Balancing Graphs Using Geometric Invariant Theory

Clayton Shonkwiler

Colorado State University

shonkwiler.org

/codex26

this talk!

CodEx

February 10, 2026

Joint Work With:

Tom Needham

Florida State University

Funding

National Science Foundation (DMS–2107700)

Anthony Caine

Arizona State University

Take-Home Messages

Symmetry + geometry sometimes tells you an optimization problem is easier than expected.
Optimization can tell us something about topology.

Some Special Matrices

Self-adjoint
Unitary
Normal (equivalently: orthogonally diagonalizable)
Self-adjoint with given spectrum (e.g., Gram matrices of Parseval frames)
Fixed singular values (e.g., Parseval frames)
Fixed row/column norms (e.g., equal-norm frames)

Can these be realized as minima of some (nice) potential?

Normal Matrices

Definition.

\(A \in \mathbb{C}^{d \times d}\) is normal if \(AA^\ast = A^\ast A\).

Equivalently,

\(0 = AA^\ast - A^\ast A = [A,A^\ast]\).

Define the non-normal energy \(\operatorname{E}:\mathbb{C}^{d \times d} \to \mathbb{R}\) by

\(\operatorname{E}(A) := \|[A,A^\ast]\|^2.\)

Obvious Fact.

The normal matrices are the global minima of \(\operatorname{E}\).

Theorem [with Needham]

The only critical points of \(\operatorname{E}\) are the global minima; i.e., the normal matrices.

Normal Matrices

\(\operatorname{E}\) is not quasiconvex!

Theorem [with Needham]

The only critical points of \(\operatorname{E}\) are the global minima; i.e., the normal matrices.

Gradient Descent

Let \(\mathcal{F}: \mathbb{C}^{d \times d} \times \mathbb{R} \to \mathbb{C}^{d \times d}\) be negative gradient descent of \(\operatorname{E}\); i.e.,

\(\mathcal{F}(A_0,0) = A_0 \qquad \frac{d}{dt}\mathcal{F}(A_0,t) = -\nabla \operatorname{E}(\mathcal{F}(A_0,t))\).

Theorem [with Needham]

For any \(A_0 \in \mathbb{C}^{d \times d}\), the matrix \(A_\infty := \lim_{t \to \infty} \mathcal{F}(A_0,t)\) exists, is normal, has the same eigenvalues as \(A_0\), and is real if \(A_0\) is.

Why?

\(\mathbb{C}^{d \times d}\) is symplectic, with symplectic form \(\omega_A(X,Y) = -\mathrm{Im}\langle X,Y \rangle = -\mathrm{Im}\mathrm{Tr}(Y^\ast X)\).

A symplectic manifold is a smooth manifold \(M\) together with a closed, non-degenerate 2-form \(\omega \in \Omega^2(M)\).

Example: \((\mathbb{R}^2,dx \wedge dy) = (\mathbb{C},\frac{i}{2}dz \wedge d\bar{z})\)

dx \wedge dy \left( \textcolor{12a4b6}{a \frac{\partial}{\partial x} + b \frac{\partial}{\partial y}}, \textcolor{d9782d}{c \frac{\partial }{\partial x} + d \frac{\partial}{\partial y}} \right) = ad - bc

(a,b) = a \vec{e}_1 + b \vec{e}_2 = a \frac{\partial}{\partial x} + b \frac{\partial}{\partial y}

(c,d) = c \vec{e}_1 + d \vec{e}_2 = c \frac{\partial}{\partial x} + d \frac{\partial}{\partial y}

Why?

\(\mathbb{C}^{d \times d}\) is symplectic, with symplectic form \(\omega_A(X,Y) = -\mathrm{Im}\langle X,Y \rangle = -\mathrm{Im}\mathrm{Tr}(Y^\ast X)\).

Consider the conjugation action of \(\operatorname{SU}(d)\) on \(\mathbb{C}^{d \times d}\): \(U \cdot A = U A U^\ast\).

This action is Hamiltonian with associated momentum map \(\mu: \mathbb{C}^{d \times d} \to \mathscr{H}_0(d)\) given by

\(\mu(A) := [A,A^\ast]\).

So \(\operatorname{E}(A) = \|\mu(A)\|^2\).

Frances Kirwan

Gert-Martin Greuel [CC BY-SA 2.0 DE], from Oberwolfach Photo Collection

Image by rawpixel.com on Freepik

This kind of function is really nice!

Geometric Invariant Theory (GIT)

The GIT quotient consists of group orbits which can be distinguished by \(G\)-invariant (homogeneous) polynomials.

\(\mathbb{C}^* \curvearrowright \mathbb{CP}^2\)

\(t \cdot [z_0:z_1:z_2] = [z_0: tz_1:\frac{1}{t}z_2]\)

Roughly: identify orbits whose closures intersect, throw away orbits on which all \(G\)-invariant polynomials vanish.

\( \mathbb{CP}^2/\!/\,\mathbb{C}^* \cong\mathbb{CP}^1\)

Abelian Version

Let \(T \simeq \operatorname{U}(1)^{d-1}\) be the diagonal subgroup of \(\operatorname{SU}(d)\). The conjugation action of \(T\) on \(\mathbb{C}^{d \times d}\) is also Hamiltonian, with momentum map

\(A \mapsto \mathrm{diag}([A,A^\ast])\).

\([A,A^\ast]_{ii} = \|A_i\|^2 - \|A^i\|^2\), where \(A_i\) is the \(i\)th row of \(A\) and \(A^i\) is the \(i\)th column.

If \(A = \left(a_{ij}\right)_{i,j} \in \mathbb{R}^{d \times d}\) such that \(\mathrm{diag}([A,A^\ast]) = 0\), then \(\widehat{A} = \left(a_{ij}^2\right)_{i,j}\) is the adjacency matrix of a balanced multigraph.

Balancing Graphs

Define the unbalanced energy \(\operatorname{B}(A) := \|\mathrm{diag}([A,A^\ast])\|^2 = \sum \left(\|A_i\|^2 - \|A^i\|^2\right)^2\).

Let \(\mathscr{F}(A_0,0) = A_0, \frac{d}{dt}\mathscr{F}(A_0,t) = - \nabla \operatorname{B}(\mathscr{F}(A_0,t))\) be negative gradient flow of \(\operatorname{B}\).

Theorem [with Needham]

For any \(A_0 \in \mathbb{C}^{d \times d}\), the matrix \(A_\infty := \lim_{t \to \infty} \mathscr{F}(A_0,t)\) exists, is balanced, has the same eigenvalues and principal minors as \(A_0\), and has zero entries whenever \(A_0\) does.