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High Dimensional Probability VII
This volume collects selected papers from the 7th High Dimensional Probability meeting held at the Institut d'Études Scientifiques de Cargèse (IESC) in Corsica, France. High Dimensional Probability (HDP) is an area of mathematics that includes the study of probability distributions and limit theorems in infinite-dimensional spaces such as Hilbert spaces and Banach spaces. The most remarkable feature of this area is that it has resulted in the creation of powerful new tools and perspectives, whose range of application has led to interactions with other subfields of mathematics, statistics, and computer science. These include random matrices, nonparametric statistics, empirical processes, statistical learning theory, concentration of measure phenomena, strong and weak approximations, functional estimation, combinatorial optimization, and random graphs. The contributions in this volume show that HDP theory continues to thrive and develop new tools, methods, techniques and perspectives to analyze random phenomena.
High-Dimensional Probability
An integrated package of powerful probabilistic tools and key applications in modern mathematical data science.
Finite Frame Theory: A Complete Introduction to Overcompleteness
Frames are overcomplete sets of vectors that can be used to stably and faithfully decompose and reconstruct vectors in the underlying vector space. Frame theory stands at the intersection of many areas in mathematics such as functional and harmonic analysis, numerical analysis, matrix theory, numerical linear algebra, algebraic and differential geometry, probability, statistics, and convex geometry. At the same time its applications in engineering, medicine, computer science, and quantum computing are motivating new research problems in applied and pure mathematics. This volume is based on lectures delivered at the 2015 AMS Short Course “Finite Frame Theory: A Complete Introduction to Over...
Multiscale Analysis and Nonlinear Dynamics
Since modeling multiscale phenomena in systems biology and neuroscience is a highly interdisciplinary task, the editor of the book invited experts in bio-engineering, chemistry, cardiology, neuroscience, computer science, and applied mathematics, to provide their perspectives. Each chapter is a window into the current state of the art in the areas of research discussed and the book is intended for advanced researchers interested in recent developments in these fields. While multiscale analysis is the major integrating theme of the book, its subtitle does not call for bridging the scales from genes to behavior, but rather stresses the unifying perspective offered by the concepts referred to in the title. It is believed that the interdisciplinary approach adopted here will be beneficial for all the above mentioned fields.
Sparse and Redundant Representations
A long long time ago, echoing philosophical and aesthetic principles that existed since antiquity, William of Ockham enounced the principle of parsimony, better known today as Ockham’s razor: “Entities should not be multiplied without neces sity. ” This principle enabled scientists to select the ”best” physical laws and theories to explain the workings of the Universe and continued to guide scienti?c research, leadingtobeautifulresultsliketheminimaldescriptionlength approachtostatistical inference and the related Kolmogorov complexity approach to pattern recognition. However, notions of complexity and description length are subjective concepts anddependonthelanguage“spoken”when...
Engineering Mathematics and Artificial Intelligence
Explains the theory behind Machine Learning and highlights how Mathematics can be used in Artificial Intelligence Illustrates how to improve existing algorithms by using advanced mathematics and discusses how Machine Learning can support mathematical modeling Captures how to simulate data by means of artificial neural networks and offers cutting-edge Artificial Intelligence technologies Emphasizes the classification of algorithms, optimization methods, and statistical techniques Explores future integration between Machine Learning and complex mathematical techniques
An Introduction to Matrix Concentration Inequalities
Random matrices now play a role in many areas of theoretical, applied, and computational mathematics. It is therefore desirable to have tools for studying random matrices that are flexible, easy to use, and powerful. Over the last fifteen years, researchers have developed a remarkable family of results, called matrix concentration inequalities, that achieve all of these goals. This monograph offers an invitation to the field of matrix concentration inequalities. It begins with some history of random matrix theory; it describes a flexible model for random matrices that is suitable for many problems; and it discusses the most important matrix concentration results. To demonstrate the value of these techniques, the presentation includes examples drawn from statistics, machine learning, optimization, combinatorics, algorithms, scientific computing, and beyond.
Mathematical Analysis of Machine Learning Algorithms
The mathematical theory of machine learning not only explains the current algorithms but can also motivate principled approaches for the future. This self-contained textbook introduces students and researchers of AI to the main mathematical techniques used to analyze machine learning algorithms, with motivations and applications. Topics covered include the analysis of supervised learning algorithms in the iid setting, the analysis of neural networks (e.g. neural tangent kernel and mean-field analysis), and the analysis of machine learning algorithms in the sequential decision setting (e.g. online learning, bandit problems, and reinforcement learning). Students will learn the basic mathematical tools used in the theoretical analysis of these machine learning problems and how to apply them to the analysis of various concrete algorithms. This textbook is perfect for readers who have some background knowledge of basic machine learning methods, but want to gain sufficient technical knowledge to understand research papers in theoretical machine learning.
Mathematical Aspects of Deep Learning
A mathematical introduction to deep learning, written by a group of leading experts in the field.
