Metric algebraic geometry

Metric algebraic geometry combines concepts from algebraic geometry and differential geometry. Building on classical foundations, it offers practical tools for the 21st century. Many applied problems center around metric questions, such as optimization with respect to distances. After a short dive i...

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Bibliographic Details
Main Authors:	Breiding, Paul, Kohn, Kathlén, Sturmfels, Bernd
Format:	eBook Book
Language:	English
Published:	Cham Springer Nature Switzerland AG 2024 Birkhäuser Springer Nature Birkhäuser Boston
Edition:	1
Series:	Oberwolfach Seminars
Subjects:	Algebraic geometry Algebraic Variety Artificial intelligence Calculus and mathematical analysis Computing and Information Technology Curvature Data Science Differential and Riemannian geometry Differential Geometry Euclidean Distance Geometry Geometry, Differential Integrals Mathematics Mathematics and Science Maximum Likelihood Numerical analysis Numerical Methods Polynomial Optimization Polynomial System Tensors
ISBN:	9783031514616, 3031514610, 9783031514623, 3031514629
Online Access:	Get full text
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Table of Contents:

15.1 Homology from Finite Samples -- 15.2 Sampling with Density Guarantees -- 15.3 Markov Chains on Varieties -- 15.4 Chow goes to Monte Carlo -- References
Intro -- Preface -- About the Authors -- Acknowledgements -- Contents -- Chapter 1 Historical Snapshot -- 1.1 Polars -- 1.2 Foci -- 1.3 Envelopes -- Chapter 2 Critical Equations -- 2.1 Euclidean Distance Degree -- 2.2 Low-Rank Matrix Approximation -- 2.3 Invitation to Polar Degrees -- Chapter 3 Computations -- 3.1 Gröbner Bases -- 3.2 The Parameter Continuation Theorem -- 3.3 Polynomial Homotopy Continuation -- Chapter 4 Polar Degrees -- 4.1 Polar Varieties -- 4.2 Projective Duality -- 4.3 Chern Classes -- Chapter 5 Wasserstein Distance -- 5.1 Polyhedral Norms -- 5.2 Optimal Transport and Independence Models -- 5.3 Wasserstein meets Segre-Veronese -- Chapter 6 Curvature -- 6.1 Plane Curves -- 6.2 Algebraic Varieties -- 6.3 Volumes of Tubular Neighborhoods -- Chapter 7 Reach and Offset -- 7.1 Medial Axis and Bottlenecks -- 7.2 Offset Hypersurfaces -- 7.3 Offset Discriminant -- Chapter 8 Voronoi Cells -- 8.1 Voronoi Basics -- 8.2 Algebraic Boundaries -- 8.3 Degree Formulas -- 8.4 Voronoi meets Eckart-Young -- Chapter 9 Condition Numbers -- 9.1 Errors in Numerical Computations -- 9.2 Matrix Inversion and Eckart-Young -- 9.3 Condition Number Theorems -- 9.4 Distance to the Discriminant -- Chapter 10 Machine Learning -- 10.1 Neural Networks -- 10.2 Convolutional Networks -- 10.3 Learning Varieties -- Chapter 11 Maximum Likelihood -- 11.1 Kullback-Leibler Divergence -- 11.2 Maximum Likelihood Degree -- 11.3 Scattering Equations -- 11.4 Gaussian Models -- Chapter 12 Tensors -- 12.1 Tensors and their Rank -- 12.2 Eigenvectors and Singular Vectors -- 12.3 Volumes of Rank-One Varieties -- Chapter 13 Computer Vision -- 13.1 Multiview Varieties -- 13.2 Grassmann Tensors -- 13.3 3D Reconstruction from Unknown Cameras -- Chapter 14 Volumes of Semialgebraic Sets -- 14.1 Calculus and Beyond -- 14.2 D-Modules -- 14.3 SDP Hierarchies -- Chapter 15 Sampling