Geometric Approaches to Frame Theory

Clayton Shonkwiler

Colorado State University

shonkwiler.org

/geotop-a-23

this talk!

Seminar GEOTOP–A, Nov. 17, 2023

Collaborators

Tom Needham

Florida State University

Funding

National Science Foundation (DMS–2107700)

Simons Foundation (#709150)

Dustin Mixon

The Ohio State University

Soledad Villar

Johns Hopkins University

Signal Analysis

Signal: \(v \in \mathbb{C}^d\).

Design: Choose \(f_1, \dots , f_n \in \mathbb{C}^d\) (a frame).

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 \(\operatorname{St}_d(\mathbb{C}^n)\).

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 white noise and erasures.

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

Frame Parameters

\(\operatorname{spec}(FF^*) = (\lambda_1, \dots , \lambda_d)\)

\((\|f_1\|^2, \dots , \|f_n\|^2)\)

choose based on noise model

detector power

Questions:

  • [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?

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

\(d(dx_1 \wedge dy_1 + y_1 dx_2 \wedge dy_2) = dy_1 \wedge dx_2 \wedge dy_2 \neq 0\)

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

What does this get you?

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 \(M\).

In particular, if \(M\) is compact, this can be normalized to give a probability measure.

Theorem [Archimedes]

\(\mathrm{d}\theta \wedge \mathrm{d}z\) is the standard area form on the unit sphere \(S^2\).

\int_{S^2} \mathrm{d}\theta \wedge \mathrm{d}z = \int_{-1}^1 \int_0^{2\pi} \mathrm{d}\theta\, \mathrm{d}z = 4\pi

A (Very) Classical Example

KoenB [   ] from Wikimedia Commons

Hamiltonian Mechanics

“Evolution of a physical system follows the symplectic gradient”

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 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 \(X_H = \frac{\partial}{\partial \theta}\).

\(H\) is constant on orbits of \(X_H\):

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

Some Mechanics

Suppose we have a particle of mass \(m\) with position \(\vec{q} = (q_1,q_2,q_3)\).

Suppose the particle is subject to a potential \(V(\vec{q})\).

Newton’s Second Law: \(-\nabla V(\vec{q}) = m \frac{d^2 \vec{q}}{dt^2}\).

Momenta \(p_i := m \frac{dq_i}{dt}\).

\((\vec{q},\vec{p})\in T^*\mathbb{R}^3 \cong \mathbb{R}^6\); canonical symplectic form \(\omega = dq_1 \wedge dp_1 + dq_2 \wedge dp_2 + dq_3 \wedge dp_3\).

The Hamiltonian Formulation

Energy function (or Hamiltonian)

\(H(\vec{q},\vec{p}) := \frac{1}{2m}|\vec{p}|^2 + V(\vec{q})\).

\(dH = \sum\left(\frac{\partial H}{\partial p_i} dp_i + \frac{\partial H}{\partial q_i}dq_i\right)\),

\((\vec{q}(t),\vec{p}(t))\) is an integral curve for \(X_H\) if and only if

\(\frac{dq_i}{dt} = \frac{\partial H}{\partial p_i} = \frac{1}{m} p_i\)

\(\frac{dp_i}{dt} = -\frac{\partial H}{\partial q_i} = -\frac{\partial V}{\partial q_i}.\)

This is Newton’s Second Law written as a first-order system!

\(X_H = \sum\left(\frac{\partial H}{\partial p_i} \frac{\partial}{\partial q_i} - \frac{\partial H}{\partial q_i} \frac{\partial}{\partial p_i}\right)\).

Noether’s Theorem

“Every continuous symmetry has a corresponding conserved quantity”

Circle Actions

A circle action on \((M,\omega)\) determines a vector field \(X\) by

X(p) = \left.\frac{d}{dt}\right|_{t=0}e^{i t} \cdot p

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

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

So \(X = \frac{\partial}{\partial \theta}\).

Symmetries and Conserved Quantities

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

\mu: M \to \mathbb{R}

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

\(X = \frac{\partial}{\partial \theta}\)

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

\(\iota_X\omega = \iota_{\frac{\partial}{\partial \theta}} d\theta \wedge dz = (d\theta \wedge dz)\left(\frac{\partial}{\partial \theta},\cdot \right) = dz \)

Generalizing Archimedes

Theorem [Atiyah and Guillemin–Sternberg]

If \((M,\omega)\) is symplectic and \(\mathbb{T}=(S^1)^k\) acts in a Hamiltonian fashion, the conserved quantities are recorded by \(\mu:M \to \mathbb{R}^k\) with:

  1. The level sets of \(\mu\) are connected.
  2. \(\mu(M)\) is the convex hull of the images of the fixed points of the \(\mathbb{T}\) action.

Theorem [Duistermaat–Heckman]

If \(\dim(M)=2n\) and \(\dim(\mathbb{T})=k\), then the pushforward measure on the moment polytope \(\mu(M)\) is absolutely continuous w.r.t. Lebesgue measure, with continuous, piecewise-polynomial density of degree \(\leq n-k\).

 

In particular, if \(n=k\), then the pushforward measure is a constant multiple of Lebesgue measure.

Example

Theorem (Archimedes)

Let \(\mu: S^2 \to \mathbb{R}\) be given by \(\mu(x,y,z) = z\). Pushing forward the uniform measure on \(S^2\) to the image \([-1,1]\) gives (\(2\pi\) times) Lebesgue measure.

Group Actions on Frames

Some nice group actions 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(\|f^1\|^2 , \dots , \|f^d\|^2\right)
\mu_{U(1)^n} \left(\begin{bmatrix} f_1 | \cdots | f_n \end{bmatrix}\right) = \left(-\|f_1\|^2 , \dots , -\|f_n\|^2\right)

Parseval frames

\(\mu_{U(d)}^{-1}(I_{d\times d})\)

equal-norm frames

\(\mu_{U(1)^N}^{-1}\left(-r,\dots , -r\right)\)

ENPs: \(\mu_{U(d)}^{-1}(I_{d \times d}) \cap \mu_{U(1)^n}^{-1}\left(-\frac{d}{n}, \dots , -\frac{d}{n}\right)\)

A Generalization of the Frame Homotopy Conjecture

Theorem [with Needham]

There is a continuous interpolation between any two frames \(f_1, \dots , f_n\) and \(g_1, \dots , g_n\) with \(\operatorname{spec}(FF^*) = \operatorname{spec}(GG^*)\) and \(\|f_i\| = \|g_i\|\) preserving these conditions.

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.

Frame Potential

Definition [Benedetto–Fickus, Casazza–Fickus]

The frame potential is

\(\operatorname{FP}(f_1,\ldots,f_n) = \|FF^\ast\|_{\operatorname{Fr}}^2\)

Proposition [cf. Welch]

When they exist, the ENP frames are exactly the global minima of \(\operatorname{FP}\).

Theorem [Benedetto–Fickus]

As a function on equal-norm frames with fixed \(d\) and \(n\), \(\operatorname{FP}\) has no spurious local minima.

Optimization

Theorem [with Mixon, Needham, and Villar]

Consider the initial value problem

\(\Gamma(F_0,0) = F_0, \qquad \frac{d}{dt}\Gamma(F_0,t) = -\operatorname{grad}\operatorname{FP}(\Gamma(F_0,t))\).

If \(F_0\) has full spark, then \(\lim_{t \to \infty} \Gamma(F_0,t)\) is an ENP frame.

Theorem [with Needham]

Same for fusion frames.

Why?

\(F \mapsto FF^*\) is the momentum map of the diagonal \(SU(d)\) action on \(\prod_{i=1}^n \mathbb{CP}^{d-1}\).

\(\Longrightarrow\) \(\operatorname{FP}(F) = \|FF^*\|_{\operatorname{Fr}}^2\) is the squared norm of a momentum map.

Frances Kirwan

Theorem [Kirwan]

Reductive algebraic group action on Kähler manifold \(\Longrightarrow\) semistable points flow to global minima.

This kind of function is really nice!

Geometric Invariant Theory (GIT)

The GIT quotient consists of group orbits which can be distinguished by \(G\)-invariant (homogeneous) polynomials.

\(\mathbb{C}^* \curvearrowright \mathbb{CP}^2\)

\(t \cdot [z_0:z_1:z_2] = [z_0: tz_1:\frac{1}{t}z_2]\)

Roughly: identify orbits whose closures intersect, throw away orbits on which all \(G\)-invariant polynomials vanish.

\( \mathbb{CP}^2/\!/\,\mathbb{C}^* \cong\mathbb{CP}^1\)

Equiangular Tight Frames

An ENP frame \(F\) is equiangular if there exists \(\alpha\) so that \(|\langle f_i , f_j \rangle| = \alpha\) for all \(i \neq j\). Equiangular ENP frames are usually called equiangular tight frames (ETFs).

If \(F \in \mathbb{C}^{d \times n}\) is an ETF, then \((f_1f_1^* - \frac{1}{n}\operatorname{Id}_{d \times d}, \dots , f_nf_n^* - \frac{1}{n}\operatorname{Id}_{d \times d})\) are the vertices of a regular \((n-1)\)-simplex in \(\mathscr{H}_0(d) \simeq \mathfrak{su}(d)^\ast\). In particular, this implies \(n \leq d^2\).

(Weak) Zauner Conjecture

For all positive integers \(d\), there exists an ETF \(F \in \mathbb{C}^{d \times d^2}\).

Such ETFs are called maximal ETFs in the math literature, and Symmetric, Informationally-Complete, Positive Operator-Valued Measures (SIC-POVMs) in quantum information theory.

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

Toric symplectic geometry and full spark frames

Tom Needham and Clayton Shonkwiler

Applied and Computational Harmonic Analysis 61 (2022), 254–287. arXiv:2110.11295

Three proofs of the Benedetto–Fickus theorem

Dustin Mixon, Tom Needham, Clayton Shonkwiler, and Soledad Villar

Sampling, Approximation, and Signal Analysis (Harmonic Analysis in the Spirit of J. Rowland Higgins), Stephen D. Casey, M. Maurice Dodson, Paulo J. S. G. Ferreira and Ahmed Zayed, eds., to appear. arXiv:2112.02916

Fusion frame homotopy and tightening fusion frames by gradient descent

Tom Needham and Clayton Shonkwiler

Journal of Fourier Analysis and Applications 29 (2023), no. 4, 51. arXiv:2208.11045

Geometric Approaches to Frame Theory

By Clayton Shonkwiler

Geometric Approaches to Frame Theory

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