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:

  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

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

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

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

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

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:

  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.

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

Real Flat ETFs

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.

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

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 \(\frac{D}{N}\approx \frac{d}{n}.\)

A New ETF

This GDD

Combined with a \(6\times 16\) ETF, like this one:

produces a complex \(266\times 1008\) ETF, which appears to be new!

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

Rocky Mountain Algebraic Combinatorics Seminar I

By John Jasper

Rocky Mountain Algebraic Combinatorics Seminar I

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