Frames as Loops
/ne23
this talk!
Special Session on Applied Knot Theory, Oct. 7, 2023
Collaborators
Florida State University
Funding
National Science Foundation (DMS–2107700)
Simons Foundation (#709150)
University of Georgia
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.
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.
Rejection Sampling
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!
References
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