Conditional Euclidean distance optimization via relative tangency

Author:

Sandra Di Rocco, Lukas Gustafsson, Luca Sodomaco

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG), Optimization and Control (math.OC)

journal:

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date:

2023-10-24 16:00:00

Abstract

We introduce a theory of relative tangency for projective algebraic varieties. The dual variety $X_Z^\vee$ of a variety $X$ relative to a subvariety $Z$ is the set of hyperplanes tangent to $X$ at a point of $Z$. We also introduce the concept of polar classes of $X$ relative to $Z$. We explore the duality of varieties of low rank matrices relative to special linear sections. In this framework, we study the critical points of the Euclidean Distance function from a data point to $X$, lying on $Z$. The locus where the number of such conditional critical points is positive is called the ED data locus of $X$ given $Z$. The generic number of such critical points defines the conditional ED degree of $X$ given $Z$. We show the irreducibility of ED data loci, and we compute their dimensions and degrees in terms of relative characteristic classes.