Rocky Mountain Algebraic Combinatorics Seminar I

Equiangular tight frames:

From algebraic tyranny to combinatorial freedom

John Jasper

Air Force Institute of Technology

Rocky Mountain Algebraic Combinatorics Seminar

The views expressed in this talk are those of the speaker and do not reflect the official policy
or position of the United States Air Force, Department of Defense, or the U.S. Government.

https://slides.com/johnjasper/rmacs1/

A Toy Packing Problem

Want: \(N\) points maximally "spread out"

Distance: \(d(x,y) = \) min. arc length

Spread out: maximize min. distance

Want: \(N\) points maximally "spread out"

Distance: \(d(x,y) = \) min. arc length

Some solutions:

\(N=2\)

\(N=3\)

\(N=4\)

\(N=5\)

A Toy Packing Problem

Spread out: maximize min. distance

How do we know these are optimal?

Take pts \(\{x_{1},\ldots,x_{N}\}\) arranged counterclockwise

\(\theta_{n}=\) arclength from \(x_{n}\) to \(x_{n+1}\)

\[\min_{n\neq m}d(x_{n},x_{m})\]

\[\leq \min_{n}\theta_{n}\]

\[\leq \frac{1}{N}\sum_{n=1}^{N}\theta_{n}\]

\[= \frac{2\pi}{N}\]

A Toy Packing Problem

\[\min_{n\neq m}d(x_{n},x_{m})\leq \frac{2\pi}{N}\]

Bound:

A Toy Packing Problem

\(N=5\)

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

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

Measuring how "spread out" vectors are

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

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

Minimizing coherence

between vectors

\(\Updownarrow\)

Maximizing min. angle

between lines

Example.

Measuring how "spread out" vectors are

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

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

Given \((d,N)\) find \(\Phi = (\varphi_{i})_{i=1}^{N}\subset\mathbb{R}^{d}\) such that \(\mu(\Phi)\) is minimal.

Goal:

Roadmap for this talk

Packing points

Vectors that are "Spread out"

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

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

\(\langle v,\Phi\Phi^{\ast}v\rangle = \)

Examples of equiangular tight frames

Example 2. Consider the (multiple of a) unitary matrix

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

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

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

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

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

Examples of equiangular tight frames

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

Roadmap for this talk

Vectors that are "Spread out"

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

Packing points

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

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

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

Let's make the orbit

For \(j=0,1,\ldots,6,\) we have the group action

\[j\cdot \varphi = U^{j}\varphi.\]

Our frame is the orbit of \(\varphi_{0}\) under this action

\[\varphi_{j} = U^{j}\varphi_{j} = \begin{bmatrix}\omega^{j} & 0 & 0\\ 0 & \omega^{2j} & 0\\ 0 & 0 & \omega^{4j}\end{bmatrix}\begin{bmatrix}1\\1\\1\end{bmatrix} = \left[\begin{array}{c} \omega^j\\ \omega^{2j}\\ \omega^{4j}\end{array}\right].\]

\[U = \begin{bmatrix}\omega^{1} & 0 & 0\\ 0 & \omega^2 & 0\\ 0 & 0 & \omega^4\end{bmatrix}\quad\text{and}\quad \varphi_{0} = \begin{bmatrix}1\\1\\1\end{bmatrix}\]

Difference sets \(\Rightarrow\) equiangular?

\(\varphi_{j} = \left[\begin{array}{c} \omega^j\\ \omega^{2j}\\ \omega^{4j}\end{array}\right]\) for \(j=0,1,\ldots,6,\) then

Note that

\[\langle \varphi_{j}\varphi_{j}^{\ast}, \varphi_{k}\varphi_{k}^{\ast}\rangle_{\text{Fro}} = \text{tr}(\varphi_{j}\varphi_{j}^{\ast}\varphi_{k}\varphi_{k}^{\ast}) = \text{tr}(\varphi_{k}^{\ast}\varphi_{j}\varphi_{j}^{\ast}\varphi_{k}) = |\langle \varphi_{j},\varphi_{k}\rangle|^{2},\]

and

\[\varphi_{j}\varphi_{j}^{\ast} = \left[\begin{array}{c} \omega^j\\ \omega^{2j}\\ \omega^{4j}\end{array}\right]\left[\omega^{-j}\ \omega^{-2j}\ \omega^{-4j}\right] = \left[\begin{array}{ccc}1 & \omega^{6 j} & \omega^{4 j}\\ \omega^{1 j} & 1 & \omega^{5 j}\\ \omega^{3 j} & \omega^{2 j} & 1 \end{array}\right]\]

Hence, for \(j\neq k\)

\[|\langle\varphi_{j},\varphi_{k}\rangle|^{2} = \langle \varphi_{j}\varphi_{j}^{\ast}, \varphi_{k}\varphi_{k}^{\ast}\rangle_{\text{Fro}} = \text{sum}\left(\left[\begin{array}{ccc}1 & \omega^{6(j-k)} & \omega^{4(j-k)}\\ \omega^{1(j-k)} & 1 & \omega^{5(j-k)}\\ \omega^{3(j-k)} & \omega^{2(j-k)} & 1 \end{array}\right]\right)=2\]

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 \(D\).

A McFarland difference set

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.

The Star Product

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

Roadmap for this talk

Vectors that are "Spread out"

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

Packing points

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