Harmonic equiangular tight               and their combinatorial generalizations

 

John Jasper

        South Dakota State University

Outline

1. Projective packings:

What do we mean by "vectors that are well spread out"?

 

2. Harmonic equiangular tight frames:

Using groups to build sets of vectors that are optimally spread out.

 

3. Getting rid of the group part 1:

That's not a group, that's a block design.

 

4. Getting rid of the group part 2:

Who needs a group? Association schemes can do the job.

Part 1.

Projective packings

What collections of vectors/lines are as spread out as possible?

Definition. Given a collection of 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

Measuring how "spread out" vectors are

Definition. Given a collection of 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 a collection of 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

Short, fat matrices

Given a collection of unit vectors \(\Phi=(\varphi_{i})_{i=1}^{N}\) in \(\mathbb{C}^d\), we will not distinguish between the sequence of vectors, and the \(d\times N\) matrix

\[\Phi = \begin{bmatrix} | & | & & |\\ \varphi_{1} & \varphi_{2} & \cdots & \varphi_{N}\\ | & | & & |\end{bmatrix}\]

Two other important matrices:

The gram matrix:

\[\Phi^{\ast}\Phi = \begin{bmatrix}\langle \varphi_{1},\varphi_{1}\rangle & \langle \varphi_{1},\varphi_{2}\rangle & \cdots & \langle \varphi_{1},\varphi_{N}\rangle\\ \langle \varphi_{2},\varphi_{1}\rangle & \langle \varphi_{2},\varphi_{2}\rangle & & \vdots\\ \vdots & & \ddots & \vdots\\ \langle \varphi_{N},\varphi_{1}\rangle & \cdots & \cdots & \langle \varphi_{N},\varphi_{N}\rangle\end{bmatrix}\]

The frame operator: \(\Phi\Phi^{\ast}\)

1. Tight:

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

\(\Leftrightarrow\quad\Phi\Phi^{\ast} = AI\) (that is, the rows of \(\Phi\) are orthogonal and equal norm)

\(\Leftrightarrow\quad\Phi^{\ast}\Phi\) is a multiple of a projection.

 

2. Equiangular: There is a constant \(\alpha\) such that \[|\langle\varphi_{i},\varphi_{j}\rangle| = \alpha\quad\text{for all }i\neq j.\]

\(\Leftrightarrow\quad\Phi^{\ast}\Phi\) has \(1\)'s on the diagonal, and modulus \(\alpha\) entries elsewhere.

Tightness and equiangularity can be rephrased in terms of the matrix \(\Phi\):

Examples of equiangular tight frames

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

Examples of equiangular tight frames

Example 3.  

Part 2.

Harmonic ETFs

Using (abelian) groups to build ETFs

Rows from a DFT

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

Rows from a DFT

Difference sets \(\Rightarrow\) equiangular?

Generating the ETF with the group

For each \(j,k\in\{0,\ldots,6\}\) set

\[\lambda_{k} = \left[\begin{array}{ccc} w^{k} & 0 & 0\\ 0 & w^{2k} & 0\\ 0 & 0 & w^{4k}\end{array}\right],\quad \varphi_{j} = \left[\begin{array}{c} \omega^j\\ \omega^{2j}\\ \omega^{4j}\end{array}\right]\]
then
\[\lambda_{k}\varphi_{j} = \varphi_{k+j}\]

and hence the ETF is \((\lambda_{k}\varphi_{0})_{k\in\Z_{7}}.\) Note that \(k\mapsto \lambda_{k}\) is a representation of the group \(\Z_{7}.\)

Part 3. Getting rid of the group I

That's not a group, that's a block design!

\[\begin{array}{c|cccccc}  & (0,0,0,1) & (1,1,0,1) & (0,0,1,0) & (1,0,1,0) & (0,0,1,1) & (0,1,1,1)\\ \hline (0,0,0,1) & (0,0,0,0) & (1,1,0,0) & (0,0,1,1) & (1,0,1,1) & (0,0,1,0) & (0,1,1,0)\\ (1,1,0,1) & (1,1,0,0) & (0,0,0,0) & (1,1,1,1) & (0,1,1,1) & (1,1,1,0) & (1,0,1,0)\\ (0,0,1,0) & (0,0,1,1) & (1,1,1,1) & (0,0,0,0) & (1,0,0,0) & (0,0,0,1) & (0,1,0,1)\\ (1,0,1,0) & (1,0,1,1) & (0,1,1,1) & (1,0,0,0) & (0,0,0,0) & (1,0,0,1)  &(1,1,0,1)\\ (0,0,1,1) & (0,0,1,0) & (1,1,1,0) & (0,0,0,1) & (1,0,0,1) & (0,0,0,0) & (0,1,0,0)\\ (0,1,1,1,) & (0,1,1,0) & (1,0,1,0) & (0,1,0,1) & (1,1,0,1) & (0,1,0,0) & (0,0,0,0) \end{array}\]

\[D=\{(0,0,0,1),(0,0,1,0),(0,0,1,1),(0,1,1,1),(1,0,1,0),(1,1,0,1)\}\]

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 \(v\) element set \(V\) together with a collection \(\mathcal{B}\) of subsets of \(V\), called blocks, with the property that each \(2\)-element subset of \(V\) is contained in exactly one block.

Example. The pair \((V,\mathcal{B})\) with

\[V = \{1,2,3,4\}\]

and

\[\mathcal{B} = \big\{\{1,4\},\{2,3\},\{1,2\},\{3,4\},\{1,3\},\{2,4\}\big\}\]

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

Steiner Systems

Definition. A \((2,k,v)\)-Steiner system \(\{0,1\}\)-matrix \(X\) with the following properties:

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 distinct pair of columns is one.

Example. The matrix

\(X = \)

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

The Star Product

Steiner ETFs

(a.k.a. Generalized McFarland Difference sets)

Theorem (Fickus, Mixon, Tremain '12). 

Steiner system with \(r\) ones per column

\(r\times (r+1)\) ETF with unimodular entries

\(=\)

"Steiner" ETF

An ETF that isn't harmonic

• There is no \(7\) element difference set in an abelian group of order \(28.\)
• But there is a \((2,3,7)\)-Steiner system

Group Divisible Designs

Definition. A \(K\)-GDD of type \(M^{U}\) is a \(\{0,1\}\)-matrix \(X\) with the following properties:

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, JJ '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

Part 4. Getting rid of the group II

Who needs groups? Association schemes can do the job!

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

Revisiting our first harmonic ETF

\[\Phi^{\ast}\Phi = \]

\[x = \omega+\omega^2+\omega^4\qquad y=\omega^3+\omega^5+\omega^6\]

Revisiting our first harmonic ETF

\[\Phi^{\ast}\Phi = \]

Association Schemes

Definition. A set of \(N\times N\) matrices \(X=\{A_{0},\ldots,A_{d}\}\) with entries in \(\{0,1\}\) is called an association scheme if the following three conditions hold:

1. \(A_{0}=I\)
2. \(A_{0}+\cdots+A_{d}=J,\) where \(J\) is the all \(1\)'s matrix
3. \(\mathcal{A}=\text{span} X\) forms a commutative \(\ast\)-algebra.

Example.

\(A_{0} = \)

\(A_{1} = \)

\(A_{2} = \)

\[A_{1}A_{2} = A_{2}A_{1} = 3A_{0}+A_{1}+A_{2}\]\[A_{1}^{\top}=A_{2}\]

Abelian Groups are Association Schemes

Let \(G\) be an abelian group.

Let \((e_{g})_{g\in G}\) be an orthonormal basis for \(\mathbb{C}^{|G|}\).

For each \(g\in G\) let \(A_{g}\) be the matrix such that \(A_{g}e_{h} = e_{gh}\).

Then the collection \((A_{g})_{g\in G}\) is an association scheme.

Example. The translation operators of \(\Z_{7}\) form an association scheme with \(7\) matrices:

\(A_{0}=\)

\(A_{1}=\)

\(A_{2}=\)

\(A_{3}=\)

\(A_{4}=\)

\(A_{5}=\)

\(A_{6}=\)

A Generalized Dual

Let \(X = \{A_{0},A_{1},\ldots,A_{d}\}\) be an association scheme.

 

The matrices in \(X\) are commuting normal operators.

 

By the spectral theorem, there is a set of mutually orthogonal projections \[\hat{X} = \{E_{0},E_{1},\ldots,E_{d}\}\] onto the maximal eigenspaces of \(X\).

 

Moreover, \(\text{span}\hat{X} = \text{span} X.\)

 

We will refer to \(\hat{X}\) as the dual of \(X\).

Hyperdifference Sets

Definition. Let \(X\) be an association scheme with dual \(\hat{X}.\) A set \(D\subset \hat{X}\) is called a hyperdifference set if 

\[\mathcal{G}_{D} := \sum_{E\in D} E\]

is the gram matrix of an ETF.

Note: For any \(D\subset \hat{X}\) the matrix \(\mathcal{G}_{D}\)  is a projection. Thus we only need to check that \(\mathcal{G}_{D}\) has equal modulus off-diagonal entries.

Example. Given the association scheme \(X = \{A_{0},A_{1},A_{2}\}\)

\(A_{0} = \)

\(A_{1} = \)

\(A_{2} = \)

The dual \(\hat{X} = \{E_{0},E_{1},E_{2}\}\) where

\[E_{1} = \frac{1}{7}\]

\[E_{0} = \frac{1}{7}J_{7}\]

\[E_{2} = I-E_{0}-E_{1}\]

Hence, \(D = \{E_{1}\}\) is a hyperdifference set.

Where can we look for hyperdifference sets?

Association schemes are all over the place, including:

1. Abelian groups
2. Schur rings over abelian groups
3. Finite Gelfand pairs
4. Non-abelian groups...

Definition. Given vector spaces \(U,V\) over a field \(\mathbb{F}\) and a bilinear map \(B:U\times U\to V.\) Let \(U\times_{B} V\) be the set \(U\times V\) with multiplication
\[(u,v)\cdot(x,y) = \big(u+x,v+y+B(u,x)\big).\]

Nonabelian groups that work

Conclusions

• Actions of abelian groups give a great way to generate optimally incoherent collections of vectors, namely, ETFs.
• Harmonic ETFs are known to be rare.
• Often the necessary ingredient is actually combinatorial, that is, some \(\{0,1\}\)-matrices: Steiner systems, group divisible designs, association schemes.
• Using these allows us to go beyond abelian groups and construct a much broader class of optimal packings.