## The (Symplectic) Geometry of Spaces of Frames

Clayton Shonkwiler

shonkwiler.org

/sissa21

this talk!

### Signal Analysis

Signal: $$v \in \mathbb{C}^d$$

Design: Choose $$f_1, \dots , f_n \in \mathbb{C}^d$$

Measurements: $$\langle f_1, v \rangle, \dots , \langle f_n, v \rangle$$.

Goal: Reconstruct the signal $$v$$ from $$F^* v$$

If $$F = \begin{bmatrix} f_1 & \cdots & f_n\end{bmatrix}$$, the measurement vector is $$F^*v$$.

### Orthonormal Bases

If $$f_1,\dots , f_d\in \mathbb{C}^d$$ form an orthonormal basis, then

$$v=\sum \langle f_k,v\rangle f_k = FF^* v$$.

$$\Leftrightarrow FF^* = \mathrm{Id}_{d \times d}$$

This is fragile! What if a measurement gets lost?

Definition.

$$\{f_1,\dots, f_n\}\subset \mathbb{C}^d$$ is a Parseval frame if $$\operatorname{Id}_{d\times d}=FF^*=f_1f_1^*+\dots+f_nf_n^*$$.

The rows of $$F$$ form an orthonormal set in $$\mathbb{C}^n$$, so the space of all length-$$n$$ Parseval frames in $$\mathbb{C}^d$$ is the Stiefel manifold $$\operatorname{St}_d(\mathbb{C}^n)$$.

real subvariety of $$\mathbb{C}^{d \times n}$$

### Dealing with Erasures

Lost measurements are still a problem:

$$F=\begin{bmatrix} 1/\sqrt{2} & 1/\sqrt{2} & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

An equal-norm Parseval frame (ENP frame) is a Parseval frame $$f_1,\dots , f_n$$ with $$\|f_i\|^2=\|f_j\|^2$$ for all $$i$$ and $$j$$.

ENP frames are optimal for signal reconstruction in the presence of additive white Gaussian noise and erasures.

real condition

$$\sum \|f_i\|^2=\operatorname{tr}F^*F=\operatorname{tr}FF^*=\operatorname{tr}\operatorname{Id}_{d \times d} = d$$, so each $$\|f_i\|^2=\frac{d}{n}$$.

### Generalizations

More generally, we may want to consider

1. frames with fixed frame vector norms: $$\|f_i\|^2=r_i$$ for specified $$r_i>0$$
2. frames with $$FF^*=S$$ for specified positive-definite, Hermitian $$S$$

real conditions

The spaces of frames satisfying 1, 2, or both are only real subvarieties of $$\mathbb{C}^{d \times n}$$.

### Some Questions

The Frame Homotopy Conjecture [Larsen, 2002]

The space of ENP frames is path-connected.

Proved by Cahill–Mixon–Strawn in 2017.

Generalized Frame Homotopy Conjecture

For given positive-definite, Hermitian $$S$$ and $$r_1,\dots , r_n>0$$, the space of frames with $$FF^*=S$$ and $$\|f_i\|^2=r_i$$ is path-connected.

Genericity of Full Spark Frames

With probability 1, a random ENP frame has full spark—meaning every size-$$d$$ subset of the $$f_i$$ spans $$\mathbb{C}^d$$.

frame with fixed $$FF^*$$ and fixed $$\|f_1\|^2,\dots, \|f_n\|^2$$

### Objects of Interest

We are interested in $$F=\begin{bmatrix}f_1 & f_2 & \cdots & f_n\end{bmatrix}\in \mathbb{C}^{d \times n}$$ with

$$FF^* = S$$

and/or

$$\|f_1\|^2=r_1, \dots , \|f_n\|^2=r_n$$.

### Symplectic Geometry

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

$$(S^2,d\theta\wedge dz)$$

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

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

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

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

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

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

The name ‘complex group’ formerly proposed by me…has become more and more embarrassing through collision with the word ‘complex’ in the connotation of complex number. I therefore propose to replace it by the corresponding Greek adjective ‘symplectic’.

— Hermann Weyl

### Volume

$$\omega^{\wedge n} = \omega \wedge \dots \wedge \omega$$ is a volume form on $$M$$, and induces a measure

m(U) := \int_U \omega^{\wedge n}

called Liouville measure on $$M$$.

If $$M$$ is compact, this can be normalized to give a probability measure.

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 the Hamiltonian vector field for $$H$$ or 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 $$X_H = \frac{\partial}{\partial \theta}$$.

$$H$$ is constant on orbits of $$X_H$$:

$$\mathcal{L}_{X_H}(H) = dH(X_H)=\omega(X_H,X_H) = 0$$

### Lie Group Actions

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

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

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

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

### 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

$$SO(3)$$ acts on $$S^2$$ by rotations.

$$= (a,b,c) \times (x,y,z)$$

For $$V_{(a,b,c)} = \begin{bmatrix} 0 & -c & b \\ c & 0 & -a \\ -b & a & 0 \end{bmatrix} \in \mathfrak{so}(3)$$,

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

$$= (bz-cy)\frac{\partial}{\partial x} + (cx - az) \frac{\partial}{\partial y} +(ay - bx) \frac{\partial}{\partial z}$$

### Symmetries and Conserved Quantities

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

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

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

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

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

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

### Moment 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 $$V \in \mathfrak{g}$$.

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

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

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

### Key Tools

Theorem [Atiyah, Guillemin–Sternberg

Let $$(M^{2n},\omega)$$ be a compact connected symplectic manifold with a Hamiltonian $$k$$-torus action with momentum map $$\mu: M \to \mathbb{R}^k$$. Then

• the nonempty level sets of $$\mu$$ are connected;
• the image of $$\mu$$ is convex (called the moment polytope);
• the image of $$\mu$$ is the convex hull of the images of fixed points of the action.

### Key Tools

Theorem [Atiyah, Guillemin–Sternberg

Let $$(M^{2n},\omega)$$ be a compact connected symplectic manifold with a Hamiltonian $$k$$-torus action with momentum map $$\mu: M \to \mathbb{R}^k$$. Then

• the nonempty level sets of $$\mu$$ are connected;
• the image of $$\mu$$ is convex (called the moment polytope);
• the image of $$\mu$$ is the convex hull of the images of fixed points of the action.

$$U(1)^2$$ acts on $$\mathbb{CP}^2$$ by

$$(e^{i\theta_1},e^{i\theta_2})\cdot [z_0:z_1:z_1]:=[z_1:e^{i\theta_1}z_1:e^{i\theta_2}z_2]$$

Moment map:

$$\mu([z_0:z_1:z_2]=-\frac{1}{2(|z_0|^2+|z_1|^2+|z_2|^2)}(|z_1|^2,|z_2|^2)$$

$$(0,0)=\mu([1:0:0])$$

$$(-\frac{1}{2},0)=\mu([0:1:0])$$

$$(0,-\frac{1}{2})=\mu([0:0:1])$$

Theorem [Duistermaat–Heckman]

If $$k=n$$, then the pushforward of Liouville measure to the moment polytope is a constant multiple of Lebesgue measure.

### Symplectic Reduction

Theorem [Meyer, Marsden–Weinstein]

Let $$\mu: M \to \mathfrak{g}^*$$ be the moment map for a Hamiltonian action of $$G$$ on $$(M,\omega)$$. If $$\xi \in \mathfrak{g}^*$$ is a regular value of $$\mu$$ and $$\mathcal{O}_\xi$$ is its coadjoint orbit, then

M /\!/\!_\xi\, G := \mu^{-1}(\mathcal{O}_\xi)/G

is a symplectic manifold with symplectic form $$\omega_{\text{red}}$$ such that

\pi^*\omega_{\text{red}} = \iota^* \omega

$$\mu^{-1}(\mathcal{O}_\xi)$$

$$M /\!/\!_\xi\, G = \mu^{-1}(\mathcal{O}_\xi)/G$$

$$M$$

$$\pi$$

$$\iota$$

### A Circle Action on $$\mathbb{C}^d$$

Let $$U(1)$$ act on $$(\mathbb{C}^d,\omega_{\text{std}})$$ by $$e^{it}\cdot \vec{v} = e^{it} \vec{v}$$. The associated vector field is

X(\vec{v}) = \left. \frac{d}{dt}\right|_{t=0} e^{it}\vec{v} = i\vec{v},

so $$(\iota_X\omega_{\text{std}})_{\vec{v}}(Y) = (\omega_{\text{std}})_{\vec{v}}(X,Y) = \langle iX,Y\rangle_{\vec{v}} = \langle -\vec{v},Y\rangle$$.

The moment map is

\mu: \mathbb{C}^d \to \mathfrak{u}(1)^* \simeq \mathbb{R}
\vec{v} \mapsto -\frac{1}{2} \left| \vec{v}\right|^2

since $$d\mu_{\vec{v}}(Y) = \left.\frac{d}{dt}\right|_{t=0} \left(-\frac{1}{2} \langle \vec{v} + tY, \vec{v} + tY\rangle\right) = \langle -\vec{v},Y\rangle$$.

The symplectic reduction over $$-\frac{1}{2} \in \mathbb{R}$$ is

$$\mathbb{C}^d/\!/\!_{-\frac{1}{2}} U(1) = \mu^{-1}\left(-\frac{1}{2}\right)\!/U(1) \simeq \mathbb{CP}^{d-1}$$.

### 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)}(F) = FF^*
\mu_{U(n)}(F) = -F^*F
\mu_{U(1)^d}\left(\begin{bmatrix} \rule[.8mm]{4mm}{.5px}\, f^1 \rule[.8mm]{4mm}{.5px}\\ \vdots \\ \rule[.8mm]{4mm}{.5px}\, f^d \rule[.8mm]{4mm}{.5px} \end{bmatrix}\right) = \left(\frac{1}{2}\|f^1\|^2 , \dots , \frac{1}{2}\|f^d\|^2\right)
\mu_{U(1)^n} \left(\begin{bmatrix} f_1 & \cdots & f_n \end{bmatrix}\right) = \left(-\frac{1}{2}\|f_1\|^2 , \dots , -\frac{1}{2}\|f_n\|^2\right)

Parseval frames

$$\mu_{U(d)}^{-1}(\operatorname{Id}_{d \times d})$$

fixed-norm frames

$$\mu_{U(1)^N}^{-1}\left(-\frac{r_1}{2},\dots , -\frac{r_n}{2}\right)$$

### Parseval Frames and a Grassmannian

$$\mu_{U(d)}(F)=FF^*$$, so

$$\mathbb{C}^{d \times n}/\!/\!_{\operatorname{Id}_{d \times d}} U(d)$$

$$= \operatorname{St}_d(\mathbb{C}^n)/U(d)$$

$$= \mu_{U(d)}^{-1}(\operatorname{Id}_{d \times d})/U(d)$$

$$=\operatorname{Gr}_d(\mathbb{C}^n)$$

Theorem [Atiyah, Guillemin–Sternberg

Let $$(M^{2n},\omega)$$ be a compact connected symplectic manifold with a Hamiltonian $$k$$-torus action with momentum map $$\mu: M \to \mathbb{R}^k$$. Then

• the nonempty level sets of $$\mu$$ are connected;

### The Generalized Frame Homotopy Conjecture

Theorem [with Needham, ’21]

Let $$S$$ be Hermitian and positive-definite and $$r_1,\dots , r_n > 0$$. The space of frames with $$FF^*=S$$ and $$\|f_1\|^2=r_1,\dots , \|f_n\|^2=r_n$$ is either empty or path-connected.

(the non-emptiness condition is explicit and known)

### Eigensteps

\begin{array}{ccccccccccc} 1&{}&1&{}&1&{}&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}&{}&{}&{}&{}\\ {}&{}&{}&{}&{}&\frac{3}{6}&{}&{}&{}&{}&{} \end{array}

Example for ENP frames with $$N=6, d=3$$:

\begin{array}{c} \sum_j \lambda_{6j} = 3\\ \sum_j \lambda_{5j} = \frac{5}{2}\\ \sum_j \lambda_{4j} = 2\\ \sum_j \lambda_{3j} = \frac{3}{2}\\ \sum_j \lambda_{2j} = 1\\ {} \end{array}

The eigensteps of a frame $$F = \begin{bmatrix}f_1 & \cdots & f_n\end{bmatrix}$$ are the eigenvalues $$\lambda_{kj}$$ of the partial frame operators

F_k := f_1 f_1^* + \dots + f_k f_k^*.

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

Corollary [using Duistermaat–Heckman]

The eigensteps of random ENP frames are distributed according to Lebesgue measure on the eigenstep polytope.

### Some Results

Proposition [with Needham; cf. Flaschka–Millson]

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

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

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

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

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

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

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

$$\lambda_{2j}$$

$$\lambda_{3j}$$

$$\lambda_{4j}$$

### Example

\begin{array}{cccccc} 1&{}&1&{}&{}&{}\\ {}&1&{}&\lambda_{42}&{}&{}\\ {}&{}&\lambda_{31}&{}&\lambda_{32}&{}\\ {}&{}&{}&\lambda_{21}&{}&\lambda_{22}\\ {}&{}&{}&{}&\frac{2}{5}&{} \end{array}
\begin{array}{c} \sum_j \lambda_{5j} = 2\\ \sum_j \lambda_{4j} = \frac{8}{5}\\ \sum_j \lambda_{3j} = \frac{6}{5}\\ \sum_j \lambda_{2j} = \frac{4}{5}\\ {} \end{array}
\begin{array}{cccccc} 1&{}&1&{}&{}&{}\\ {}&1&{}&\frac{3}{5}&{}&{}\\ {}&{}&\lambda_{31}&{}&\frac{6}{5}-\lambda_{31}&{}\\ {}&{}&{}&\lambda_{21}&{}&\frac{4}{5}-\lambda_{21}\\ {}&{}&{}&{}&\frac{2}{5}&{} \end{array}

$$d=2,n=5$$

### Genericity of Full Spark Frames

Proof sketch

The first $$d$$ columns of $$F$$ fail to be full spark only if

$$f_1f_1^*+\dots+f_df_d^*$$

has a zero eigenvalue; i.e., $$\lambda_{dd}=0$$.

This is the fiber over a facet of the eigenstep polytope, and hence has measure zero.

By permutation invariance of the measure, the same is true for any $$d$$ columns.

Theorem* [with Needham]

The spark-deficient frames have measure zero inside the space of ENP frames.

Similarly for any space with fixed $$FF^*$$ and $$\|f_1\|^2,\dots,\|f_n\|^2$$.

### Questions

Real Frames. The symplectic machinery only seems to apply to complex frames; is there some analogous story for frames in $$\mathbb{R}^d$$?

Symplectic reductions correspond to GIT quotients; e.g.,

$$\mathbb{C}^{d \times n}/\!/\!_{\operatorname{Id}_{d \times d}} U(d) \simeq \mathbb{C}^{d \times n}/\!/\!_{\mathscr{L}}GL_n(\mathbb{C})$$.

RIP Problem. What is the probability that a random ENP frame satisfies a Restricted Isometry Property?

$$(1-\delta) \|v\|^2 \leq \|F^* v \|^2 \leq (1+\delta) \|v\|^2$$ for all sufficiently sparse $$v$$

# Thank you!

Funding: Simons Foundation

### References

Symplectic geometry and connectivity of spaces of frames

Tom Needham and Clayton Shonkwiler

Advances in Computational Mathematics 47 (2021), no. 1, 5

arXiv:1804.05899

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

Clayton Shonkwiler

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

#### The (Symplectic) Geometry of Spaces of Frames

By Clayton Shonkwiler

• 636