Harmonic equi-isoclinic tight fusion frames
John Jasper
Air Force Institute of Technology
(w/ Matthew Fickus, Joseph W. Iverson, Dustin G. Mixon)
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.
October 22, 2022
How do we arrange \(N\) different \(r\)-dimensional subspaces of \(\mathbb{F}^{\ }\) so that the smallest principal angle between any two is as large as possible?
\(d\)
A subspace packing problem
Q:
Equiangular Tight Frames (ETF)
Definition. Let \[\Phi = \big[\varphi_{1}\ \ \varphi_{2}\ \ \cdots\ \ \varphi_{N}\big]\in \mathbb{F}^{d\times N},\]
where each column \(\varphi_{n}\) is unit norm
\[\|\varphi_{n} \|^{2}=1.\]
1) (Tightness) \(\exists\,A>0\) such that \((\Phi^{\ast}\Phi)^{2} = A\Phi^{\ast}\Phi\).
2) (Equiangular) \(\exists\,B>0\) such that \(|\frac{1}{B}\varphi_{m}^{\ast}\varphi_{n}^{}|=1\) for \(m\neq n\).
If both 1) and 2) hold, then \(\{\varphi_{n}\}_{n=1}^{N}\) is an ETF(\(d,N)\).
\[\Phi^{\ast}\Phi = \left[\begin{array}{cccc} 1 & \varphi_{1}^{\ast}\varphi_{2} & \cdots & \varphi_{1}^{\ast}\varphi_{N}\\[1ex] \varphi_{2}^{\ast}\varphi_{1} & 1 & \cdots & \varphi_{2}^{\ast}\varphi_{N}\\[1ex] \vdots & \vdots & \ddots & \vdots\\[1ex] \varphi_{N}^{\ast}\varphi_{1} & \varphi_{N}^{\ast}\varphi_{2} & \cdots & 1\end{array}\right]\]
\(1\)'s down the diagonal
1) \(\Phi^{\ast}\Phi \propto\) projection
2) \(|\varphi_{m}^{\ast}\varphi_{n}^{}|\) constant
Equi-isoclinic tight fusion frame (EITFF)
Definition. Let \[\Phi = \big[\Phi_{1}\ \ \Phi_{2}\ \ \cdots\ \ \Phi_{N}\big]\in(\mathbb{F}^{d\times r})^{1\times N},\]
where the columns of each \(\Phi_{n}\) form ONB for a subspace (w/ dim\(=r\))
\[\Phi_{n}^{\ast}\Phi_{n} = \boldsymbol{I}.\]
1) (Tightness) \(\exists\,A>0\) such that \((\Phi^{\ast}\Phi)^{2} = A\Phi^{\ast}\Phi\).
2) (Equi-isoclinic) \(\exists\,B>0\) such that \(\frac{1}{B}\Phi_{m}^{\ast}\Phi_{n}\) is unitary for \(m\neq n\).
If both 1) and 2) hold, then \(\{\Phi_{n}\}_{n=1}^{N}\) is an EITFF(\(d,N,r)\).
\[\Phi^{\ast}\Phi = \left[\begin{array}{cccc} \boldsymbol{I} & \Phi_{1}^{\ast}\Phi_{2} & \cdots & \Phi_{1}^{\ast}\Phi_{N}\\[1ex] \Phi_{2}^{\ast}\Phi_{1} & \boldsymbol{I} & \cdots & \Phi_{2}^{\ast}\Phi_{N}\\[1ex] \vdots & \vdots & \ddots & \vdots\\[1ex] \Phi_{N}^{\ast}\Phi_{1} & \Phi_{N}^{\ast}\Phi_{2} & \cdots & \boldsymbol{I}\end{array}\right]\]
Identities down the diagonal
1) \(\Phi^{\ast}\Phi \propto\) projection
2) \(\Phi_{m}^{\ast}\Phi_{n}\propto\) unitaries
Examples
Take my favorite ETF, an ETF\((3,4)\):
\[\Phi = \frac{1}{\sqrt{3}}\left[\begin{array}{cccc} + & - & + & -\\ + & + & - & -\\ + & - & - & +\end{array}\right]\]
1) (ETFs) \(\Phi\) is an EITFF\((3,4,1)\)
2) (Tensors) \(\Phi\otimes\boldsymbol{I} = \dfrac{1}{\sqrt{3}}\left[\begin{array}{cc|cc|cc|cc} + & 0 & - & 0 & + & 0 & - & 0\\ 0 & + & 0 & - & 0 & + & 0 & -\\ + & 0 & + & 0 & - & 0 & - & 0\\ 0 & + & 0 & + & 0 & - & 0 & -\\ + & 0 & - & 0 & - & 0 & + & 0\\ 0 & + & 0 & - & 0 & - & 0 & + \end{array}\right]\)
\[\exists\,\operatorname{EITFF}(d,N,r)\quad\Rightarrow\quad \exists\,\operatorname{EITFF}(md,N,mr)\]
\[\exists\,\operatorname{ETF}(d,N)\quad\Leftrightarrow\quad\exists\,\operatorname{EITFF}(d,N,1)\]
We call these "tensor-sized"
is an EITFF\((6,4,2)\)
Examples
\[\Phi^{\ast}\Phi=\frac{1}{4}\left[\begin{array}{lllllllll}4&1&1&1&\omega&\omega^2&1&\omega&\omega^2\\1&4&1&\omega^2&1&\omega&\omega&\omega^2&1\\1&1&4&\omega&\omega^2&1&\omega^2&1&\omega\\1&\omega&\omega^2&4&1&1&1&\omega^2&\omega\\\omega^2&1&\omega&1&4&1&\omega&1&\omega^2\\\omega&\omega^2&1&1&1&4&\omega^2&\omega&1\\1&\omega^2&\omega&1&\omega^2&\omega&4&1&1\\\omega^2&\omega&1&\omega&1&\omega^2&1&4&1\\\omega&1&\omega^2&\omega^2&\omega&1&1&1&4\end{array}\right]\]
3) (\(\mathbb{C}\) to \(\mathbb{R}\) trick) Take \(\Phi\) to be an \(\operatorname{ETF}(6,9)\) with
(\(\omega=\exp(\frac{2\pi i}{3})\))
Replace: \(R\exp(\theta i)\mapsto R\left[\begin{array}{cc}\cos(\theta)&-\sin(\theta)\\\sin(\theta)&\cos(\theta)\end{array}\right]\)
We obtain the Gram matrix of a real EITFF\((12,9,2)\)
\[\frac{1}{4}\]
Tensor-sized, but no real ETF\((6,9)\) exists.
Goal: Find
Example. Let \(\mathcal{G} = \mathbb{Z}_{7} = \{0,1,2,3,4,5,6\}\),
\[\pi(g) = \begin{bmatrix} 1 & 0 & 0\\ 0 & e^{2\pi i g/7} & 0\\ 0 & 0 & e^{2 \pi i 3g/7}\end{bmatrix},\]
and \(\Phi_{0}= [1\ \ 1\ \ 1]^{\top}/\sqrt{3}\).
\(\Phi=\)
- a group \(\mathcal{G}\)
- a unitary representation \(\pi:\mathcal{G}\to U(d)\)
- an isometry \(\Phi_{0}\in\mathbb{F}^{d\times r}\)
such that \(\{\pi(g)\Phi_{0}\}_{g\in\mathcal{G}}\) is an EITFF.
Example 2. Let \(\mathcal{G} = \mathbb{Z}_{5} = \{0,1,2,3,4\}\) and \(\omega=\exp(2\pi i/5)\).
\[\Psi = \big[\begin{array}{c|c|c|c|c} \Psi_{0} & \Psi_{1} & \Psi_{2} & \Psi_{3} & \Psi_{4}\end{array}\big] = \frac{1}{\sqrt{2}}\left[\begin{array}{c|c|c|c|c}\cdot & 1 & 1 & 1 & 1\\ \cdot & 1 & i & -i & -1\end{array}\right]\]
\[\Phi_{g}\doteq\pi(g)\Psi^{\ast}\doteq\left[\begin{array}{ccccc}\hspace{-5px}1 \boldsymbol{I}_{0}&&&&\\&\hspace{-5px}\omega^{g} \boldsymbol{I}_{1}&&&\\&&\hspace{-5px}\omega^{2g} \boldsymbol{I}_{1}&&\\&&&\hspace{-5px}\omega^{3g} \boldsymbol{I}_{1}&\\&&&&\hspace{-5px}\omega^{4g} \boldsymbol{I}_{1}\end{array}\right]\left[\begin{array}{cc}\cdot & \cdot\\\hline 1 & 1\\\hline 1 & -i\\\hline 1 & i\\\hline 1 & -1 \end{array}\right]\]
\[=\frac{1}{2}\left[\begin{array}{cc|cc|cc|cc|cc}1&1&\mu^{4}&\mu^{4}&\mu^{8}&\mu^{8}&\mu^{12}&\mu^{12}&\mu^{16}&\mu^{16}\\1&\mu^{15}&\mu^{8}&\mu^{3}&\mu^{16}&\mu^{11}&\mu^{4}&\mu^{19}&\mu^{12}&\mu^{7}\\1&\mu^{5}&\mu^{12}&\mu^{17}&\mu^{4}&\mu^{9}&\mu^{16}&\mu^{1}&\mu^{8}&\mu^{13}\\1&\mu^{10}&\mu^{16}&\mu^{6}&\mu^{12}&\mu^{2}&\mu^{8}&\mu^{18}&\mu^{4}&\mu^{14}\end{array}\right]\]
\[\Phi_{g}\doteq\pi(g)\Psi^{\ast}\doteq\left[\begin{array}{cccc}&\hspace{-5px}\omega^{g} \boldsymbol{I}_{1}&&&\\&&\hspace{-5px}\omega^{2g} \boldsymbol{I}_{1}&&\\&&&\hspace{-5px}\omega^{3g} \boldsymbol{I}_{1}&\\&&&&\hspace{-5px}\omega^{4g} \boldsymbol{I}_{1}\end{array}\right]\left[\begin{array}{cc} 1 & 1\\ 1 & -i\\ 1 & i\\ 1 & -1 \end{array}\right]\]
\[\Phi = \big[\ \Phi_{0}\ \ \vert\ \ \Phi_{1}\ \ \vert\ \ \Phi_{2}\ \ \vert\ \ \Phi_{3}\ \ \vert\ \ \Phi_{4}\ \big]\]
\(\big(\mu = \exp(2\pi i/20)\big)\)
Harmonic Tight Fusion Frames
- Abelian group: \(\mathcal{G} = \{g_{0},g_{1},\ldots,g_{N-1}\}\)
- Tight fusion frame: \(\Psi = \big[\begin{array}{c|c|c|c|c} \Psi_{\chi_{0}} & \Psi_{\chi_{1}} & \Psi_{\chi_{2}} & \cdots & \Psi_{\chi_{N-1}}\end{array}\big] \)
For each \(\chi\in\hat{\mathcal{G}}\) the matrix \(\Psi_{\chi}\) is an \(r\times D_{\chi}\) matrix. (\(D_{\chi}\) could be zero!)
Start with:
Characters: \(\hat{\mathcal{G}} = \{\chi_{0},\chi_{1},\ldots,\chi_{N-1}\}\)
\[\pi(g) := \bigoplus_{\chi\in\hat{\mathcal{G}}}\chi(g)\boldsymbol{I}_{D_{\chi}} = \left[\begin{array}{ccccc}\hspace{-2px}\chi_{0}(g) \boldsymbol{I}_{D_{\chi_{0}}}&&&&\\&\hspace{-10px}\chi_{1}(g) \boldsymbol{I}_{D_{\chi_{1}}}&&&\\&&\hspace{-10px}\chi_{2}(g) \boldsymbol{I}_{D_{\chi_{2}}}&&\\&&&\hspace{-15px}\ &\\&&&&\hspace{-10px}\chi_{N-1}(g) \boldsymbol{I}_{D_{\chi_{N-1}}}\hspace{-2px}\end{array}\right]\]
Harmonic TFF generated by \(\Psi\):
\[\Phi \doteq\left[\begin{array}{c|c|c|c|c} \pi(g_{0})\Psi^{\ast} & \pi(g_{1})\Psi^{\ast} & \pi(g_{2})\Psi^{\ast} & \cdots & \pi(g_{N-1})\Psi^{\ast}\end{array}\right]\]
\(\ddots\)
Cross-Grams
Harmonic TFF generated by \(\Psi\):
\[\Phi \doteq \left[\begin{array}{c|c|c|c|c} \pi(g_{0})\Psi^{\ast} & \pi(g_{1})\Psi^{\ast} & \pi(g_{2})\Psi^{\ast} & \cdots & \pi(g_{N-1})\Psi^{\ast}\end{array}\right]\]
Set \(\Phi_{g} \doteq \pi(g)\Psi^{\ast}\).
\[\Phi_{g}^{\ast}\Phi_{h}^{} \doteq (\pi(g)\Psi^{\ast})^{\ast}(\pi(h)\Psi^{\ast}) = \Psi\pi(g^{-1})\pi(h)\Psi^{\ast} \doteq \Phi_{g_{0}}^{\ast}\Phi_{h^{-1}g}^{}\]
Cross-Grams:
The Gram matrix is block circulant.
Also,
\[\Phi_{g_{0}}^{\ast}\Phi_{g} \doteq \sum_{\chi\in\hat{\mathcal{G}}}\chi(g)\Psi_{\chi}\Psi_{\chi}^{\ast} = \sum_{\chi\in\hat{\mathcal{G}}}\chi(g)\mathbf{P}_{\chi}=:\mathbf{M}_{g}\]
\(\mathbf{P}_{\chi}\) is the projection onto \(\operatorname{ran}\Psi_{\chi}\)
\(\mathbf{M}_{g}\) is the entrywise Fourier transform of the \(\mathbf{P}_{\chi}\)'s.
Harmonic EITFFs
Theorem (Fickus,Iverson,J,Mixon '21) Suppose
- \(\mathcal{G}\) is a finite abelian group
- \(\Psi=[\Psi_{\chi}]_{\chi\in\mathcal{\hat{G}}}\) is a TFF
- \(\Phi = [\Phi_{g}]_{g\in\mathcal{G}}\) is the harmonic frame generated by \(\Psi\)
Then,
- \(\Phi\) is a TFF
- \(\Phi\) is an EITFF if and only if \[\mathbf{M}_{g}:=\sum_{\chi\in\hat{\mathcal{G}}}\chi(g)\Psi_{\chi}\Psi_{\chi}^{\ast} \] are a common multiple of unitaries for \(g\in \mathcal{G}\setminus\{g_{0}\}\).
Example 2 (redux). Let \(\mathcal{G} = \mathbb{Z}_{5} = \{0,1,2,3,4\}\) and \(\omega=\exp(2\pi i/5)\).
\[\Psi = \big[\begin{array}{c|c|c|c|c} \Psi_{0} & \Psi_{1} & \Psi_{2} & \Psi_{3} & \Psi_{4}\end{array}\big] = \frac{1}{\sqrt{2}}\left[\begin{array}{c|c|c|c|c}\cdot & 1 & 1 & 1 & 1\\ \cdot & 1 & i & -i & -1\end{array}\right]\]
\[=\frac{1}{2}\left[\begin{array}{cc}-1 & \omega^{-k}-i\omega^{-2k}+i\omega^{-3k}-\omega^{-4k}\\\omega^{-k}+i\omega^{-2k}-i\omega^{-3k}-\omega^{-4k} & -1\end{array}\right]\]
\[\mathbf{P}_{0} = \begin{bmatrix}0&0\\0&0\end{bmatrix}\]
\[\mathbf{P}_{1} = \frac{1}{2}\begin{bmatrix}1&1\\1&1\end{bmatrix}\]
\[\mathbf{P}_{2} = \frac{1}{2}\begin{bmatrix}1&-i\\i&1\end{bmatrix}\]
\[\mathbf{P}_{3} = \frac{1}{2}\begin{bmatrix}1&i\\-i&1\end{bmatrix}\]
\[\mathbf{P}_{4} = \frac{1}{2}\begin{bmatrix}1&-1\\-1&1\end{bmatrix}\]
\[\mathbf{M}_{k} = \sum_{j=0}^{4}\omega^{-jk}\mathbf{P}_{j}\]
Indeed, \(\mathbf{M}_{k}^{\ast}\mathbf{M}_{k}^{} = \frac{3}{2}\boldsymbol{I}_{2}\) for each nonzero \(k\in\mathbb{Z}_{5}\).
Therefore \(\Phi\) is a EITFF\((4,5,2)\) (five two-dimensional subspaces of \(\mathbb{C}^{4}\))
Example 1 (redux). Let \(\mathcal{G} = \mathbb{Z}_{7} = \{0,1,2,3,4,5,6\}\) and \(\omega=\exp(2\pi i/7)\)
\[\Psi = \big[\begin{array}{c|c|c|c|c|c|c} \Psi_{0} & \Psi_{1} & \Psi_{2} & \Psi_{3} & \Psi_{4} & \Psi_{5} & \Psi_{6}\end{array}\big] = \big[\begin{array}{c|c|c|c|c|c|c} 1 & 1 & \cdot & 1 & \cdot & \cdot & \cdot\end{array}\big]\]
\[\mathbf{P}_{0} = \begin{bmatrix}1\end{bmatrix},\ \mathbf{P}_{1} = \begin{bmatrix}1\end{bmatrix},\ \mathbf{P}_{2} = \begin{bmatrix}0\end{bmatrix},\ \mathbf{P}_{3} = \begin{bmatrix}1\end{bmatrix},\ \mathbf{P}_{4} = \begin{bmatrix}0\end{bmatrix},\ \mathbf{P}_{5} = \begin{bmatrix}0\end{bmatrix},\ \mathbf{P}_{6} = \begin{bmatrix}0\end{bmatrix}\]
\[\mathbf{M}_{k} = \omega^{0k}+\omega^{-k}+\omega^{-3k}\]
\(\mathbf{M}_{k}\) has the same modulus for all \(k\neq 0\)
The Fourier transform of \([1\ \ 1\ \ 0\ \ 1\ \ 0\ \ 0\ \ 0]^{\top}\) is "spike+flat"
\(\{0,1,3\}\) is a difference set in \(\mathbb{Z}_{7}\).
\(\Updownarrow\)
\(\Updownarrow\)
New Harmonic EITFFs
Theorem. (Fickus, Iverson, J, Mixon '21) Write \(\mathbb{F}_{q}^{\times} = \langle \alpha\rangle = \{\alpha^{j}\}_{j=0}^{q-2}\)
- If \(q\) is an odd prime power, and \(\chi\) is an additively odd multiplicitive character, then the harmonic TFF generated by \[\Psi = \frac{1}{\sqrt{2}}\left[\begin{array}{c|c|c|c|c}\cdot & 1 & 1 & \cdots & 1\\ \cdot & \chi(\alpha^{0}) & \chi(\alpha^{1}) & \cdots & \chi(\alpha^{q-2})\end{array}\right]\] is an EITFF\((q-1,q,2)\). (\(19\leq q\equiv 1\mod 4\) seem to be new.)
- If \(q\geq 4\) is any prime power and \(\chi\neq 1\) is a multiplicative character of \(\mathbb{F}_{q}\), then the harmonic TFF generated by \[\Psi = \frac{1}{\sqrt{2q-2}}\left[\begin{array}{c|c|c|c} \sqrt{2q-2} & \sqrt{q-2} & \cdots & \sqrt{q-2}\\ 0 & \sqrt{q}\chi(\alpha^{0}) & \cdots & \sqrt{q}\chi(\alpha^{q-2})\end{array}\right]\] is an EITFF\((q,q,2)\). (\(q\equiv 3\mod 4\) seem to be new.)
Ideas for future research
Are there more harmonic EITFFs?
- We found a harmonic EITFF(11,11,3), but it was...complicated. Does it generalize in some way?
- All examples we know of:
Thanks for your attention!
Questions?
- Difference sets: \(\cdot\)'s and \(1\)'s in the generating frame \(\Psi\).
- Nontrivial subspaces in \(\Psi\) indexed by \(\hat{G}\) or \(\hat{G}\setminus\{\chi_{0}\}\).
Can we combine these?