### Hamiltonian Group Actions on Frame Spaces

Clayton Shonkwiler

http://shonkwiler.org

/jmm20

This talk!

### Take Home Message

Symplectic geometry is a powerful set of tools which is useful for the study of frames

### Frames

A frame in $$\mathbb{C}^d$$ is an ordered spanning set $$\{\phi_1,\dots ,\phi_N\}$$, which we often represent as a short, fat matrix

\Phi = \begin{bmatrix} \phi_1 | \phi_2 | \dots | \phi_N \end{bmatrix} \in \mathbb{C}^{d \times N}

### Symplectic Geometry

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

Example: $$(S^2,d\theta\wedge dz)$$

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

Example. $$(S^2,\omega)$$, where $$\omega_p(u,v) = (u \times v) \cdot p$$

Example. $$(\mathbb{R}^2,\omega)$$ where $$\omega(u,v) = \langle i u, v \rangle$$

Example. $$(\mathbb{C}^n, \frac{i}{2} \sum dz_k \wedge d\overline{z}_k)$$

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

Example. $$(\mathbb{C}^{d \times N}, \omega)$$ with $$\omega(X_1,X_2) = -\operatorname{Im} \operatorname{trace}(X_1^* X_2)$$.

### Maps

If $$H: M \to \mathbb{R}$$ is smooth, then there exists a unique vector field $$X_H$$ so that $${dH = \iota_{X_H}\omega}$$, i.e.,

dH(\cdot) = \omega(X_H, \cdot)

($$X_H$$ is called the Hamiltonian vector field for $$H$$, or sometimes the symplectic gradient of $$H$$)

Example. $$H: (S^2, d\theta\wedge dz) \to \mathbb{R}$$ given by $$H(\theta,z) = z$$.

$$dH = dz = \iota_{\frac{\partial}{\partial \theta}}(d\theta\wedge dz)$$, so $$X_H = \frac{\partial}{\partial \theta}$$.

### Lie Group Actions

Let $$G$$ be a Lie group, and let $$\mathfrak{g}$$ be its Lie algebra. If $$G$$ acts on $$(M,\omega)$$, then each $$V \in \mathfrak{g}$$ determines a vector field $$X_V$$ on $$M$$ by

X_V(p) = \left.\frac{d}{dt}\right|_{t=0}\exp(t V) \cdot p

$$S^1=U(1)$$ acts on $$(S^2,d\theta \wedge dz)$$ by

e^{it} \cdot(\theta, z) = (\theta + t, z)

For $$r \in \mathbb{R} \simeq \mathfrak{u}(1)$$, $$X_r = r \frac{\partial}{\partial \theta}$$.

### Momentum Maps

Definition. An action of $$U(1)$$ on $$(M,\omega)$$ is Hamiltonian if there exists a map

\mu: M \to \mathfrak{u}(1)^* \simeq \mathbb{R}

so that $$d\mu = \iota_{X}\omega$$, where $$X$$ is the vector field generated by the circle action.

$$X = \frac{\partial}{\partial \theta}$$

$$\iota_X\omega = \iota_{\frac{\partial}{\partial \theta}} d\theta \wedge dz = dz$$

$$\mu(\theta,z) = z$$

### Momentum Maps

Definition. An action of $$G$$ on $$(M,\omega)$$ is Hamiltonian if each one-parameter subgroup action is Hamiltonian. Equivalently, there exists a map

\mu: M \to \mathfrak{g}^*

so that $$\omega_p(X_V, X) = D_p \mu(X)(V)$$ for each $$p \in M$$, $$X \in T_pM$$, and $$V \in \mathfrak{g}$$.

$$X_{V_{(a,b,c)}}(x,y,z) = (a,b,c) \times (x,y,z)$$

$$(\iota_{X_{V_{(a,b,c)}}}\omega)_{(x,y,z)} = a dx + b dy + c dz$$

$$\mu(x,y,z)(V_{(a,b,c)}) = (x,y,z)\cdot(a,b,c)$$

### Group Actions on Frames

What compact Lie groups act nicely on $$\mathbb{C}^{d \times N}$$?

1. $$U(d)$$ acts on the left
2. $$U(N)$$ acts on the right
3. $$U(1)^d$$ acts on the left
4. $$U(1)^N$$ acts on the right
\mu_{U(d)}(\Phi) = \Phi \Phi^*
\mu_{U(N)}(\Phi) = -\Phi^*\Phi
\mu_{U(1)^d}\left(\begin{bmatrix} \rule[.8mm]{4mm}{.5px}\, \phi^1 \rule[.8mm]{4mm}{.5px}\\ \vdots \\ \rule[.8mm]{4mm}{.5px}\, \phi^d \rule[.8mm]{4mm}{.5px} \end{bmatrix}\right) = \left(\frac{1}{2}\|\phi^1\|^2 , \dots , \frac{1}{2}\|\phi^d\|^2\right)
\mu_{U(1)^N} \left(\begin{bmatrix} \phi_1 | \cdots | \phi_N \end{bmatrix}\right) = \left(-\frac{1}{2}\|\phi_1\|^2 , \dots , -\frac{1}{2}\|\phi_N\|^2\right)

Parseval frames

$$\mu_{U(d)}^{-1}(I_d)$$

unit-norm frames

$$\mu_{U(1)^N}^{-1}\left(-\frac{1}{2},\dots , -\frac{1}{2}\right)$$

### FUNTFs

The (finite) unit-norm tight frames (FUNTFs) are the level set

Let $$\mu$$ be the momentum map of the product group

U(d) \times U(1)^N
\mu^{-1}\left(\frac{N}{d} I_d, \left(-\frac{1}{2},\dots , -\frac{1}{2}\right)\right)= \mu^{-1}\left(\frac{N}{d}I_d, -\vec{\frac{1}{2}}\right)

### The Generalized Frame Homotopy Conjecture

Theorem [Cahill–Mixon–Strawn ’17]

The space of length-$$N$$ FUNTFs in $$\mathbb{C}^d$$ is path-connected for all $$N \geq d\geq 1$$.

is either empty or path-connected.

and

\|\phi_i\|^2 = r_i
\Phi \Phi^* = S

Theorem [with Needham]

For any invertible, Hermitian matrix $$S$$ and any $$r_1, \dots , r_N \geq 0$$, the space of frames $$\Phi = [\phi_1 | \cdots | \phi_N]$$ with

### Eigensteps

\begin{array}{ccccccccccc} \frac{6}{3}&{}&\frac{6}{3}&{}&\frac{6}{3}&{}&0&{}&0&{}&0\\ {}&\lambda_{51}&{}&\lambda_{52}&{}&\lambda_{53}&{}&0&{}&0&{}\\ {}&{}&\lambda_{41}&{}&\lambda_{42}&{}&\lambda_{43}&{}&0&{}&{}\\ {}&{}&{}&\lambda_{31}&{}&\lambda_{32}&{}&\lambda_{33}&{}&{}&{}\\ {}&{}&{}&{}&\lambda_{21}&{}&\lambda_{22}&{}&{}&{}&{}\\ {}&{}&{}&{}&{}&1&{}&{}&{}&{}&{} \end{array}

Example for FUNTFs with $$N=6, d=3$$:

\begin{array}{c} \sum_j \lambda_{6j} = 6\\ \sum_j \lambda_{5j} = 5\\ \sum_j \lambda_{4j} = 4\\ \sum_j \lambda_{3j} = 3\\ \sum_j \lambda_{2j} = 2\\ {} \end{array}

The eigensteps of a frame $$\Phi = [\phi_1 | \dots | \phi_N]$$ are the eigenvalues $$\lambda_{ij}$$ of the partial frame operators

\Phi_i := \phi_1\phi_1^* + \dots + \phi_i \phi_i^*.

Must satisfy $$\ldots \geq \lambda_{i,j} \geq \lambda_{i-1,j} \geq \lambda_{i,j+1} \geq \dots$$

Proposition [with Needham; cf. Flaschka–Millson]

The eigensteps of a FUNTF are coordinates of the moment map associated to a maximal torus action on (an open, dense subset of a quotient of) FUNTF space.

Corollary [using Duistermaat–Heckman]

The eigensteps of uniform random FUNTFs are distributed according to Lebesgue measure on the eigenstep polytope.

### Some Results

Proposition [with Needham; cf. Flaschka–Millson]

The eigensteps of a FUNTF are coordinates of the moment map associated to a maximal torus action on FUNTF space.

### Consequences

Bounds: The fraction of FUNTFs satisfying any condition is no bigger than the fraction of possible eigensteps compatible with the condition.

Sampling:

Cylindrical coordinates $$(\theta,z)$$ are uniform on the sphere.

### Geometry in traceless Hermitian matrices

$$\phi_1 \phi_1^* - \frac{1}{d}I_d$$

$$\phi_2 \phi_2^* - \frac{1}{d}I_d$$

$$\phi_3 \phi_3^* - \frac{1}{d}I_d$$

$$\phi_4 \phi_4^* - \frac{1}{d}I_d$$

$$\phi_5 \phi_5^* - \frac{1}{d}I_d$$

$$\phi_6 \phi_6^* - \frac{1}{d}I_d$$

$$\lambda_{21}$$

$$\lambda_{31}$$

$$\lambda_{41}$$

Sampling algorithm: $$((\lambda_{ij})_{i,j},(\theta_{ij})_{i,j}) \mapsto \Phi$$

# Thank you!

Funding: Simons Foundation

### References

The geometry of constrained random walks and an application to frame theory

Clayton Shonkwiler

2018 IEEE Statistical Signal Processing Workshop (SSP), 343–347

#### Hamiltonian Group Actions on Frame Spaces

By Clayton Shonkwiler

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