Normal Matrices

Definition.

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

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 \(\omega_A(X,Y) = -\mathrm{Im}\langle X,Y \rangle = -\mathrm{Im}\mathrm{Tr}(Y^\ast X)\).

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 \(\mathbb{C}^{d \times d}\): \(U \cdot A  = U A U^\ast\).

Geometric Invariant Theory (GIT)

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

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.

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 \(\operatorname{B}\).

Balancing Graphs

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.

If \(A_0\) is real, so is \(A_\infty\), and if \(A_0\) has all non-negative entries, then so does \(A_\infty\).

Balancing Graphs

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.

If \(A_0\) is real, so is \(A_\infty\), and if \(A_0\) has all non-negative entries, then so does \(A_\infty\).

Balancing Graphs

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.

If \(A_0\) is real, so is \(A_\infty\), and if \(A_0\) has all non-negative entries, then so does \(A_\infty\).

\(\|A\|^2=1\)

\(\|A\|^2=0.569\)

Balancing Graphs

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.

If \(A_0\) is real, so is \(A_\infty\), and if \(A_0\) has all non-negative entries, then so does \(A_\infty\).

Balancing Graphs

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.

If \(A_0\) is real, so is \(A_\infty\), and if \(A_0\) has all non-negative entries, then so does \(A_\infty\).

Preserving Weights

By doing gradient flow \(\overline{\mathscr{F}}\) on the unit sphere, we can preserve weights:

Theorem [with Needham]

For any non-nilpotent \(A_0 \in \mathbb{C}^{d \times d}\) with \(\|A\|^2=1\), the matrix \(A_\infty := \lim_{t \to \infty} \overline{\mathscr{F}}(A_0,t)\) exists, is balanced, has Frobenius norm 1, and has zero entries whenever \(A_0\) does.

If \(A_0\) is real, so is \(A_\infty\), and if \(A_0\) has all non-negative entries, then so does \(A_\infty\).

Preserving Weights

By doing gradient flow \(\overline{\mathscr{F}}\) on the unit sphere, we can preserve weights:

Theorem [with Needham]

For any non-nilpotent \(A_0 \in \mathbb{C}^{d \times d}\) with \(\|A\|^2=1\), the matrix \(A_\infty := \lim_{t \to \infty} \overline{\mathscr{F}}(A_0,t)\) exists, is balanced, has Frobenius norm 1, and has zero entries whenever \(A_0\) does.

If \(A_0\) is real, so is \(A_\infty\), and if \(A_0\) has all non-negative entries, then so does \(A_\infty\).

Preserving Weights

By doing gradient flow \(\overline{\mathscr{F}}\) on the unit sphere, we can preserve weights:

Theorem [with Needham]

For any non-nilpotent \(A_0 \in \mathbb{C}^{d \times d}\) with \(\|A\|^2=1\), the matrix \(A_\infty := \lim_{t \to \infty} \overline{\mathscr{F}}(A_0,t)\) exists, is balanced, has Frobenius norm 1, and has zero entries whenever \(A_0\) does.

If \(A_0\) is real, so is \(A_\infty\), and if \(A_0\) has all non-negative entries, then so does \(A_\infty\).

Preserving Weights

By doing gradient flow \(\overline{\mathscr{F}}\) on the unit sphere, we can preserve weights:

Theorem [with Needham]

For any non-nilpotent \(A_0 \in \mathbb{C}^{d \times d}\) with \(\|A\|^2=1\), the matrix \(A_\infty := \lim_{t \to \infty} \overline{\mathscr{F}}(A_0,t)\) exists, is balanced, has Frobenius norm 1, and has zero entries whenever \(A_0\) does.

If \(A_0\) is real, so is \(A_\infty\), and if \(A_0\) has all non-negative entries, then so does \(A_\infty\).

Equal-Norm Parseval Frames

A spanning set \(f_1, \dots , f_n \in \mathbb{C}^d\) is a frame.

\(\Rightarrow F = [f_1 \cdots f_n] \in \mathbb{C}^{d \times n}\)

Frame Potential

Definition [Benedetto–Fickus, Casazza–Fickus]

The frame potential is

\(\operatorname{FP}(F) = \|FF^\ast\|_{\operatorname{Fr}}^2\)

Frame Potential

Optimization

Theorem [with Mixon, Needham, and Villar; FFT 2021 video]

On the space of equal-norm frames, consider the initial value problem

\(\Gamma(F_0,0) = F_0, \qquad \frac{d}{dt}\Gamma(F_0,t) = -\operatorname{grad}\operatorname{FP}(\Gamma(F_0,t))\).

If \(F_0\) has full spark, then \(\lim_{t \to \infty} \Gamma(F_0,t)\) is an ENP frame.

Frame Energy

Definition [Bodmann–Casazza]

The frame energy is

\(\operatorname{FE}(F) = \sum_{j,k} \left( \|f_j\|^2 - \|f_k\|^2\right)^2 = 2n \sum_j \|f_j\|^4 -2d^2\)

Frame potential: \(\operatorname{FP}(F) = \|FF^\ast\|_{\operatorname{Fr}}^2\)

Total Frame Energy

Definition.

For \(\vec{r} \in \mathbb{R}_+^n\), the total frame energy of a frame is

\(E_{\vec{r}}(F) :=\|FF^* - \mathbb{I}_d \|_{\text{Fr}}^2 + \frac{1}{4}\sum_j \left(\frac{\|f_j\|^2}{r_j}-1 \right)^2\)

Proposition.

The Parseval frames with \(\|f_j\|^2 = r_j\) for \(j=1,\dots , n\) are exactly the global minima of \(E_{\vec{r}}\).

Variations on a Theme

Definition.

For \(\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}\) and \(\vec{r} \in \mathbb{R}_+^n\), let \(\operatorname{PF}^{\mathbb{K}}_d(\vec{r})\) be the space of Parseval frames \(F \in \mathbb{K}^{d \times n}\) with \(\|f_i\|^2 = r_i\).

Variations on a Theme

Theorem [with Caine and Needham]

Let \(\vec{r} \in \mathbb{Q}_+^n\) such that \(\operatorname{PF}_d^{\mathbb{K}}(\vec{r}) \neq \emptyset\). Consider

\(\Gamma_{\vec{r}}(F_0,0) = F_0,\quad \frac{d}{dt} \Gamma_{\vec{r}}(F_0,t) = -\nabla E_{\vec{r}}(\Gamma_{\vec{r}}(F_0,t))\).

If \(F_0 \in \mathbb{K}^{d \times n}\) is full spark, then \(\lim_{t \to \infty} \Gamma_{\vec{r}}(F_0,t)\) is in \(\operatorname{PF}_d^{\mathbb{K}}(\vec{r})\).

Topology of Frame Spaces

Frame Homotopy Conjecture [Larson 2002]

The space of ENP frames is path-connected.

Topology from Optimization

Meta-Theorem

Gradient flow of \(E_{\vec{r}}\) gives a deformation retract of \(\mathbb{K}^{d \times n} \backslash \mathcal{U}_{\vec{r}}\) onto \(\operatorname{PF}_d^{\mathbb{K}}(\vec{r})\).