Notes on Zauner’s conjecture

Last week, I visited Joey Iverson at the University of Maryland, and we spent a lot of time working through different approaches to Zauner’s conjecture. In general, my relationship with this problem is very similar to Steve Flammia‘s description, as paraphrased by smerkel on Physics Stack Exchange:

[Flammia described] the SIC-POVM problem as a “heartbreaker” because every approach you take seems super promising but then inevitably fizzles out without really giving you a great insight as to why.

Case in point, Joey and I identified a promising approach involving ideas from our association schemes paper. We were fairly optimistic, and Joey even bet me $5 that our approach would work. Needless to say, I now have this keepsake from Joey:

5dollars.png

While our failure didn’t offer any great insights (as Flammia predicted), the experience forced me to review the literature on Zauner’s conjecture a lot more carefully. A few things caught my eye, and I’ll discuss them here. Throughout, SIC denotes “symmetric informationally complete line set” and WH denotes “the Weyl-Heisenberg group.”

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Introduction to the polynomial method (and other similar things)

The polynomial method has made some waves recently (see this and that, for instance), and last week, Boris Alexeev gave a very nice talk on various applications of this method. This post is loosely based on his talk. All errors are my own.

It’s hard to pin down what exactly the polynomial method is. It’s a technique in algebraic extremal combinatorics, where the goal is to provide bounds on the sizes of objects with certain properties. The main idea is to identify the desired cardinality with some complexity measure of an algebraic object (e.g., the dimension of a vector space, the degree of a polynomial, or the rank of a tensor), and then use algebraic techniques to estimate that complexity measure. If at some point you use polynomials, then you might say you applied the polynomial method.

What follows is a series of instances of this meta-method.

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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:

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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.

Let’s start with two motivating applications:

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<p. 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.

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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
  • Quadratic or bilinear problems
  • Clustering in graphs or Euclidean space

Examples of application-driven CS include theoretical results for radar-inspired sensing matrices and model-based CS for quantitative MRI. Readers of this blog are probably familiar with the prototypical quadratic problem (phase retrieval), and bilinear problems include blind deconvolution and self-calibration. Recently, I have blogged quite a bit about clustering in Euclidean space (specifically, k-means clustering), but I haven’t written much about clustering in graphs (other than its application to phase retrieval). For the remainder of this entry, I will discuss two of the talks from CSA2015 that covered different aspects of graph clustering.

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Zero to One: Notes on Startups, or How to Build the Future

I’ve been thinking a lot about my place in the world lately. I’m interested in doing math that makes a difference, and considering much of the breakthroughs in our society have come from various startups, I decided to investigate the startup culture. How might academia benefit from startup culture? One could easily imagine a hip research environment adorned with beanbag chairs and foosball tables, but these perks aren’t the stuff that makes a startup successful. To catch a glimpse, I turned to a book recently written by Peter Thiel (of PayPal fame):

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Probably certifiably correct algorithms

This post is based on two papers (one and two). The task is to quickly solve typical instances of a given problem, and to quickly produce a certificate of that solution. Generally, problems of interest are NP-hard, and so we consider a random distribution on problem instances with the philosophy that real-world instances might mimic this distribution. In my community, it is common to consider NP-hard optimization problems:

minimize f(x) subject to x\in S.     (1)

In some cases, f is convex but S is not, and so one might relax accordingly:

minimize f(x) subject to x\in T,     (2)

where T\supseteq S is some convex set. If the minimizer of (2) happens to be a member of S, then it’s also a minimizer of (1) — when this happens, we say the relaxation is tight. For some problems (and distributions on instances), the relaxation is typically tight, which means that (1) can be typically solved by instead solving (2); for example, this phenomenon occurs in phase retrieval, in community detection, and in geometric clustering. Importantly, strong duality ensures that solving the dual of the convex relaxation provides a certificate of optimality.

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