2015
DOI: 10.1007/s10208-014-9240-x
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The Euclidean Distance Degree of an Algebraic Variety

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 general point outside the variety. Focusi… Show more

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Cited by 185 publications
(312 citation statements)
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“…It is called the generic Euclidean distance degree of X, and denoted by gED(X). We refer to [8] for more details about Euclidean distance degree. For determinantal varieties, we have Proposition 5.5.…”
Section: Proof Of the Main Theorem For Each Coefficientmentioning
confidence: 99%
“…It is called the generic Euclidean distance degree of X, and denoted by gED(X). We refer to [8] for more details about Euclidean distance degree. For determinantal varieties, we have Proposition 5.5.…”
Section: Proof Of the Main Theorem For Each Coefficientmentioning
confidence: 99%
“…The bulk of this note is devoted to counting the number of critical points on G of the function d u , in the general framework of the Euclidean distance degree (ED degree) [2]. In Section 2 we specialise this framework to matrix groups.…”
Section: The Distance To a Matrix Groupmentioning
confidence: 99%
“…), then the number of solutions to (1) will not depend on u, provided that u is sufficiently general. Following [2], we call this number the Euclidean distance degree (ED degree for short) of G. This number gives an algebraic measure for the complexity of writing down the solution to the minimisation Problem 1.1. We now distinguish two classes of groups: those that preserve the inner product (.|.)…”
Section: The Ed Degree and Critical Equationsmentioning
confidence: 99%
“…The following theorem justifies that the ML-degree and ED-degree are well defined. The corresponding proofs can be found in [8] and [7] respectively. Theorem 2.4.…”
Section: Critical Pointsmentioning
confidence: 99%
“…There are already some fundamental results about these algebraic degrees and the geometry behind them. A general degree theory for the ED-degree was introduced in [7] and for the ML-degree in [6,17,19]. For some special cases, it is possible to find formulas for d using techniques from algebraic geometry [1,18,19,21,23,24,25,27], some are summarized in [26].…”
Section: Introductionmentioning
confidence: 99%