Frames as Loops

Clayton Shonkwiler

Colorado State University

shonkwiler.org

/ne23

this talk!

Special Session on Applied Knot Theory, Oct. 7, 2023

Collaborators

Tom Needham

Florida State University

Funding

National Science Foundation (DMS–2107700)

Simons Foundation (#709150)

Jason Cantarella

University of Georgia

Henrik Schumacher

University of Georgia

Signal Analysis

Signal: \(v \in \mathbb{K}^d\), where \(\mathbb{K}=\mathbb{R}\) or \(\mathbb{C}\) or \(\mathbb{H}\).

Design: Choose \(f_1, \dots , f_n \in \mathbb{K}^d\) (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 \(f_1,\dots , f_d\in \mathbb{K}^d\) 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.

\(\{f_1,\dots, f_n\}\subset \mathbb{K}^d\) 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 \(\mathbb{K}^n\), so the space of all length-\(n\) Parseval frames in \(\mathbb{K}^d\) is the Stiefel manifold \(\operatorname{St}_d(\mathbb{K}^n)\).

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 \(\|f_i\|^2=\frac{d}{n}\).

ENP Frames as Loops

Defining Conditions: \(f_1, \dots , f_n \in \mathbb{K}^d\) so that

\(f_1f_1^* + \dots + f_n f_n^* = \mathrm{Id}_{d \times d}\)
\(\frac{d}{n} = \|f_i\|^2 = \|f_i f_i^*\|_{\mathrm{Fr}}\)

Think of \(f_if_i^* \in \mathscr{H}(d)\), the space of Hermitian \(d \times d\) matrices.

ENP Frames as Loops

Defining Conditions: \(f_1, \dots , f_n \in \mathbb{K}^d\) so that

\((f_1f_1^*-\frac{1}{n}\mathrm{Id}_{d \times d}) + \dots + (f_n f_n^*-\frac{1}{n}\mathrm{Id}_{d \times d}) = \mathbb{0}_{d \times d}\)
\(\frac{\sqrt{d(d-1)}}{n} = \|f_i f_i^*-\frac{1}{n}\mathrm{Id}_{d \times d}\|_{\mathrm{Fr}}\)

Think of \(f_if_i^*-\frac{1}{n}\mathrm{Id}_{d \times d} \in \mathscr{H}_0(d)\), the space of traceless Hermitian \(d \times d\) matrices.

\begin{bmatrix} 0.415739\, -0.00650983 i & 0.398881\, +0.0366811 i \\ -0.194377+0.100081 i & 0.484179\, -0.226066 i \\ -0.521298-0.21948 i & 0.0890816\, -0.0739928 i \\ -0.331832+0.232676 i & 0.384705\, +0.145206 i \\ 0.335624\, -0.196393 i & 0.207641\, -0.372833 i \\ -0.286823-0.248326 i & 0.394717\, -0.1833 i \end{bmatrix}

ENP frames in \(\mathbb{C}^2\) are equilateral polygons in \(\mathbb{R}^3\)!

Interpolation: Improved Polygon Sampling

Theorem [with Cantarella]

The reconstruction map \(\mathcal{P}_n \times (S^1)^{n-3} \to \mathrm{Pol}(n)/SO(3)\) is measure-preserving.

0 \leq d_i \leq 2

1 \leq d_i + d_{i-1}

|d_i - d_{i-1}| \leq 1

0 \leq d_{n-3} \leq 2

Rejection Sampling

0 \leq d_i \leq 2

1 \leq d_i + d_{i-1}

|d_i - d_{i-1}| \leq 1

0 \leq d_{n-3} \leq 2

Theorem [with Cantarella, Duplantier, and Uehara]

Rejection sampling differences of \(d_i\)’s from the hypercube yields random \(n\)-gons in expected time \(\Theta(n^{5/2})\).

The Progressive Action-Angle Method

Theorem [with Cantarella and Schumacher]

The Progressive Action-Angle Method yields random \(n\)-gons in expected time \(\Theta(n^2)\).

Some Results

Generalized Frame Homotopy Theorem [with Needham]

Let \(\mathbb{K} = \mathbb{C}\) or \(\mathbb{H}\). Let \(\boldsymbol{\lambda} = (\lambda_1,\dots , \lambda_d) \in \mathbb{R}_+^{d}\) and let

\(\boldsymbol{r} = (r_1, \dots , r_n) \in \mathbb{R}_+^n\). Then the space \(\mathscr{F}_{\boldsymbol{\lambda}}^{\mathbb{K}}(\boldsymbol{r})\) of \(F \in \mathbb{K}^{d \times n}\) with \(\operatorname{spec}(FF^\ast) = \boldsymbol{\lambda}\) and \(\|f_i\|^2 = r_i\) for all \(i=1, \dots , n\) is path-connected.

Theorem [with Needham]

\(\mathscr{F}_{\boldsymbol{\lambda}}^{\mathbb{C}}(\boldsymbol{r})\) either (i) is empty; (ii) contains no full-spark frames; or (iii) the subset of full-spark frames has full measure.

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.