## Zauner’s conjecture is true in dimensions 18, 20, 21, 30, 31, 37, 39 and 43

Two years ago, I blogged about Tuan-Yow Chien’s PhD thesis, which proved Zauner’s conjecture in dimension 17. The idea was to exploit certain conjectures on the field structure of SIC-POVM fiducial vectors so as to round numerical solutions to exact solutions. This week, the arXiv announced Chien’s latest paper (coauthored with Appleby, Flammia and Waldron), which extends this work to find exact solutions in 8 new dimensions.

The following line from the introduction caught my eye:

For instance the print-out for exact fiducial 48a occupies almost a thousand A4 pages (font size 9 and narrow margins).

As my previous blog entry illustrated, the description length of SIC-POVM fiducial vectors appears to grow rapidly with $d$. However, it seems that the rate of growth is much better than I originally thought. Here’s a plot of the description lengths of the known fiducial vectors (the new ones due to ACFW17available here — appear in red):

## Optimal line packings from association schemes

Joey Iverson recently posted our paper with John Jasper on the arXiv. As the title suggests, this paper explains how to construct optimal line packings (specifically, equiangular tight frames) using association schemes. In particular, we identify many schemes whose adjacency algebra contains the Gram matrix of an ETF. This unifies several existing constructions, and also enabled us to construct the first known infinite family of ETFs that arise from nonabelian groups.

John is on the job market this year, and when reading his research statement, I was struck by his discussion of our paper, so I asked him to expand his treatment to a full blown blog entry. Without further ado, here is John’s guest blog post (which I’ve lightly edited for hyperlinks and formatting):

## Recent developments in equiangular tight frames, II

Equiangular tight frames (ETFs) are optimal packings of lines through the origin. At the moment, they are the subject of a rapidly growing literature. In fact, there have been quite a few updates since my last post on this subject (less than five months ago), and I’ve revamped the table of ETFs accordingly. What follows is a brief discussion of the various developments:

1. There is an ETF of 76 vectors in $\mathbb{C}^{19}$

See this paper. Last time, I mentioned a recent proof that there is no ETF of 76 vectors in $\mathbb{R}^{19}$. It turns out that a complex ETF of this size does exist. To prove this, it actually seems more natural to view the vectors as columns of a $20\times 76$ matrix whose row vectors sum to zero. As a lower-dimensional example, consider the following matrix:

$\displaystyle{\left[\begin{array}{cccccccccc}+&+&+&+&+&+&+&+&+&+\\+&+&+&+&-&-&-&-&-&-\\+&-&-&-&+&+&+&-&-&-\\-&+&-&-&+&-&-&+&+&-\\-&-&+&-&-&+&-&+&-&+\\-&-&-&+&-&-&+&-&+&+\end{array}\right]}$

## SOFT 2016: Summer of Frame Theory

This month, several experts in frame theory will be visiting my department, and so Matt Fickus and I decided to organize a workshop in the style of AIM. Considering the recent progress we’ve made on equiangular tight frames (ETFs) — namely, one, two, three, and four — we are hoping this workshop will spur further progress in this area. To kick off the month, I asked a few people to prepare hour-long chalk talks, and what follows are the extended abstracts:

1. Introduction to ETFs (Dustin G. Mixon)

Given a $d$-dimensional Hilbert space space $\mathbb{H}_d$ and a positive integer $n$, we are interested in packing $n$ lines through the origin so that the interior angle between any two is as large as possible. It is convenient to represent each line by a unit vector that spans the line, and in doing so, the problem amounts to finding unit vectors $\{\varphi_i\}_{i=1}^n$ that minimize coherence:

$\displaystyle{\mu\Big(\{\varphi_i\}_{i=1}^n\Big)=\max_{i\neq j}|\langle \varphi_i,\varphi_j\rangle|.}$

This minimization amounts to a nonconvex optimization problem. To construct provably optimal packings, one must prove a lower bound on $\mu$ for a given $n$ and spatial dimension $d$, and then construct an ensemble which meets equality in that bound. To date, we know of three bounds that are sharp:

## The Voronoi Means Conjecture

UPDATE (July 26, 2016): Boris Alexeev recently disproved the Voronoi Means Conjecture! In particular, he found that certain stable isogons fail to exhibit the conjectured behavior, and his solution suggests a certain refinement of the conjecture. I asked him to write a guest blog entry about his solution, so expect to hear more in the coming weeks.

Suppose you’re given a sample from an unknown balanced mixture of $k$ spherical Gaussians of equal variance in dimension $d$:

In the above example, $k=3$ and $d=2$. How do you estimate the centers $\{\gamma_i\}_{i=1}^k$ of each Gaussian from the data? In this paper, Dasgupta provides an algorithm in which you project the data onto a randomly drawn subspace of some carefully selected dimension so as to concentrate the data points towards their respective centers. After doing so, there will be $k$ extremely popular regions of the subspace, and for each region, you can average the corresponding points in the original dataset to estimate the corresponding Gaussian center. With this algorithm, Dasgupta proved that

$\displaystyle{\mathrm{MSE}:=\frac{1}{k}\sum_{i=1}^k\|\hat{\gamma}_i-\gamma_i\|^2\lesssim d\sigma^2 \qquad\text{whp}}$

provided $\mathrm{SNR}:=\frac{1}{\sigma^2}\min_{i\neq j}\|\gamma_i-\gamma_j\|^2\gtrsim d$.

## The HRT Conjecture

A couple of weeks ago, I attended a workshop hosted by Darrin Speegle on the HRT Conjecture. First posed twenty years ago by Chris Heil, Jay Ramanathan, and Pankaj Topiwala in this paper, the conjecture states that every finite collection of time-frequency shifts of any nonzero function in $L^2(\mathbb{R})$ is linearly independent. In this post, I will discuss some of the key ideas behind some of the various attempts at chipping away at HRT, and I’ll describe what appear to be fundamental barriers to a complete solution. For more information, see these survey papers: one and two.

First, some notation: Denote the translation-by-$a$ and modulation-by-$b$ operators by

$(T_af)(x):=f(x-a),\qquad (M_bf)(x)=e^{2\pi i bx}f(x),$

respectively. Then the formal conjecture statement is as follows:

The HRT Conjecture. For every $0\neq f\in L^2(\mathbb{R})$ and every finite $\Lambda\subset\mathbb{R}^2$, the collection $\{M_bT_af\}_{(a,b)\in\Lambda}$ is linearly independent.

What follows are some of the popular methods for tackling HRT:

## Recent developments in equiangular tight frames

Equiangular tight frames (ETFs) are optimal packings of lines through the origin. Last year, Matt and I tabulated all known existence results for ETFs. Since then, there have been a few interesting developments on the subject, so I’ll have to update the table soon. In the meantime, this blog entry covers the highlights.

First, some context: In the real case, ETFs are in one-to-one correspondence with certain strongly regular graphs (SRGs). Waldron’s paper on the subject is the standard modern treatment of this correspondence. SRGs have been an active area of research for a few decades, and Brouwer’s table provides a survey of the various known constructions. This suggests a straightforward program for constructing real ETFs: Go to Brouwer’s table, find an SRG of the requisite size, investigate the reference that constructs that graph, follow Waldron’s treatment to construct the corresponding Gram matrix, and then decompose the Gram matrix to get the desired ETF.

Now for the news: