Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the minimal algebraic complexity of the rank-one approximation problem for general inner products

Khazhgali Kozhasov, Alan Muniz, Yang Qi, Luca Sodomaco

TL;DR

This work studies how the algebraic complexity of the rank-one approximation problem, measured by the Euclidean Distance (ED) degree, varies with the choice of inner product. By formulating the ED degree as a map $\Phi_{\mathcal{X}}(Q)$ over inner products and using tools from Singularity Theory, Morse Theory, and Intersection Theory, the authors derive explicit generic ED degrees for Segre-Veronese varieties and establish a local (and in some cases global) minimality of the Frobenius inner product. They prove the conjectured minimality for matrices and symmetric $3\times3\times3$ tensors, and provide a detailed analysis of ED defects for quadric hypersurfaces, rational normal curves, and Veronese surfaces, highlighting how inner products influence the number and nature of critical points. A key technical advance is the extended ED polynomial, whose leading coefficient determines loci of inner products that reduce the ED degree, with these loci identified as dual varieties under the second Veronese embedding, tying geometric transversality to optimization complexity. Overall, the paper advances understanding of how choosing an inner product shapes the computational hardness of low-rank approximation problems and offers a framework to identify inner products that minimize algebraic complexity in practical settings.

Abstract

We study the algebraic complexity of Euclidean distance minimization from a generic tensor to a variety of rank-one tensors. The Euclidean Distance (ED) degree of the Segre-Veronese variety counts the number of complex critical points of this optimization problem. We regard this invariant as a function of inner products. We prove that Frobenius inner product is a local minimum of the ED degree, and conjecture that it is a global minimum. We prove our conjecture in the case of matrices and symmetric binary and $3\times 3\times 3$ tensors. We discuss the above optimization problem for other algebraic varieties, classifying all possible values of the ED degree. Our approach combines tools from Singularity Theory, Morse Theory, and Algebraic Geometry.

On the minimal algebraic complexity of the rank-one approximation problem for general inner products

TL;DR

This work studies how the algebraic complexity of the rank-one approximation problem, measured by the Euclidean Distance (ED) degree, varies with the choice of inner product. By formulating the ED degree as a map over inner products and using tools from Singularity Theory, Morse Theory, and Intersection Theory, the authors derive explicit generic ED degrees for Segre-Veronese varieties and establish a local (and in some cases global) minimality of the Frobenius inner product. They prove the conjectured minimality for matrices and symmetric tensors, and provide a detailed analysis of ED defects for quadric hypersurfaces, rational normal curves, and Veronese surfaces, highlighting how inner products influence the number and nature of critical points. A key technical advance is the extended ED polynomial, whose leading coefficient determines loci of inner products that reduce the ED degree, with these loci identified as dual varieties under the second Veronese embedding, tying geometric transversality to optimization complexity. Overall, the paper advances understanding of how choosing an inner product shapes the computational hardness of low-rank approximation problems and offers a framework to identify inner products that minimize algebraic complexity in practical settings.

Abstract

We study the algebraic complexity of Euclidean distance minimization from a generic tensor to a variety of rank-one tensors. The Euclidean Distance (ED) degree of the Segre-Veronese variety counts the number of complex critical points of this optimization problem. We regard this invariant as a function of inner products. We prove that Frobenius inner product is a local minimum of the ED degree, and conjecture that it is a global minimum. We prove our conjecture in the case of matrices and symmetric binary and tensors. We discuss the above optimization problem for other algebraic varieties, classifying all possible values of the ED degree. Our approach combines tools from Singularity Theory, Morse Theory, and Algebraic Geometry.
Paper Structure (13 sections, 36 theorems, 126 equations, 2 tables)

This paper contains 13 sections, 36 theorems, 126 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Algebraic complexity of Euclidean Distance optimization
  4. Chern classes, duality, and polar classes
  5. Generic ED degree and ED defect
  6. Morse theory
  7. ED degrees for Segre-Veronese varieties
  8. Minimum ED degrees for varieties of rank-one matrices
  9. ED defects of special varieties
  10. Quadric hypersurfaces
  11. Rational normal curves
  12. Veronese surfaces
  13. Nontransversality conditions and leading coefficient of the ED polynomial

Key Result

Proposition 2.5

The roots $\varepsilon^2$ of $\mathrm{EDpoly}_{\mathcal{X}}(u,Q,\varepsilon^2)$ are precisely of the form $\varepsilon^2=q(u-x)$, where $x$ is a critical point of the polynomial function $x\mapsto q(u-x)$ on $\mathcal{X}_{\mathrm{sm}}$. In particular, when restricting to positive definite symmetric

Theorems & Definitions (75)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Example 2.7
  • Definition 2.8
  • Definition 2.9
  • Theorem 2.10
  • ...and 65 more