Hamiltonian Group Actions on Frame Spaces

Clayton Shonkwiler

Colorado State University

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

Hamiltonian Group Actions on Frame Spaces

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