Equiangular Tight Frames

John Jasper

South Dakota State University

MRI: Resolution v. Scan Time

Pixels

Time

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:

$$=$$

Image

Transformed Image

• 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:

$$=$$

Transformed Image

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

Compressed Sensing

Equiangular tight frames

ETFs from groups

$$\Z_{2}\times\Z_{2}\times\Z_{2}\times\Z_{2}$$

The design theory underneath

Real Flat ETFs

Some binary codes

Group divisible designs

ETFs and graphs

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

$\mu(\Phi) = \cos(\theta)$

$$\mu(\Phi) = \cos(\theta)$$??

$$\mu(\Phi) = \cos(\theta)$$

Measuring how "spread out" vectors are

Definition. Given unit vectors $$\Phi=(\varphi_{i})_{i=1}^{N}$$, we define the coherence

$\mu(\Phi) = \max_{i\neq j}|\langle \varphi_{i},\varphi_{j}\rangle|.$

Example.

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:

1. Tight: There is a constant $$A>0$$ such that $\sum_{i=1}^{N}|\langle v,\varphi_{i}\rangle|^{2} = A\|v\|^{2} \quad\text{for all } v.$
2. 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

Example 1. Consider the (multiple of a) unitary matrix

$\left[\begin{array}{rrrr}1 & -1 & 1 & -1\\ 1 & 1 & -1 & -1\\ 1 & -1 & -1 & 1\end{array}\right]$

$\left[\begin{array}{rrrr} 1 & 1 & 1 & 1\\ 1 & -1 & 1 & -1\\ 1 & 1 & -1 & -1\\ 1 & -1 & -1 & 1\end{array}\right]$

$\left[\begin{array}{rrrr} 1 & 1 & 1\\ \sqrt{2} & -\sqrt{\frac{1}{2}} & -\sqrt{\frac{1}{2}}\\ 0 & \sqrt{\frac{3}{2}} & -\sqrt{\frac{3}{2}}\end{array}\right]$

$\left[\begin{array}{rrrr}\sqrt{2} & -\sqrt{\frac{1}{2}} & -\sqrt{\frac{1}{2}}\\ 0 & \sqrt{\frac{3}{2}} & -\sqrt{\frac{3}{2}}\end{array}\right]$

Example 3.

Theme of the talk:

$$\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$$

Some ETFs

arise from

groups...

a lot more ETFs

arise from

combinatorial designs!

M

Compressed Sensing

Equiangular tight frames

ETFs from groups

$$\Z_{2}\times\Z_{2}\times\Z_{2}\times\Z_{2}$$

The design theory underneath

Real Flat ETFs

Some binary codes

Group divisible designs

ETFs and graphs

$\Z_{7}\left\{\begin{array}{c} 0\\ 1\\ 2\\ 3\\ 4\\ 5\\ 6 \end{array}\right. \left[\begin{array}{ccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & \omega & \omega^2 & \omega^3 & \omega^4 & \omega^5 & \omega^6\\ 1 & \omega^2 & \omega^4 & \omega^6 & \omega & \omega^3 & \omega^5\\ 1 & \omega^3 & \omega^6 & \omega^2 & \omega^5 & \omega & \omega^4\\ 1 & \omega^4 & \omega & \omega^5 & \omega^2 & \omega^6 & \omega^3\\ 1 & \omega^5 & \omega^3 & \omega & \omega^6 & \omega^4 & \omega^2\\ 1 & \omega^6 & \omega^5 & \omega^4 & \omega^3 & \omega^2 & \omega \end{array}\right]$

$\begin{array}{c} 1\\ 2\\ 4 \end{array}\left[\begin{array}{ccccccc} 1 & \omega & \omega^2 & \omega^3 & \omega^4 & \omega^5 & \omega^6\\ 1 & \omega^2 & \omega^4 & \omega^6 & \omega & \omega^3 & \omega^5\\ 1 & \omega^4 & \omega & \omega^5 & \omega^2 & \omega^6 & \omega^3 \end{array}\right]$

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:

$$\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$$

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 $$D$$.

Steiner Systems

Definition. A $$(2,k,v)$$-Steiner system is a $$\{0,1\}$$-matrix $$X$$ such that:

1. Each row of $$X$$ has exactly $$k$$ ones.
2. Each column of $$X$$ has exactly $$r=\frac{v-1}{k-1}$$ ones.
3. The dot product of any pair of distinct columns is one.

Example. The matrix

$$X =$$

is a $$(2,2,4)$$-Steiner system.

A way to construct lots of ETFs

$$=$$

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

Compressed Sensing

Equiangular tight frames

ETFs from groups

$$\Z_{2}\times\Z_{2}\times\Z_{2}\times\Z_{2}$$

The design theory underneath

Real Flat ETFs

Some binary codes

Group divisible designs

ETFs and graphs

Real Flat ETFs

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

$$=$$

Real Flat ETFs

Example. A $$276\times 576$$ real ETF

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.

Compressed Sensing

Equiangular tight frames

ETFs from groups

$$\Z_{2}\times\Z_{2}\times\Z_{2}\times\Z_{2}$$

The design theory underneath

Real Flat ETFs

Some binary codes

Group divisible designs

ETFs and graphs

Meme of the talk:

$$\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{3}\times\mathbb{Z}_{3}$$

Some ETFs

arise from

Spence Difference Sets...

a lot more ETFs

arise from

Group Divisible

Designs!

Ex:

$$G$$

$$\mathbb{Z}_{2}$$

$$\times$$

$$\mathbb{Z}_{2}$$

$$\times$$

$$\mathbb{Z}_{3}$$

$$\times$$

$$\mathbb{Z}_{3}$$

$$=$$

$$\bigotimes$$

$\left[\begin{array}{l} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \end{array}\right.$

$\left.\begin{array}{l} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \end{array}\right]$

$\cong$

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:

1. $$X$$ has $$UM$$ columns.
2. Each row of $$X$$ has $$K$$ ones.
3. $$X^{\top}X = R\cdot I_{UM}+J_{UM}-(I_{U}\otimes J_{M})$$ for some $$R\in\N$$

Example. The following is a $$3$$-GDD of type $$3^3$$:

$$X =$$

$$X^{\top}X =$$

ETFs from GDDs

Theorem (Fickus, J '19). Given a

$$d\times n$$ ETF

$$k$$-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

$$\Phi=$$

• Given a real ETF $$\Phi$$
• Normalize so that all dot products with the first vector are positive.

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 $$\Phi^{\top}\Phi$$

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

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

Real ETFs and Graphs

A. E. Brouwer maintains a table of known strongly regular graphs.

Our approach:

Real

ETFs

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:

$v=h(2h+1),\quad k=h^2-1,\quad \lambda=\frac{1}{2}(h^2-4),\quad \mu = \frac{1}{2}h(h-1)$

There exists a $$20\times 20$$ Hadamard matrix, and hence an SRG(820,399,198,190), which is new!

From Brouwer's table online:

Overall: Five new infinite families!

Compressed Sensing

Equiangular tight frames

ETFs from groups

$$\Z_{2}\times\Z_{2}\times\Z_{2}\times\Z_{2}$$

The design theory underneath

Real Flat ETFs

Some binary codes

Group divisible designs

ETFs and graphs

Single-pixel Camera

• $$75\times 100$$ image $$=7500$$ pixels
• We took $$4096$$ measurements

Results

• $$75\times 100$$ image $$=7500$$ pixels
• We took $$4096$$ measurements

Results

$$L^{1}$$ Minimization Solution

$$L^{2}$$ Minimization Solution

Questions?

Future work:

• Subspaces that are "spread out"
• Frames that saturate the Levenshtein bound
• SRGs from finite geometries