Metric Algebraic Geometry (Oberwolfach Seminars, 53)

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 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