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The Christoffel-Darboux kernel, a central object in approximation theory, is shown to have many potential uses in modern data analysis, including applications in machine learning. This is the first book to offer a rapid introduction to the subject, illustrating the surprising effectiveness of a simple tool. Bridging the gap between classical mathematics and current evolving research, the authors present the topic in detail and follow a heuristic, example-based approach, assuming only a basic background in functional analysis, probability and some elementary notions of algebraic geometry. They cover new results in both pure and applied mathematics and introduce techniques that have a wide range of potential impacts on modern quantitative and qualitative science. Comprehensive notes provide historical background, discuss advanced concepts and give detailed bibliographical references. Researchers and graduate students in mathematics, statistics, engineering or economics will find new perspectives on traditional themes, along with challenging open problems.
List of contents
Foreword Francis Bach; Preface; 1. Introduction; Part I. Historical and Theoretical Background: 2. Positive definite kernels and moment problems; 3. Univariate Christoffel-Darboux analysis; 4. Multivariate Christoffel-Darboux analysis; 5. Singular supports; Part II. Statistics and Applications to Data Analysis: 6. Empirical Christoffel-Darboux analysis; 7. Applications and occurrences in data analysis; Part III. Complementary Topics: 8. Further applications; 9. Transforms of Christoffel-Darboux kernels; 10. Spectral characterization and extensions of the Christoffel function; References; Index.
About the author
Jean Bernard Lasserre is Emeritus Directeur de Recherche at the LAAS-CNRS and the Institute of Mathematics at the Université Fédérale Toulouse Midi-Pyrénées. He is Chair of 'Polynomial Optimization' at the Artificial & Natural Intelligence Toulouse Institute. He has won numerous awards for his contributions to the fields of applied mathematics, control, operations research and probability, including the 2015 John von Neumann Theory prize and the 2015 Khachiyan Prize of the INFORMS Optimization Society, for lifetime achievements in the area of optimization. He is the author and co-author of eight books and about 200 articles in international journals.Edouard Pauwels is Associate Professor at the Institut de Recherche en Informatique de Toulouse, Université Paul Sabatier, and a member of the Artificial & Natural Intelligence Toulouse Institute. His research focuses on numerical optimization and applications in machine learning and data analysis. He received the Bronze Medal of the CNRS in 2021.Mihai Putinar is Professor in the Mathematics Department at the University of California at Santa Barbara and Professor of Pure Mathematics in the School of Mathematics, Statistics and Physics at Newcastle University. Working in the fields of spectral theory, complex analytic geometry and moment problems, he is the author of three books and more than 200 scientific articles. Over the course of his career, he has been awarded prizes including the Simion Stoilow Prize of the Romanian Academy (1987) and the Romanian National Order of Merit with the rank of Knight (2011).
Summary
This book is the first to give an accessible overview of the Christoffel–Darboux kernel as a novel, powerful, yet simple, tool in statistical data analysis. It offers non-expert, graduate-level readers a rapid and informative introduction to an inter-disciplinary subject, with numerous ramifications and intriguing open questions.
