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...

Ausführliche Beschreibung

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Bibliographische Detailangaben
Hauptverfasser: Breiding, Paul, Kohn, Kathlén, Sturmfels, Bernd
Format: E-Book Buch
Sprache:Englisch
Veröffentlicht: Cham Springer Nature Switzerland AG 2024
Birkhäuser
Springer Nature
Birkhäuser Boston
Ausgabe:1
Schriftenreihe:Oberwolfach Seminars
Schlagworte:
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-Zugang:Volltext
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Inhaltsangabe:
  • 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