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Hermitian Distance Degree of Unitary-Invariant Matrix Varieties

Nikhil Ken

TL;DR

The paper defines and analyzes the Hermitian distance degree (HDdeg) for matrix varieties invariant under left-right unitary actions, proving that for such a variety $M$ the invariant equals the real Euclidean distance degree of the associated absolutely symmetric singular-value set $S$, i.e., $\mathrm{HDdeg}(M)=\mathrm{\mathbb{R}EDdeg}(S)$. It shows that for generic data with SVD $Y=U\mathrm{diag}(y)V^*$, Hermitian distance critical points on $M$ lift from real ED critical points on $S$ via the same singular vectors, providing a concrete geometric correspondence. A Hermitian slicing theorem is established, allowing the critical-point count to be computed on a diagonal slice $V_0^{\mathbb{R}}$, paralleling Bik-Draisma's slicing principle in the Hermitian setting. The framework yields a Hermitian analogue of Eckart-Young, clarifying closest Hermitian distance points to rank-k determinantal varieties and enabling real-enumerative analysis of critical points in unitary-invariant matrix problems, with potential applications in quantum information geometry and signal processing. All key results are expressed through SVD-reduction and diagonal-slice reductions, enabling practical computation of critical points via the singular-value data.

Abstract

We study the Hermitian distance degree, a real enumerative invariant counting critical points of the squared Hermitian distance function, for matrix varieties invariant under left and right unitary actions. For such a variety \(M \subset \mathbb{C}^{n\times t}\), we prove that its Hermitian distance degree equals the real Euclidean distance degree of the associated absolutely symmetric variety of singular values. Equivalently, for a generic data matrix, Hermitian distance critical points on \(M\) are obtained by lifting Euclidean distance critical points from the singular-value slice. We also establish a Hermitian slicing theorem, paralleling the Bik--Draisma principle, which reduces the critical point count to a diagonal slice. As a motivating example, we recover a geometric Hermitian analogue of the Eckart-Young theorem.

Hermitian Distance Degree of Unitary-Invariant Matrix Varieties

TL;DR

The paper defines and analyzes the Hermitian distance degree (HDdeg) for matrix varieties invariant under left-right unitary actions, proving that for such a variety the invariant equals the real Euclidean distance degree of the associated absolutely symmetric singular-value set , i.e., . It shows that for generic data with SVD , Hermitian distance critical points on lift from real ED critical points on via the same singular vectors, providing a concrete geometric correspondence. A Hermitian slicing theorem is established, allowing the critical-point count to be computed on a diagonal slice , paralleling Bik-Draisma's slicing principle in the Hermitian setting. The framework yields a Hermitian analogue of Eckart-Young, clarifying closest Hermitian distance points to rank-k determinantal varieties and enabling real-enumerative analysis of critical points in unitary-invariant matrix problems, with potential applications in quantum information geometry and signal processing. All key results are expressed through SVD-reduction and diagonal-slice reductions, enabling practical computation of critical points via the singular-value data.

Abstract

We study the Hermitian distance degree, a real enumerative invariant counting critical points of the squared Hermitian distance function, for matrix varieties invariant under left and right unitary actions. For such a variety , we prove that its Hermitian distance degree equals the real Euclidean distance degree of the associated absolutely symmetric variety of singular values. Equivalently, for a generic data matrix, Hermitian distance critical points on are obtained by lifting Euclidean distance critical points from the singular-value slice. We also establish a Hermitian slicing theorem, paralleling the Bik--Draisma principle, which reduces the critical point count to a diagonal slice. As a motivating example, we recover a geometric Hermitian analogue of the Eckart-Young theorem.
Paper Structure (4 sections, 24 theorems, 130 equations, 1 figure)

This paper contains 4 sections, 24 theorems, 130 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Matrices invariant under unitary actions and critical points
  4. Slicing Theorem for Hermitian Distance Degree

Key Result

Theorem 1

golub2013matrix(2.4.4) Let $A$ be a matrix in $\mathbb{C}^{m \times n}$. Then $A$ can be factored as where $U \in \mathbb{C}^{m \times m}$ is unitary, $V \in \mathbb{C}^{n \times n}$ is unitary, and $\Sigma \in \mathbb{R}^{m \times n}$ has the form where $p = \min(m, n)$. The $s_i's$ are called the singular values of $A$, and they are the square roots of the eigenvalues of $AA^*$

Figures (1)

  • Figure 1: The absolutely symmetric set $S=\{(x_1,x_2):(x_2-x_1^2)(x_1-x_2^2)=0\}$ (union of two parabolas) together with the evolutes.

Theorems & Definitions (53)

  • Theorem 1
  • Definition 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 5: Eckart-Young for HD
  • Definition 6
  • Lemma 7: Terracini
  • proof : Proof of Theorem \ref{['gg']}
  • ...and 43 more