CMUC


Book Description




Graph Representation Learning


Book Description

Graph-structured data is ubiquitous throughout the natural and social sciences, from telecommunication networks to quantum chemistry. Building relational inductive biases into deep learning architectures is crucial for creating systems that can learn, reason, and generalize from this kind of data. Recent years have seen a surge in research on graph representation learning, including techniques for deep graph embeddings, generalizations of convolutional neural networks to graph-structured data, and neural message-passing approaches inspired by belief propagation. These advances in graph representation learning have led to new state-of-the-art results in numerous domains, including chemical synthesis, 3D vision, recommender systems, question answering, and social network analysis. This book provides a synthesis and overview of graph representation learning. It begins with a discussion of the goals of graph representation learning as well as key methodological foundations in graph theory and network analysis. Following this, the book introduces and reviews methods for learning node embeddings, including random-walk-based methods and applications to knowledge graphs. It then provides a technical synthesis and introduction to the highly successful graph neural network (GNN) formalism, which has become a dominant and fast-growing paradigm for deep learning with graph data. The book concludes with a synthesis of recent advancements in deep generative models for graphs—a nascent but quickly growing subset of graph representation learning.




Real Analysis (Classic Version)


Book Description

This text is designed for graduate-level courses in real analysis. Real Analysis, 4th Edition, covers the basic material that every graduate student should know in the classical theory of functions of a real variable, measure and integration theory, and some of the more important and elementary topics in general topology and normed linear space theory. This text assumes a general background in undergraduate mathematics and familiarity with the material covered in an undergraduate course on the fundamental concepts of analysis.




Random Walks and Electric Networks


Book Description

Probability theory, like much of mathematics, is indebted to physics as a source of problems and intuition for solving these problems. Unfortunately, the level of abstraction of current mathematics often makes it difficult for anyone but an expert to appreciate this fact. Random Walks and electric networks looks at the interplay of physics and mathematics in terms of an example—the relation between elementary electric network theory and random walks —where the mathematics involved is at the college level.




Random Graphs and Complex Networks


Book Description

This classroom-tested text is the definitive introduction to the mathematics of network science, featuring examples and numerous exercises.




Mathematical Reviews


Book Description




Variational Analysis


Book Description

From its origins in the minimization of integral functionals, the notion of variations has evolved greatly in connection with applications in optimization, equilibrium, and control. This book develops a unified framework and provides a detailed exposition of variational geometry and subdifferential calculus in their current forms beyond classical and convex analysis. Also covered are set-convergence, set-valued mappings, epi-convergence, duality, and normal integrands.




Advanced Calculus (Revised Edition)


Book Description

An authorised reissue of the long out of print classic textbook, Advanced Calculus by the late Dr Lynn Loomis and Dr Shlomo Sternberg both of Harvard University has been a revered but hard to find textbook for the advanced calculus course for decades.This book is based on an honors course in advanced calculus that the authors gave in the 1960's. The foundational material, presented in the unstarred sections of Chapters 1 through 11, was normally covered, but different applications of this basic material were stressed from year to year, and the book therefore contains more material than was covered in any one year. It can accordingly be used (with omissions) as a text for a year's course in advanced calculus, or as a text for a three-semester introduction to analysis.The prerequisites are a good grounding in the calculus of one variable from a mathematically rigorous point of view, together with some acquaintance with linear algebra. The reader should be familiar with limit and continuity type arguments and have a certain amount of mathematical sophistication. As possible introductory texts, we mention Differential and Integral Calculus by R Courant, Calculus by T Apostol, Calculus by M Spivak, and Pure Mathematics by G Hardy. The reader should also have some experience with partial derivatives.In overall plan the book divides roughly into a first half which develops the calculus (principally the differential calculus) in the setting of normed vector spaces, and a second half which deals with the calculus of differentiable manifolds.




Physics Avoidance


Book Description

Mark Wilson presents a series of explorations of our strategies for understanding the world. "Physics avoidance" refers to the fact that we frequently cannot reason about nature in the straightforward manner we anticipate, but must seek alternative policies that allow us to address the questions we want answered in a tractable way. Within both science and everyday life, we find ourselves relying upon thought processes that reach useful answers in opaque and roundabout manners. Conceptual innovators are often puzzled by the techniques they develop, when they stumble across reasoning patterns that are easy to implement but difficult to justify. But simple techniques frequently rest upon complex foundations—a young magician learns how to execute a card-guessing trick without understanding how its progressive steps squeeze in on a proper answer. As we collectively improve our inferential skills in this gradually evolving manner, we often wander into unfamiliar explanatory landscapes in which simple words encode physical information in complex and unanticipated ways. Like our juvenile conjurer, we fail to recognize the true strategic rationales underlying our achievements and may turn instead to preposterous rationalizations for our policies. We have learned how to reach better conclusions in a more fruitful way, but we remain baffled by our own successes. At its best, philosophical reflection illuminates the natural developmental processes that generate these confusions and explicates their complexities. But current thinking within philosophy of science and language works to opposite effect by relying upon simplistic conceptions of "cause", "law of nature", "possibility", and "reference" that ignore the strategic complexities in which these concepts become entangled within real life usage. To avoid these distortions, better descriptive tools are required in philosophy. The nine new essays within this volume illustrate this need for finer discriminations through a range of revealing cases, of both historical and contemporary significance.




CMS Technical Summary Report


Book Description