## 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:

## On the low-rank approach for semidefinite programs arising in synchronization and community detection

From MaxCut to PhaseLift, semidefinite programming has proven to be rather powerful, especially for convex relaxation. SDP solvers take polynomial time, but the exponent is large, and anyone who’s run an SDP on CVX has experienced some frustration with the runtime. In practice, the SDP-optimal matrix tends to have extremely low rank, and so one may apply a rank constraint to facilitate the search for the SDP’s solution. This heuristic was first introduced by Burer and Monteiro, and it works well in practice, but the rank-constrained program is nonconvex and the theory is scant. Recently, the theory gap started to close with this paper:

On the low-rank approach for semidefinite programs arising in synchronization and community detection

Afonso S. Bandeira, Nicolas Boumal, Vladislav Voroninski

As the title suggests, this paper provides strong performance guarantees for the Burer-Monteiro heuristic in the particular cases of $\mathbb{Z}_2$ synchronization and community detection. I was very excited to see this paper, and so I interviewed one of the authors (Nicolas Boumal). I’ve lightly edited his responses for formatting and hyperlinks:

## Clustering noisy data with semidefinite relaxations

Soledad Villar recently posted our latest paper on the arXiv (this one coauthored by her advisor, Rachel Ward). The paper provides guarantees for the k-means SDP when the points are drawn from a subgaussian mixture model. This blog entry will discuss one of the main ideas in our analysis, which we borrowed from Guedon and Vershynin’s recent paper.

The first application comes from graph clustering. Consider the stochastic block model, in which the $n$ vertices are secretly partitioned into two communities, each of size $n/2$, and edges between vertices of a common community are drawn iid with some probability $p$, and all other edges are drawn with probability $q. The goal of community estimation is to estimate the communities given a random draw of the graph. For this task, you might be inclined to find the maximum likelihood estimator for this model, but this results in an integer program. Relaxing the program leads to a semidefinite program, and amazingly, this program is tight and recovers the true communities with high probability when $p=(\alpha\log n)/n$ and $q=(\beta\log n)/n$ for good choices of $(\alpha,\beta)$. (See this paper.) These edge probabilities scale like the threshold for connected Erdos-Renyi graphs, and this makes sense since we wouldn’t know how to assign vertices in isolated components. If instead, the probabilities were to scale like $1/n$, then we would be in the “giant component” regime, so we’d still expect enough signal to correctly assign a good fraction of the vertices, but the SDP is not tight in this regime.

## 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:

## Genius at Play: The Curious Mind of John Horton Conway

I got exactly what I wanted for Christmas this year! This book is great, and I highly recommend it:

True story: One evening in 1996, I remember watching the news with my parents, and the program concluded with a “Persons of the Week” segment, in which the winner of the Westinghouse Science Talent Search was interviewed. Jacob Lurie‘s winning research investigated a certain collection of numbers that, at the time, didn’t seem terribly exciting to me. I asked my parents, “What’s so interesting about serial numbers?” After laughing at my honest confusion, my parents offered some explanation: “He’s talking about surreal numbers, not serial numbers.” But in the absence of wikipedia, no further explanation could be provided.

## Compressed Sensing and its Applications 2015

Last week, I attended this conference in Berlin, and much like the last CSA conference, it was very nice. This year, most of the talks followed one of three themes:

• Application-driven compressed sensing