Frames as Loops

Clayton Shonkwiler

Colorado State University


this talk!

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


Tom Needham

Florida State University


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?


\(\{f_1,\dots, f_n\}\subset \mathbb{K}^d\) 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.

Thank you!


A faster direct sampling algorithm for equilateral closed polygons

Jason Cantarella, Henrik Schumacher, and Clayton Shonkwiler


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

Linear Algebra and its Applications 645 (2022), 237–255. arXiv:2108.02275

Toric symplectic geometry and full spark frames

Tom Needham and Clayton Shonkwiler

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

Frames as Loops

By Clayton Shonkwiler

Frames as Loops

  • 187