Papers › Conditional Euclidean distance optimization via relative tangency
Conditional Euclidean distance optimization via relative tangency
Sandra Di Rocco, Lukas Gustafsson, Luca Sodomaco
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We introduce a theory of relative tangency for projective algebraic varieties. The dual variety X_Z^∨ 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.
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