Hermitian Distance Degree of Unitary-Invariant Matrix Varieties

1. Problem Statement

The paper addresses the problem of computing the Hermitian Distance Degree (HD degree) for algebraic varieties of matrices that are invariant under left and right unitary actions.

• Context: In algebraic geometry, the Euclidean Distance Degree (ED degree) counts the number of critical points of the squared distance function from a generic data point to a variety. This is a fundamental real enumerative invariant.
• The Hermitian Setting: The author considers the space of complex matrices $V = \mathbb{C}^{n \times t}$ equipped with the Hermitian inner product $\langle A, B \rangle = \text{Tr}(AB^*)$.
• The Real Problem: The HD degree is defined not as a count of complex critical points, but as the number of real critical points of the squared distance function $d_Y(X) = \|X - Y\|^2 = \text{Re}\langle X-Y, X-Y \rangle$, where the variety $M$ is viewed as a real algebraic set in $\mathbb{R}^{2nt}$.
• The Challenge: Directly computing these critical points for high-dimensional matrix varieties is computationally difficult. The paper seeks to reduce this complex matrix problem to a simpler problem involving singular values.

2. Methodology

The author employs a combination of Lie group actions, Singular Value Decomposition (SVD), and slicing principles to reduce the dimensionality of the problem.

• Unitary Invariance: The study focuses on varieties $M \subset \mathbb{C}^{n \times t}$ such that $UMV^* = M$ for all $U \in U(n)$ and $V \in U(t)$.
• Singular Value Reduction:
  • Any matrix $A$ can be decomposed as $A = U \Sigma V^*$, where $\Sigma$ is a rectangular diagonal matrix of singular values.
  • The author defines a map $\sigma(M) = \{x \in \mathbb{R}^n : \text{diag}(x) \in M\}$.
  • It is proven that if $M$ is unitary-invariant, the set of singular values $S = \sigma(M)$ forms an absolutely symmetric set (invariant under signed permutations) in $\mathbb{R}^n$.
• Lifting Critical Points: The core methodological insight is that critical points on the matrix variety $M$ can be "lifted" from critical points on the singular value slice $S$. If $Y = U \text{diag}(y) V^*$ is a generic data matrix, the critical points $X$ on $M$ take the form $X = U \text{diag}(x) V^*$, where $x$ is a critical point of $y$ on $S$.
• Hermitian Slicing Theorem: The paper establishes a slicing principle analogous to the Bik–Draisma theorem (originally for ED degrees). It proves that for a unitary representation, the ambient space $\mathbb{R}^{2nt}$ decomposes orthogonally (with respect to the real part of the Hermitian product) into the diagonal slice and the tangent space of the group orbit. This allows the critical point count to be reduced to the slice.

3. Key Contributions and Results

A. Main Theorem (Theorem 23)

The central result establishes a numerical and structural equality between the Hermitian distance degree of a matrix variety and the real Euclidean distance degree of its singular value slice.

• Statement: For a unitary-invariant matrix variety $M$ and its associated absolutely symmetric set of singular values $S$:
  $$\text{HDdeg}(M) = \text{REDdeg}(S)$$
• Structural Insight: For a generic data matrix $Y = U \text{diag}(y) V^*$, the set of Hermitian distance critical points on $M$ is exactly:
  $$\{ U \text{diag}(x) V^* \mid x \text{ is an ED critical point of } y \text{ on } S \}$$
  This means the critical points share the same singular vectors ($U, V$) as the data matrix, and only the singular values change according to the optimization on $S$.

B. Hermitian Slicing Theorem (Theorem 30)

The paper generalizes the Bik–Draisma slicing principle to the Hermitian setting.

• It proves that if a real algebraic variety $X$ is stable under a unitary group action $G$, and $X_0$ is the intersection of $X$ with a generic diagonal slice $V_0$, then the HD degree of $X$ equals the RED degree of $X_0$.
• This relies on the fact that unitary representations act as orthogonal transformations on the underlying real vector space equipped with the real inner product $q = \text{Re}\langle \cdot, \cdot \rangle$.

C. Geometric Recovery of Eckart-Young Theorem (Theorem 5)

As a motivating example, the paper provides a geometric proof of the Eckart-Young theorem for the Hermitian distance.

• It characterizes the critical points of the distance from a matrix $A$ to the variety of rank-$k$ matrices.
• The critical points are shown to be matrices formed by keeping exactly $k$ singular values of $A$ (specifically, selecting a subset of the rank-1 summands in the SVD).
• If singular values are distinct, the number of critical points is $\binom{r}{k}$, where $r$ is the rank of $A$.

D. Examples of Chamber-Dependent Counts

The paper illustrates that, unlike the complex ED degree, the real HD degree depends on the "chamber" (connected component of the complement of the discriminant) where the data point lies.

• Example 24 (Singular Value Relation): For matrices where singular values satisfy $\{\sigma_1, \sigma_2\} = \{a, a^2\}$, the number of critical points jumps between 2, 4, and 6 depending on the position of the data point relative to the evolutes of the defining parabolas.
• Example 25 (Determinant Magnitude): For the variety $|\det A| = 1$, the critical count jumps between 4 and 6 based on the discriminant of a quartic polynomial derived from the singular values.

4. Significance

1. Computational Reduction: The paper transforms a high-dimensional optimization problem over complex matrices ($\mathbb{R}^{2nt}$) into a lower-dimensional problem over real vectors ($\mathbb{R}^n$). This makes the computation of distance degrees for unitary-invariant varieties tractable.
2. Unification of Fields: It bridges the gap between Euclidean Distance Degree theory (typically studied for real varieties) and Hermitian Distance problems (arising in quantum information, signal processing, and matrix analysis).
3. Real Enumerative Geometry: The work emphasizes the importance of counting real critical points. It demonstrates that for unitary-invariant varieties, the real critical points are structurally simple (diagonalizable with the same basis as the data), but their number is sensitive to the data's location, a phenomenon captured by the chamber-wise constant.
4. Applications: The results have direct implications for:
   • Low-rank matrix approximation: Generalizing the Eckart-Young theorem to the Hermitian setting.
   • Quantum Information: The critical points of Hermitian distance functions relate to the study of closest product states and geometric measures of entanglement.
   • Schatten Norms: The framework applies to unit spheres of Schatten $d$-norms, providing explicit counts for their distance degrees.

In summary, Nikhil Ken's paper provides a rigorous framework showing that the complexity of finding closest matrices in unitary-invariant varieties is entirely governed by the geometry of their singular values, effectively reducing a matrix problem to a vector problem while preserving the enumerative invariants.