Equiangular Tight Frames
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6755976/JustSD.png)
John Jasper
South Dakota State University
MRI: Resolution v. Scan Time
Pixels
Time
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9360826/pasted-from-clipboard.png)
High resolution \(\Longleftrightarrow\) Long scan time
Real world signals are sparse.
\(\Downarrow\)
The conventional tradeoff is a lie!
We can solve "underdetermined"
\[Ax=b\] by \(L^{1}\) minimization provided:
- \(x\) is sparse, and
- Columns of \(A\) are "spread out"
Pixels
Time
Compressed Sensing
We need an image with many pixels:
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7153670/x.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7153671/ColorSensingMatrix1Rows.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7153672/ColorSensingMatrix2Rows.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7153674/ColorSensingMatrix3Rows.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7153675/ColorSensingMatrix15Rows.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7153676/ColorSensingMatrix.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7153677/y1Rows.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7153679/y2Rows.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7153680/y3Rows.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7153682/y15Rows.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7153684/y.png)
\(=\)
Image
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7153723/Tx.png)
Transformed Image
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7153676/ColorSensingMatrix.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7153745/Ty.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7153676/ColorSensingMatrix.png)
- A measurement is a single dot product
- Using old linear algebra we need as many dot products as pixels
- Under an invertible transformation the image is sparse (lots of zeros)
Compressed Sensing
We need an image with many pixels:
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7153675/ColorSensingMatrix15Rows.png)
\(=\)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7153723/Tx.png)
Transformed Image
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7153814/Ty15Rows.png)
Column vectors need to be "spread out" in space
- For sparse signals we need a lot fewer measurements
- But the columns of our sensing matrix must be "spread out" in space
Roadmap for this talk
Compressed Sensing
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6932431/symmetry2_3.png)
Vectors that are "Spread out"
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6932415/symmetry1_3.png)
Equiangular tight frames
ETFs from groups
\(\Z_{2}\times\Z_{2}\times\Z_{2}\times\Z_{2}\)
The design theory underneath
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7154657/pasted-from-clipboard.png)
Real Flat ETFs
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6952674/3x4Simplex.png)
Some binary codes
Group divisible designs
ETFs and graphs
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7154412/6x16WaldronGraphPicture.png)
Definition. Given unit vectors \(\Phi=(\varphi_{i})_{i=1}^{N}\), we define the coherence
\[\mu(\Phi) = \max_{i\neq j}|\langle \varphi_{i},\varphi_{j}\rangle|.\]
Measuring how "spread out" vectors are
\[\mu(\Phi) = \cos(\theta)\]
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6932431/symmetry2_3.png)
\(\mu(\Phi) = \cos(\theta)\)??
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6932433/symmetry3_3.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6932436/symmetry3_4.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6932437/symmetry3_5.png)
\(\mu(\Phi) = \cos(\theta)\)
Measuring how "spread out" vectors are
Definition. Given unit vectors \(\Phi=(\varphi_{i})_{i=1}^{N}\), we define the coherence
\[\mu(\Phi) = \max_{i\neq j}|\langle \varphi_{i},\varphi_{j}\rangle|.\]
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6932415/symmetry1_3.png)
Example.
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6932424/symmetry1_4.png)
Measuring how "spread out" vectors are
Definition. Given unit vectors \(\Phi=(\varphi_{i})_{i=1}^{N}\), we define the coherence
\[\mu(\Phi) = \max_{i\neq j}|\langle \varphi_{i},\varphi_{j}\rangle|.\]
Vectors that are as spread out as possible
Theorem (the Welch bound). Given a collection of unit vectors
\(\Phi=(\varphi_{i})_{i=1}^{N}\) in \(\mathbb{C}^d\), the coherence satisfies
\[\mu(\Phi)\geq \sqrt{\frac{N-d}{d(N-1)}}.\]
Equality holds if and only if the following two conditions hold:
- Tight: There is a constant \(A>0\) such that \[\sum_{i=1}^{N}|\langle v,\varphi_{i}\rangle|^{2} = A\|v\|^{2} \quad\text{for all } v.\]
- Equiangular: There is a constant \(\alpha\) such that \[|\langle\varphi_{i},\varphi_{j}\rangle| = \alpha\quad\text{for all }i\neq j.\]
Welch bound equality \(\Longleftrightarrow\) equiangular tight frame (ETF)
Tightness and short, fat matrices
Useful matrix representation: \(\quad\Phi = \begin{bmatrix} | & | & & |\\ \varphi_{1} & \varphi_{2} & \cdots & \varphi_{N}\\ | & | & & |\end{bmatrix}\)
Tightness: There is a constant \(A>0\) such that \[\sum_{i=1}^{N}|\langle v,\varphi_{i}\rangle|^{2} = A\|v\|^{2} \quad\text{for all } v.\]
( )
\(\Leftrightarrow\quad\Phi\Phi^{\ast} = AI\)
\(\Leftrightarrow\quad\) the rows of \(\Phi\) are orthogonal and equal norm
\(\langle v,\Phi\Phi^{\ast}v\rangle = \)
\(\langle v,\Phi\Phi^{\ast}v\rangle = \)
Examples of equiangular tight frames
Example 2. Consider the (multiple of a) unitary matrix
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6932612/symmetry1_3.png)
Example 1. Consider the (multiple of a) unitary matrix
\[\left[\begin{array}{rrrr}1 & -1 & 1 & -1\\ 1 & 1 & -1 & -1\\ 1 & -1 & -1 & 1\end{array}\right]\]
\[\left[\begin{array}{rrrr} 1 & 1 & 1 & 1\\ 1 & -1 & 1 & -1\\ 1 & 1 & -1 & -1\\ 1 & -1 & -1 & 1\end{array}\right]\]
\[\left[\begin{array}{rrrr} 1 & 1 & 1\\ \sqrt{2} & -\sqrt{\frac{1}{2}} & -\sqrt{\frac{1}{2}}\\ 0 & \sqrt{\frac{3}{2}} & -\sqrt{\frac{3}{2}}\end{array}\right]\]
\[\left[\begin{array}{rrrr}\sqrt{2} & -\sqrt{\frac{1}{2}} & -\sqrt{\frac{1}{2}}\\ 0 & \sqrt{\frac{3}{2}} & -\sqrt{\frac{3}{2}}\end{array}\right]\]
Examples of equiangular tight frames
Example 3.
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6932993/Ex30.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6932994/Ex31.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6932996/Ex32.png)
Theme of the talk:
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9357677/Drake-Hotline-Bling.jpg)
\(\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}\)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7154657/pasted-from-clipboard.png)
Some ETFs
arise from
groups...
a lot more ETFs
arise from
combinatorial designs!
M
Roadmap for this talk
Compressed Sensing
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6932431/symmetry2_3.png)
Vectors that are "Spread out"
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6932415/symmetry1_3.png)
Equiangular tight frames
ETFs from groups
\(\Z_{2}\times\Z_{2}\times\Z_{2}\times\Z_{2}\)
The design theory underneath
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7154657/pasted-from-clipboard.png)
Real Flat ETFs
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6952674/3x4Simplex.png)
Some binary codes
Group divisible designs
ETFs and graphs
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7154412/6x16WaldronGraphPicture.png)
\[\Z_{7}\left\{\begin{array}{c} 0\\ 1\\ 2\\ 3\\ 4\\ 5\\ 6 \end{array}\right. \left[\begin{array}{ccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & \omega & \omega^2 & \omega^3 & \omega^4 & \omega^5 & \omega^6\\ 1 & \omega^2 & \omega^4 & \omega^6 & \omega & \omega^3 & \omega^5\\ 1 & \omega^3 & \omega^6 & \omega^2 & \omega^5 & \omega & \omega^4\\ 1 & \omega^4 & \omega & \omega^5 & \omega^2 & \omega^6 & \omega^3\\ 1 & \omega^5 & \omega^3 & \omega & \omega^6 & \omega^4 & \omega^2\\ 1 & \omega^6 & \omega^5 & \omega^4 & \omega^3 & \omega^2 & \omega \end{array}\right]\]
\[\begin{array}{c} 1\\ 2\\ 4 \end{array}\left[\begin{array}{ccccccc} 1 & \omega & \omega^2 & \omega^3 & \omega^4 & \omega^5 & \omega^6\\ 1 & \omega^2 & \omega^4 & \omega^6 & \omega & \omega^3 & \omega^5\\ 1 & \omega^4 & \omega & \omega^5 & \omega^2 & \omega^6 & \omega^3 \end{array}\right]\]
Rows from a DFT
(\(\omega = e^{2\pi i/7}\))
\[\Phi = \left[\begin{array}{ccccccc} 1 & \omega & \omega^2 & \omega^3 & \omega^4 & \omega^5 & \omega^6\\ 1 & \omega^2 & \omega^4 & \omega^6 & \omega & \omega^3 & \omega^5\\ 1 & \omega^4 & \omega & \omega^5 & \omega^2 & \omega^6 & \omega^3 \end{array}\right]\]
Rows from a DFT
\(\Phi\) is tight, since it is rows out of a unitary.
\(\Phi\) is equiangular, since \(D=\{1,2,4\}\subset\Z_{7}\) is a difference set.
That is, if we look at the difference table
\[\begin{array}{r|rrr} - & 1 & 2 & 4\\ \hline 1 & 0 & 6 & 4\\ 2 & 1 & 0 & 5\\ 4 & 3 & 2 & 0 \end{array}\]
every nonidentity group element shows up the same number of times
Difference sets \(\Rightarrow\) equiangular?
\(\Phi^{\ast}\Phi = \operatorname{circ}(\hat{\mathbf{1}}_{D})\)
\(|\Phi^{\ast}\Phi|^{2} = \operatorname{circ}\big(|\hat{\mathbf{1}}_{D}|^{2}\big)\)
\(|\hat{\mathbf{1}}_{D}|^{2} = \hat{\mathbf{1}}_{D}\odot\overline{\hat{\mathbf{1}}_{D}} = \mathcal{F}\big(\mathbf{1}_{D}\ast \mathbf{1}_{-D}\big) \)
Want \(\Phi=\) ETF, i.e., \(|\hat{\mathbf{1}}_{D}|^{2} = a\delta_{0}+b\mathbf{1}_{G} = \textit{spike + flat}\)
\(\mathcal{F}(\textit{spike + flat}) = \textit{spike + flat}\)
\(\mathbf{1}_{D}\ast \mathbf{1}_{-D}=\textit{spike + flat}\quad\Longleftrightarrow\quad D\) is a difference set
Suppose: \(\Phi=(\text{rows of DFT indexed by }D)\)
Meme of the talk:
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9357677/Drake-Hotline-Bling.jpg)
\(\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}\)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7154657/pasted-from-clipboard.png)
Some ETFs
arise from
McFarland Difference Sets...
a lot more ETFs
arise from
Steiner systems!
\[\begin{array}{c|cccccc} & (0,0,0,1) & (1,1,0,1) & (0,0,1,0) & (1,0,1,0) & (0,0,1,1) & (0,1,1,1)\\ \hline (0,0,0,1) & (0,0,0,0) & (1,1,0,0) & (0,0,1,1) & (1,0,1,1) & (0,0,1,0) & (0,1,1,0)\\ (1,1,0,1) & (1,1,0,0) & (0,0,0,0) & (1,1,1,1) & (0,1,1,1) & (1,1,1,0) & (1,0,1,0)\\ (0,0,1,0) & (0,0,1,1) & (1,1,1,1) & (0,0,0,0) & (1,0,0,0) & (0,0,0,1) & (0,1,0,1)\\ (1,0,1,0) & (1,0,1,1) & (0,1,1,1) & (1,0,0,0) & (0,0,0,0) & (1,0,0,1) &(1,1,0,1)\\ (0,0,1,1) & (0,0,1,0) & (1,1,1,0) & (0,0,0,1) & (1,0,0,1) & (0,0,0,0) & (0,1,0,0)\\ (0,1,1,1,) & (0,1,1,0) & (1,0,1,0) & (0,1,0,1) & (1,1,0,1) & (0,1,0,0) & (0,0,0,0) \end{array}\]
\[D=\{(0,0,0,1),(0,0,1,0),(0,0,1,1),(0,1,1,1),(1,0,1,0),(1,1,0,1)\}\]
is a (McFarland) difference set in \(G=\Z_{2}\times\Z_{2}\times\Z_{2}\times\Z_{2}\)
A McFarland difference set
The subgroup \[H=\Z_{2}\times \Z_{2}\times 0\times 0\leqslant G\] is disjoint from \(D\).
A McFarland difference set
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6950683/McFarland160.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6950684/McFarland161.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6950685/McFarland162.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6950686/McFarland163.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6950688/McFarland164.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6950690/McFarland165.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6950691/McFarland166.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6950719/Kirkman160.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6955994/Kirkman161.png)
Steiner Systems
Definition. A \((2,k,v)\)-Steiner system is a \(\{0,1\}\)-matrix \(X\) such that:
- Each row of \(X\) has exactly \(k\) ones.
- Each column of \(X\) has exactly \(r=\frac{v-1}{k-1}\) ones.
- The dot product of any pair of distinct columns is one.
Example. The matrix
\(X = \)
is a \((2,2,4)\)-Steiner system.
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6952589/Kirkman2_2_4.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9357489/pasted-from-clipboard.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6948542/fasttest1.png)
The Star Product
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6948584/gif1.gif)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6948590/gif2.gif)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6948593/gif3.gif)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6948599/gif4.gif)
A way to construct lots of ETFs
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6952640/7x28ETF.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6952672/S2_3_7.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6952674/3x4Simplex.png)
\(=\)
Take a Steiner system with \(r\) ones per column
and an \(r\times (r+1)\) ETF with unimodular entries
The Star product is a "Steiner" ETF
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7154657/pasted-from-clipboard.png)
Roadmap for this talk
Compressed Sensing
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6932431/symmetry2_3.png)
Vectors that are "Spread out"
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6932415/symmetry1_3.png)
Equiangular tight frames
ETFs from groups
\(\Z_{2}\times\Z_{2}\times\Z_{2}\times\Z_{2}\)
The design theory underneath
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7154657/pasted-from-clipboard.png)
Real Flat ETFs
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6952674/3x4Simplex.png)
Some binary codes
Group divisible designs
ETFs and graphs
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7154412/6x16WaldronGraphPicture.png)
Real Flat ETFs
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7154027/KirkmanETF6x16.png)
ETFs with all \(\pm 1\) entries: Real Flat ETFs
Why real flat ETFs?
- Waveform design: maximize \(\|x\|_{2}\) subject to \(\|x\|_{\infty}\leq B\) (minimal peak-to-average power ratio)
- Quasi-symmetric designs
- Grey-Rankin equality binary codes
Real Flat ETFs
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7153914/Steiner6x16.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6948542/fasttest1.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7153914/Steiner6x16.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7153914/Steiner6x16.png)
Real Flat ETFs
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7153914/Steiner6x16.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7154004/6x6Rotation.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7154027/KirkmanETF6x16.png)
\(=\)
Real Flat ETFs
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7154336/276x576.png)
Example. A \(276\times 576\) real ETF
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9357713/276x576Steiner.png)
Real Flat ETFs
Theorem (J '13)
\(N\times N\) Hadamard matrix \(\Longrightarrow\) \(N(2N-1)\times 4N^2\) real flat ETF
Previously known real flat ETFs:
- \(N = 2^k\) harmonic ETFs on \(\mathbb{Z}_{2}^{2k+2}\)
- \(N = 6\) [Bracken, McGuire, & Ward, 2006]
Theorem (Mixon, J, Fickus '13)
Real Flat
ETFs
Grey-Rankin
equality
binary codes
1-1 correspondence
We can also construct a real flat \(317886556\times 1907416992\) ETF.
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9357677/Drake-Hotline-Bling.jpg)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9357677/Drake-Hotline-Bling.jpg)
Roadmap for this talk
Compressed Sensing
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6932431/symmetry2_3.png)
Vectors that are "Spread out"
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6932415/symmetry1_3.png)
Equiangular tight frames
ETFs from groups
\(\Z_{2}\times\Z_{2}\times\Z_{2}\times\Z_{2}\)
The design theory underneath
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7154657/pasted-from-clipboard.png)
Real Flat ETFs
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6952674/3x4Simplex.png)
Some binary codes
Group divisible designs
ETFs and graphs
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7154412/6x16WaldronGraphPicture.png)
Meme of the talk:
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9357677/Drake-Hotline-Bling.jpg)
\(\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{3}\times\mathbb{Z}_{3}\)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7154657/pasted-from-clipboard.png)
Some ETFs
arise from
Spence Difference Sets...
a lot more ETFs
arise from
Group Divisible
Designs!
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6956062/BigDFT_w_labels_alpha_test_intensity10.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6956199/BigDFT_w_labels_diff_set_highlighted_alpha_intensity10.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6956218/Harmonic_unsorted_w_labels_highlighted_alpha_intensity10.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6956277/Harmonic_sorted_cols_alpha_intensity10.png)
Ex:
\(G\)
\( \mathbb{Z}_{2}\)
\(\times\)
\(\mathbb{Z}_{2}\)
\(\times\)
\(\mathbb{Z}_{3}\)
\(\times\)
\(\mathbb{Z}_{3}\)
\(=\)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6957801/3GDDM_3U_3.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6957810/3x4Simplex.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6957828/Top_Kirkmaned.png)
\(\bigotimes\)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6957836/OneXFour1s.png)
\[\left[\begin{array}{l} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \end{array}\right.\]
\[\left.\begin{array}{l} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \end{array}\right]\]
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6956280/Harmonic_sorted.png)
\[\cong\]
Unitary transformation
\(I_{3}\otimes\)(\(2\times 3\) ETF)
\(3\times 4\) ETF with unimodular entries
???
Group Divisible Designs
Definition. A \(K\)-GDD of type \(M^{U}\) is a \(\{0,1\}\)-matrix \(X\) such that:
- \(X\) has \(UM\) columns.
- Each row of \(X\) has \(K\) ones.
- \(X^{\top}X = R\cdot I_{UM}+J_{UM}-(I_{U}\otimes J_{M})\) for some \(R\in\N\)
Example. The following is a \(3\)-GDD of type \(3^3\):
\(X = \)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6957801/3GDDM_3U_3.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6958397/3GDDM_3U_3Gram.png)
\(X^{\top}X = \)
ETFs from GDDs
Theorem (Fickus, J '19). Given a
\(d\times n\) ETF
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6958061/GDD2HQ.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6958236/Steiner6x16.png)
\(k\)-GDD of type \(M^{U}\)
and
provided certain integrality conditions hold, there exists a \(D\times N\) ETF with \(D>d\), \(N>n\) and \(\frac{D}{N}\approx \frac{d}{n}.\)
Real ETFs and Graphs
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6958236/Steiner6x16.png)
\(\Phi=\)
- Given a real ETF \(\Phi\)
- Normalize so that all dot products with the first vector are positive.
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7154380/WaldromSteinerETF6x16.png)
Real ETFs and Graphs
\(\Phi^{\top}\Phi=\)
- Given a real ETF \(\Phi\)
- Normalize so that all dot products with the first vector are positive.
- Look at the gram matrix \(\Phi^{\top}\Phi\)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7154385/WaldromSteinerETF6x16Gram.png)
Real ETFs and Graphs
- Given a real ETF \(\Phi\)
- Normalize so that all dot products with the first vector are positive.
- Look at the gram matrix \(\Phi^{\top}\Phi\)
- Remove the first row and column and zero out the diagonal
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7154389/WaldromSteinerETF6x16SeidelGraph.png)
Real ETFs and Graphs
- Given a real ETF \(\Phi\)
- Normalize so that all dot products with the first vector are positive.
- Look at the gram matrix \(\Phi^{\top}\Phi\)
- Remove the first row and column and zero out the diagonal
- This is the Seidel adjacency matrix of a strongly regular graph
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7154389/WaldromSteinerETF6x16SeidelGraph.png)
Real ETFs and Graphs
A. E. Brouwer maintains a table of known strongly regular graphs.
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9360583/pasted-from-clipboard.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9360590/brouwer_table_image.png)
Our approach:
Real
ETFs
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6952640/7x28ETF.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7154657/pasted-from-clipboard.png)
Combinatorial
designs
Strongly
regular graph
Construct
object
Certify
novelty
New Graphs!
Theorem (J, '21). If there exists an
\(h\times h\) Hadamard matrix with \(h\equiv 1\) or \(2\) \(\text{mod}\ 3\),
then there exists a strongly regular graph with parameters:
\[v=h(2h+1),\quad k=h^2-1,\quad \lambda=\frac{1}{2}(h^2-4),\quad \mu = \frac{1}{2}h(h-1)\]
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6952674/3x4Simplex.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/8849541/OneXFour1s.png)
There exists a \(20\times 20\) Hadamard matrix, and hence an SRG(820,399,198,190), which is new!
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/8850388/online_table.png)
From Brouwer's table online:
Overall: Five new infinite families!
Roadmap for this talk
Compressed Sensing
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6932431/symmetry2_3.png)
Vectors that are "Spread out"
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6932415/symmetry1_3.png)
Equiangular tight frames
ETFs from groups
\(\Z_{2}\times\Z_{2}\times\Z_{2}\times\Z_{2}\)
The design theory underneath
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7154657/pasted-from-clipboard.png)
Real Flat ETFs
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6952674/3x4Simplex.png)
Some binary codes
Group divisible designs
ETFs and graphs
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/7154412/6x16WaldronGraphPicture.png)
Single-pixel Camera
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9356770/AFIT_logo_800x600.jpg)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9356775/NoMasks.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9356780/masks1.gif)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9356785/NoMasks.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9356725/test1.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9356731/test43.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9356733/test44.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9356735/test45.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9356738/test46.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9356739/test47.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9356740/test48.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9356742/test49.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9356744/test50.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9356745/test51.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9356748/test52.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9356958/raster3.gif)
- \(75\times 100\) image \(=7500\) pixels
- We took \(4096\) measurements
Results
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9356883/logo_bw.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9356890/LRI.png)
- \(75\times 100\) image \(=7500\) pixels
- We took \(4096\) measurements
Results
\(L^{1}\) Minimization Solution
\(L^{2}\) Minimization Solution
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9356895/LRI.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9356919/L2min.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/9356922/L1min.png)
Thanks for your attention!
Questions?
![](https://s3.amazonaws.com/media-p.slid.es/uploads/1109442/images/6958918/jaspertoon.png)
Future work:
- Subspaces that are "spread out"
- Frames that saturate the Levenshtein bound
- SRGs from finite geometries
- Diagonals of self-adjoint operators
AFIT Talk
By John Jasper
AFIT Talk
- 292