Learning algebraic varieties from samples

2018/02/28 by Paul Breiding, Sara Kališnik, Sara Kalisnik Verovsek +2 · 39 citations
Computer Science · Mathematics · #Algebra over a field #Algebraic number #Algebraic variety #Computational Geometry and Mesh Generation #Dimension (graph theory) #Dimension of an algebraic variety #Focus (optics) #Polynomial and algebraic computation #Real algebraic geometry #Topological and Geometric Data Analysis #Variety (cybernetics) #math.AG

paper · pdf · doi:10.1007/s13163-018-0273-6

published in Revista Matemática Complutense 31(3), 545-593 (Springer Science+Business Media)

openalex publication_date 2018/08/13 · arxiv created 2018/08/16 · arxiv updated 2018/08/17 · openalex created_date 2019/06/27 · openalex updated_date 2026/08/05

Abstract

We seek to determine a real algebraic variety from a fixed finite subset of points. Existing methods are studied and new methods are developed. Our focus lies on aspects of topology and algebraic geometry, such as dimension and defining polynomials. All algorithms are tested on a range of datasets and made available in a Julia package.

Citations

• Quartic curves and their bitangents
• Topology and data
• The kinematic formula in Riemannian homogeneous spaces
• Julia: A Fresh Approach to Numerical Computing
• Geometry of Cuts and Metrics
• Moment Varieties of Gaussian Mixtures
• Curvature measures
• Ideals, Varieties, and Algorithms
• Applied Numerical Linear Algebra