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.
Inhaltsverzeichnis
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.
