elliptic and sub-elliptic operators. Each chapter is dedicated to a different method, highlighting the diversity and richness of approaches available in this field of study.
Stochastic Calculus serves as a foundational tool for various scientific fields involving random fluctuations, such as signal processing, financial markets, and population dynamics. The book emphasizes the necessity of a solid mathematical background, particularly in probability, analysis, and measure theory, to effectively understand and apply these concepts. It acknowledges the complexity of Stochastic Calculus, highlighting the time and effort required to master its theoretical framework and practical applications across diverse disciplines.
This bestselling textbook for higher-level courses was extensively revised in 1990 to accommodate developments in model theoretic methods. Topics include models constructed from constants, ultraproducts, and saturated and special models. 1990 edition.
