Frames as Loops

Clayton Shonkwiler

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 $$j$$.

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}

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

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!

References

A faster direct sampling algorithm for equilateral closed polygons

Jason Cantarella, Henrik Schumacher, and Clayton Shonkwiler

arXiv:2309.10163

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

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