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Conditional Euclidean distance optimization via relative tangency

Sandra Di Rocco, Lukas Gustafsson, Luca Sodomaco

TL;DR

The paper builds a comprehensive framework for relative tangency in projective geometry, introducing X_Z^∨ and relative polar classes to analyze tangency along a subvariety Z. It then develops ED data loci theory for both affine and projective varieties, connecting conditional Euclidean distance degrees to relative invariants via characteristic classes and duality formulas. The authors derive explicit degree relations for the projective ED data loci, including complete-intersection and Kalman-type cases, and extend the theory to determinantal and low-rank matrix varieties. They also study multiple ED data loci and singularities to understand when the conditional ED degree simplifies to 1, providing tools for conditional optimization on algebraic models. Overall, the work unifies relative duality, polar theory, and ED geometry to yield computable invariants with practical relevance to optimization and sampling on algebraic sets.

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.

Conditional Euclidean distance optimization via relative tangency

TL;DR

The paper builds a comprehensive framework for relative tangency in projective geometry, introducing X_Z^∨ and relative polar classes to analyze tangency along a subvariety Z. It then develops ED data loci theory for both affine and projective varieties, connecting conditional Euclidean distance degrees to relative invariants via characteristic classes and duality formulas. The authors derive explicit degree relations for the projective ED data loci, including complete-intersection and Kalman-type cases, and extend the theory to determinantal and low-rank matrix varieties. They also study multiple ED data loci and singularities to understand when the conditional ED degree simplifies to 1, providing tools for conditional optimization on algebraic models. Overall, the work unifies relative duality, polar theory, and ED geometry to yield computable invariants with practical relevance to optimization and sampling on algebraic sets.

Abstract

We introduce a theory of relative tangency for projective algebraic varieties. The dual variety of a variety relative to a subvariety is the set of hyperplanes tangent to at a point of . We also introduce the concept of polar classes of relative to . 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 , lying on . The locus where the number of such conditional critical points is positive is called the ED data locus of given . The generic number of such critical points defines the conditional ED degree of given . We show the irreducibility of ED data loci, and we compute their dimensions and degrees in terms of relative characteristic classes.
Paper Structure (10 sections, 45 theorems, 157 equations, 4 figures)

This paper contains 10 sections, 45 theorems, 157 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on duality and polar classes
  3. Relative duality and relative polar classes
  4. Relative tangency in determinantal varieties
  5. Duality of low-rank matrices relative to special column and row spaces
  6. Duality of low-rank matrices relative to symmetric and skew-symmetric low-rank matrices
  7. Euclidean Distance data loci of affine varieties
  8. Euclidean Distance data loci of projective varieties
  9. Degrees of projective ED data loci
  10. Multiple ED data loci and singularities of ED data loci

Key Result

Theorem 1.3

Assume that $X$ is dual regular relative to $Z$. The following properties are equivalent:

Figures (4)

  • Figure 1: The Cayley's nodal cubic surface $X$ (in yellow) and the green ED data locus $\mathrm{DL}_{X|Z}$ with respect to the blue circle $Z\subset X$.
  • Figure 2: On the left, in green the ED data locus $\mathrm{DL}_{X|Z}$ given the line $Z\subset X$ joining the singular points $P_1$ and $P_2$ of $X$. In the center and on the right, in orange the normal cones of $X$ at $P_1$ and $P_2$.
  • Figure 3: On the left, the green plane $x_3=0$ is the ED data locus of the unit sphere $S$ through the origin given the blue unit circle $Z=V(x_1^2+x_2^2-1,x_3)$. On the right, the green quadric cone $V(x_1^2+x_2^2-3x_3^2)$ is the ED data locus of $S$ given the blue circle $Z=V(x_1^2+x_2^2-1/4,x_3-1/2)$.
  • Figure 4: The yellow quadric surface $X$ and the green ED data locus $\mathrm{DL}_{X|Z}$ given the line $Z\subset X$, ruled by the red lines normal to $X$.

Theorems & Definitions (119)

  • Example 1.1
  • Example 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • ...and 109 more