## Geometric Approaches to Frame Theory

Clayton Shonkwiler

shonkwiler.org

/isu22

this talk!

Geometric Analysis Seminar, Feb. 28, 2022

### Collaborator

Tom Needham

Florida State University

### Funding

National Science Foundation (DMS–2107700)

Simons Foundation (#709150)

### Finite Frames

$$\mathfrak{H} = \mathbb{R}^d$$ or $$\mathbb{C}^d$$. A (finite) frame in $$\mathfrak{H}$$ is a spanning set $$f_1, \dots, f_N \in \mathfrak{H}$$.

### The Model Problem

Signal: $$v \in \mathfrak{H}$$

Design: $$f_1, \dots , f_N \in \mathfrak{H}$$

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

Parseval’s Theorem

If $$f_1, \dots , f_d \in \mathfrak{H}$$ is an orthonormal basis,

$$\|v\|^2 = \sum |\langle f_i, v \rangle |^2 = \|F^* v\|^2$$

for any $$v \in \mathfrak{H}$$.

Even better,

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

If $$F = [f_1 \, f_2 \dots f_N]$$, the measurement vector is $$F^*v$$.

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

Parseval frame

Fact: For any $$n \geq d$$, there exist Parseval frames $$f_1, \dots , f_N \in \mathfrak{H}$$ so that $$\|f_i\|=\|f_j\|$$ for all $$i,j$$; these are the equal-norm Parseval frames (ENPs).

This is fragile! What if a measurement gets lost?

### Questions

We are often interested in specifying $$\operatorname{spec}(FF^\ast) = (\lambda_1, \dots , \lambda_d)$$ and/or $$(\|f_1\|^2,\dots , \|f_N\|^2) = (r_1, \dots, r_N)$$.

• [Admissibility] Is the space of frames with this data non-empty? Is it smooth?
• [Frame homotopy] Is it connected?
• [Sampling] Can we randomly sample it?
• [Robustness] Is a typical frame with this data well-suited to the reconstruction problem?

### Gram Matrices

Both $$\operatorname{spec}(FF^\ast) = (\lambda_1, \dots , \lambda_d)$$ and $$(\|f_1\|^2,\dots , \|f_N\|^2) = (r_1, \dots, r_N)$$ are present in the Gram matrix

$$F^\ast F = \left[\langle f_i, f_j\rangle \right]_{i,j} \in \mathscr{H}(N)$$

since $$F^\ast F$$ and $$F F^\ast$$ have the same rank and nonzero eigenvalues.

Unitary equivalence classes of frames have the same Gram matrix: if $$U \in O(d)$$ or $$U(d)$$, then

$$(UF)^\ast(UF) = F^\ast U^\ast U F = F^\ast F$$.

For $$\boldsymbol{\lambda} = (\lambda_1, \dots , \lambda_d, 0, \dots 0)$$,

$$\mathcal{O}_{\boldsymbol{\lambda}} := \{U \operatorname{diag}(\boldsymbol{\lambda}) U^\ast | U \in U(N) \text{ or } O(N)\}\subset \mathscr{H}(N)$$

is a flag manifold.

### Lie Groups

Informal Definition. A Lie group is a group that is also a manifold (a “continuous group of symmetries”).

Examples.

• $$\mathbb{R}^n$$, $$\mathbb{C}^n$$
• $$GL_n(\mathbb{R})$$, $$GL_n(\mathbb{C})$$
• $$SL_n(\mathbb{R})$$, $$SL_n(\mathbb{C})$$
• $$O(n)$$, $$SO(n)$$, $$U(n)$$, $$SU(n)$$ (e.g., $$SO(3)\simeq \mathbb{RP}^3$$, $$SU(2) \simeq S^3$$)
• Products (e.g. the compact torus $$U(1)^n$$)

### Lie Algebras

Informal Definition. A Lie algebra $$\mathfrak{g}$$ (or $$\operatorname{Lie}(G)$$) associated to a Lie group $$G$$ is the tangent space at the identity (the “infinitesimal symmetries”).

Examples.

• $$\operatorname{Lie}(\mathbb{R}^n) \simeq \mathbb{R}^n$$
• $$\mathfrak{gl}_n(\mathbb{R}) \simeq \mathbb{R}^{n \times n}$$, $$\mathfrak{gl}_n(\mathbb{C}) \simeq \mathbb{C}^{n \times n}$$
• $$\mathfrak{sl}_n(\mathbb{R}) \simeq \{\text{traceless matrices}\}$$, $$\mathfrak{sl}_n(\mathbb{C}) \simeq \{\text{traceless matrices}\}$$
• $$\mathfrak{o}(n) = \mathfrak{so}(n) \simeq \{\text{skew-symmetric matrices}\}$$, $$\mathfrak{u}(n) \simeq \{\text{skew-Hermitian matrices}\}$$, $$\mathfrak{su}(n) \simeq \{\text{traceless skew-Hermitian}\}$$

Suppose $$\gamma(t) \in U(n)$$ with $$\gamma(0)=I$$. Since $$I = \gamma(t)\gamma(t)^\ast$$,

$$0 = \left. \frac{d}{dt}\right|_{t=0} \gamma(t)\gamma(t)^\ast = \gamma'(0)\gamma(0)^\ast + \gamma(0)\gamma'(t)^\ast = \gamma'(0) + \gamma'(0)^\ast$$

The Lie bracket $$[\cdot , \cdot ]$$ is the matrix commutator on matrix groups: $$[A,B] = AB-BA$$.

A Lie group $$G$$ acts on itself by conjugation: $$g \cdot h := g h g^{-1}$$.

This fixes the identity, so linearizing gives an action of $$G$$ on $$\mathfrak{g}$$, called the adjoint action.

For matrix groups, the adjoint action is the conjugation action.

Example. $$\mathfrak{so}(3) \simeq \left\{\begin{pmatrix} 0 & -z & y \\ z & 0 & -x \\ -y & x & 0\end{pmatrix}:x,y,z \in \mathbb{R}\right\}\simeq (\mathbb{R}^3, \times)$$

\begin{pmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & -z & y \\ z & 0 & -x \\ -y & x & 0 \end{pmatrix} \begin{pmatrix} \cos \theta & \sin \theta & 0 \\ -\sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix} \\ = \begin{pmatrix} 0 & -z & y\cos \theta + x \sin \theta \\ z & 0 & -x\cos \theta + y \sin \theta \\ -y\cos \theta - x\sin \theta & x \cos \theta - y \sin \theta & 0 \end{pmatrix}

If $$G$$ is a Lie group, each orbit of the adjoint action is a symplectic manifold.$$^1$$

1. Strictly speaking, coadjoint orbits in $$\mathfrak{g}^\ast$$ are symplectic.

$$i\mathscr{H}(N) = \mathfrak{u}(N)$$, so conjugation orbits of Hermitian matrices can be interpreted as adjoint orbits.

### Symplectic Geometry

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}

Non-Example: $$(\mathbb{R}^3, dx \wedge dy)$$

For any vector $$v$$, $$dx \wedge dy\left(\frac{\partial}{\partial z}, v\right) = 0$$, so $$dx \wedge dy$$ is degenerate.

Non-Example: $$(\mathbb{R}^4, dx_1 \wedge dy_1 + y_1 dx_2 \wedge dy_2)$$

Because of the coefficient $$y_1$$, this is not closed (“not divergence-free”).

### Examples

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

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

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

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

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

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

$$(T^* \mathbb{R}^n,\sum dq_i \wedge dp_i)$$

phase space

position

momentum

### Symplectic Geometry and Frame Theory

$$\mathcal{O}_{\boldsymbol{\lambda}} = \{U \operatorname{diag}(\boldsymbol{\lambda})U^\ast | U \in U(N)\} \simeq \{F \in \mathbb{C}^{d \times N} | \operatorname{spec}(FF^\ast) = (\lambda_1, \dots , \lambda_d)\}/U(d)$$ is symplectic.

Proposition. The map recording diagonal entries is the momentum map of a Hamiltonian torus action.

Theorem [Atiyah and Guillemin–Sternberg]

Momentum maps of Hamiltonian torus actions have convex image and connected level sets.

Theorem [with Needham, 2021]

The space $$\mathscr{F}_{\boldsymbol{\lambda}}^{\mathbb{C}^d,N}(\boldsymbol{r})\subset \mathbb{C}^{d \times N}$$ of frames with $$\operatorname{spec}(FF^\ast) = (\lambda_1, \dots , \lambda_d)$$ and $$\|f_i\|^2 = r_i$$ for $$i=1, \dots , N$$ is path-connected.

This solves the general frame homotopy problem for complex frames.

### Robustness

For compressed sensing, it is desirable to require every minor of the $$d \times N$$ matrix $$F$$ to be invertible. Such an $$F$$ is said to have full spark.

Theorem [with Needham]

Fix the spectrum of $$FF^\ast$$ and fix $$\|f_1\|,\dots,\|f_N\|$$. There are three possibilities:

• It is impossible for a frame to have these data
• It is impossible for a frame with these data to have full spark
• A random frame with these data will have full spark with probability 1.

### How Could This Generalize?

$$\dim(\mathfrak{o}(N)) = \frac{N(N-1)}{2}$$

$$\dim(\{\text{symmetric } N \times N \text{ matrices}\})=\frac{N(N+1)}{2}$$

The analog of the unitary group for symplectic matrices is the (compact) symplectic group $$\operatorname{Sp}(N)$$; its Lie algebra $$\mathfrak{sp}(N)$$ consists of quaternionic skew-Hermitian matrices.

$$\dim(\mathfrak{sp}(N)) = 2N^2+N$$

$$\dim(\{\text{Hermitian quaternionic } N \times N \text{ matrices}\}) = 2N^2-N$$

Can’t identify these!

### Isoparametric Submanifolds

Definition. A submanifold $$M$$ of a Riemannian manifold $$N$$ is isoparametric if its normal bundle $$\nu(M)$$ is flat and the principal curvatures along any parallel normal field are constant.

Proposition/Example. If $$\boldsymbol{\lambda}$$ is generic (i.e., $$\lambda_i \neq \lambda_j$$ for $$i \neq j$$), then $$\mathcal{O}_{\boldsymbol{\lambda}}\subset \mathscr{H}(N)$$ is isoparametric.

If $$\boldsymbol{\lambda}$$ is not generic, then $$\mathcal{O}_{\boldsymbol{\lambda}}$$ is parallel to an isoparametric submanifold.

If $$\xi$$ is a parallel section of $$\nu(M)$$, then $$M_\xi :=\{p+\xi(p):p \in M\}$$ is parallel to $$M$$

### Nice Features

Theorem [Terng]

Let $$M_\xi\subset \mathbb{R}^n$$ be parallel to an isoparametric submanifold $$M$$. Let $$p \in M$$ so that $$p+\xi(p) \in M_\xi$$, let $$\nu_p(M)$$ be the normal space to $$M$$ at $$p$$, and let $$P:M_\xi \to \nu_p(M)$$ be orthogonal projection. Then $$P(M_\xi)$$ is a convex polytope.

Theorem [Mare]

Let $$M\subset \mathbb{R}^n$$ be isoparametric with all multiplicities $$\geq 2$$ and let $$M_\xi$$ be parallel to $$M$$. If $$p \in M$$ and $$b \in \nu_p(M)$$ is in the image of the projection $$P: M_\xi \to \nu_p(M)$$, then $$P^{-1}(b)$$ is connected.

Theorem [Atiyah and Guillemin–Sternberg]

Momentum maps of Hamiltonian torus actions have convex image and connected level sets.

For adjoint orbits, this is the difference in dimension between principal and sub-principal orbits.

### Cartan Decomposition

An involution $$\theta: \mathfrak{g} \to \mathfrak{g}$$ is a Cartan involution if $$-B(\cdot , \theta \cdot)$$ is positive-definite, where $$B(\cdot , \cdot)$$ is the Killing form.

Example. $$\theta: \mathfrak{sl}_n(\mathbb{R}) \to \mathfrak{sl}_n(\mathbb{R})$$ defined by $$\theta(X) = -X^T$$. This is the derivative of the involution on $$SL_n(\mathbb{R})$$ given by $$U \mapsto (U^{-1})^T$$.

$$E_{+1} =: \mathfrak{k}$$ consists of skew-symmetric matrices

$$E_{-1} =: \mathfrak{p}$$ consists of (traceless) symmetric matrices

$$\mathfrak{sl}_n(\mathbb{R}) = \mathfrak{k} \oplus \mathfrak{p}$$ is an orthogonal decomposition — the Cartan decomposition

$$[\mathfrak{k},\mathfrak{p}]\subseteq \mathfrak{p}$$, so the adjoint action of $$SO(n) = \exp(\mathfrak{k})$$ restricts to an action on $$\mathfrak{p}$$

$$[\mathfrak{p},\mathfrak{p}]\subseteq\mathfrak{k}$$, so any Lie subalgebra of $$\mathfrak{p}$$ is abelian.

Theorem. $$G$$ non-compact, semisimple, and $$\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$$ a Cartan decomposition with $$K = \exp(\mathfrak{k})$$ and $$P = \exp(\mathfrak{p})$$, then $$G = KP$$ and $$T_{[I]} G/K \simeq \mathfrak{p}$$.

The adjoint action of $$K$$ on $$\mathfrak{p}$$ is often called the isotropy representation.

### Kostant’s Convexity Theorem

Proposition [see, e.g., Palais–Terng Example 6.5.6]

Let $$G$$ be a semisimple Lie group with Cartan decomposition $$\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$$.

The isotropy orbits are (parallel to) isoparametric submanifolds of $$\mathfrak{p}$$, and, if $$\mathfrak{a} \subset \mathfrak{p}$$ is a maximal abelian subspace and $$a \in \mathfrak{a}$$, then $$\nu_a(\mathcal{O}_a) = \mathfrak{a}$$.

Example.

$$G = SL_n(\mathbb{C})$$, $$K=SU(N)$$, $$\mathfrak{k} = \mathfrak{su}(N)$$, $$\mathfrak{p} = \mathscr{H}_0(N)$$, $$a = \operatorname{diag}(a_1, \dots , a_N)$$,

$$\mathcal{O}_a = \{UaU^\ast | U \in SU(N)\}$$

$$P(\mathcal{O}_a) = \operatorname{conv}(S_N \cdot a)$$ is essentially Schur–Horn.

This gives a non-symplectic proof of frame homotopy for complex frame spaces!

In particular, Terng and Mare’s theorems imply the projection of $$\mathcal{O}_a$$ onto $$\mathfrak{a}$$ has convex image and connected level sets.

### Quaternions

$$\mathbb{H} = \{a + b \boldsymbol{i} + c \boldsymbol{j} + d \boldsymbol{k}\}$$

$$\boldsymbol{i}^2 = \boldsymbol{j}^2 = \boldsymbol{k}^2 = -1$$

$$\boldsymbol{i}\boldsymbol{j}\boldsymbol{k} = -1$$

$$z+w \boldsymbol{j} \mapsto \begin{pmatrix} z & w \\ -\overline{w} & \overline{z}\end{pmatrix}$$

$$\mathbb{H}^d$$ is a right vector space over the skew-field $$\mathbb{H}$$.

### Quaternionic Frames

A spanning set $$\{f_1, \dots , f_N\} \subset \mathbb{H}^d$$ is a (finite) frame for $$\mathbb{H}^d$$.

Much of standard frame theory goes through for quaternionic frames (see, e.g., Waldron).

In particular, the frame operator $$FF^\ast$$ and the Gram matrix $$F^\ast F$$ are well-defined, and $$\operatorname{Sp}(d)$$ equivalence classes of frames in $$\mathbb{H}^{d \times N}$$ are uniquely determined by their Gram matrices.

### Realizing Quaternion Matrices as Complex Matrices

$$\Psi:\mathbb{H}^{N \times N} \to \mathbb{C}^{2N \times 2N}$$ given by

$$\Psi(Z + W\boldsymbol{j}) = \begin{pmatrix} Z & W \\ -\overline{W} & \overline{Z} \end{pmatrix}$$.

$$\Psi(\operatorname{Sp}(N)) \subset SU(2N)$$ is the fixed point set of the involution

$$U \mapsto \Omega^\ast \overline{U} \Omega$$,

where $$\Omega = \begin{pmatrix} 0 & I \\ -I & 0 \end{pmatrix}$$.

Corresponding Cartan decomposition of the Lie algebra:

$$\mathfrak{su}(2N) = \mathfrak{k} \oplus \mathfrak{p} \simeq \Psi(\mathfrak{sp}(N)) \oplus \Psi(\mathscr{H}_0(N))$$.

### Results

Admissibility for Quaternionic Frames [with Needham]

$$\mathscr{F}_{\boldsymbol{\lambda}}^{\mathbb{H}^d,N}(\boldsymbol{r})\neq \emptyset$$ if and only if $$\boldsymbol{r} \in \operatorname{conv}(S_N \cdot (\lambda_1, \dots , \lambda_d, 0, \dots , 0))$$ (cf. Casazza–Leon).

$$\mathscr{F}_{\boldsymbol{\lambda}}^{\mathbb{H}^d,N}(\boldsymbol{r}):= \{F \in \mathbb{H}^{d \times N} | \operatorname{spec}(FF^\ast)=\boldsymbol{\lambda}, \|f_i\|^2 = r_i \text{ for all } i=1,\dots, N\}$$

$$\mathcal{O}_{\boldsymbol{\lambda}} = \{U \operatorname{diag}(\lambda_1, \dots , \lambda_d, 0, \dots , 0)U^\ast | U \in \operatorname{Sp}(N)\}$$

Key Idea. Terng/Kostant convexity.

Generalized Frame Homotopy for Quaternionic Frames [with Needham]

$$\mathscr{F}_{\boldsymbol{\lambda}}^{\mathbb{H}^d,N}(\boldsymbol{r})$$ is always path-connected.

Key Idea. Mare’s connectedness theorem.

### Questions

Isotropy orbits of real symmetric matrices do not satisfy the technical hypothesis in Mare’s theorem; indeed, some $$\mathscr{F}_{\boldsymbol{\lambda}}^{\mathbb{R}^d,N}(\boldsymbol{r})$$ spaces are disconnected. Can we characterize the $$(\boldsymbol{\lambda},\boldsymbol{r})$$ which lead to connectedness?

Does the Riemannian geomery of isoparametric submanifolds give any insight into the measure on real or quaternionic frame spaces? Does this lead to a sampling algorithm?

# Thank you!

### 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

Admissibility and frame homotopy for quaternionic frames

Tom Needham and Clayton Shonkwiler

Preprint, 2021

arXiv:2108.02275

Toric symplectic geometry and full spark frames

Tom Needham and Clayton Shonkwiler

Preprint, 2021

arXiv:2110.11295

#### Geometric Approaches to Frame Theory

By Clayton Shonkwiler

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