The Euclidean distance degree of an algebraic variety

J. Draisma, E. Horobet, G. Ottaviani, B. Sturmfels, R.R. Thomas

Research output: Contribution to journalArticleAcademicpeer-review

105 Citations (Scopus)
3 Downloads (Pure)


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 general point outside the variety. Focusing on varieties seen in applications, we present numerous tools for exact computations.
Original languageEnglish
Pages (from-to)99-149
JournalFoundations of Computational Mathematics
Issue number1
Publication statusPublished - Feb 2016


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