The (Symplectic) Geometry of Spaces of Frames

Clayton Shonkwiler

Colorado State University

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 \(j\).

Theorem [Casazza–Kovačevic, Goyal–Kovačevic–Kelner, Holmes–Paulsen]

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.

Functions and Symplectic Gradients

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

The (Symplectic) Geometry of Spaces of Frames

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