Abstract
The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low rank matrices, the Eckart-Young Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest point maps from the perspective of computational algebraic geometry. The Euclidean distance degree of a variety is the number of critical points of the squared distance to a generic point outside the variety. Focusing on varieties seen in applications, we present numerous tools for computation.
Keywords: Nearest point map, Euclidean distance, polynomial optimization, computing critical points, dual variety, Chern class
| Original language | English |
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| Title of host publication | 2014 Symposium on Symbolic-Numeric Computation (SNC'14, Shanghai, China, July 28-31, 2014) |
| Editors | L. Zhi, M. Watt |
| Place of Publication | New York |
| Publisher | Association for Computing Machinery, Inc |
| Pages | 9-16 |
| ISBN (Print) | 978-1-4503-2963-7 |
| DOIs | |
| Publication status | Published - 2014 |
| Event | conference; 2014 Symposium on Symbolic-Numeric Computation; 2014-07-28; 2014-07-31 - Duration: 28 Jul 2014 → 31 Jul 2014 |
Conference
| Conference | conference; 2014 Symposium on Symbolic-Numeric Computation; 2014-07-28; 2014-07-31 |
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| Period | 28/07/14 → 31/07/14 |
| Other | 2014 Symposium on Symbolic-Numeric Computation |