Two well studied invariants of a complex projective variety are the unit Euclidean distance degree and the generic Euclidean distance degree. These numbers give a measure of the algebraic complexity for "nearest" point problems of the algebraic variety. It is well known that the latter is an upper bound for the former. While this bound may be tight, many varieties appearing in optimization, engineering, statistics, and data science, have a significant gap between these two numbers. We call this difference the defect of the ED degree of an algebraic variety. In this paper we compute this defect by classical techniques in Singularity Theory, thereby deriving a new method for computing ED degrees of smooth complex projective varieties.

中文翻译：

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欧氏距离度的缺陷

复杂射影变体的两个经过充分研究的不变量是单位欧几里得距离度和通用欧几里得距离度。这些数字给出了代数种类的“最近”点问题的代数复杂度的度量。众所周知，后者是前者的上限。虽然这个界限可能很紧，但优化、工程、统计和数据科学中出现的许多变体在这两个数字之间存在显着差距。我们称这种差异为代数簇的 ED 度缺陷。在本文中，我们通过奇异理论中的经典技术来计算这个缺陷，从而推导出一种计算光滑复射影簇的ED度的新方法。