Polymath16, twelfth thread: Year in review and future plans

This is the twelfth “research” thread of the Polymath16 project to make progress on the Hadwiger–Nelson problem, continuing this post. This project is a follow-up to Aubrey de Grey’s breakthrough result that the chromatic number of the plane is at least 5. Discussion of the project of a non-research nature should continue in the Polymath proposal page. We will summarize progress on the Polymath wiki page.

Activity on this project has slowed considerably, as we’ve gone 6 months without having to roll over to a new thread. As mentioned in the original thread, the deadline for this project is April 15, 2019, so we only have a couple of weeks remaining. Dömötör and Aubrey took the time to summarize the highlights of what we’ve accomplished in the last year (see below). While we don’t have a single killer result to publish, there are several branches of minor results that warrant publication. Feel free to comment on additional results that were missed in the summaries below, as well as possible venues for publication.

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MATH 8610: Mathematics of Data Science

This spring, I’m teaching a graduate-level special topics course called “Mathematics of Data Science” at the Ohio State University. This will be a research-oriented class, and in lecture, I plan to cover some of the important ideas from convex optimization, probability, dimensionality reduction, clustering, and sparsity.

Click here for a draft of my lecture notes.

The current draft consists of a chapter on convex optimization. I will update the above link periodically. Feel free to comment below.

UPDATE #1: Lightly edited Chapter 1 and added a chapter on probability.

UPDATE #2: Lightly edited Chapter 2 and added a section on PCA.

UPDATE #3: Added a section on random projection.

UPDATE #4: Lightly edited Chapter 3. The semester is over, so I don’t plan to update these notes again until I teach a complementary special topics course next year.

A few paper announcements

This last semester, I was a long-term visitor at the Simons Institute for the Theory of Computing. My time there was rather productive, resulting in a few (exciting!) arXiv preprints, which I discuss below.

1. SqueezeFit: Label-aware dimensionality reduction by semidefinite programming.

Suppose you have a bunch of points in high-dimensional Euclidean space, some labeled “cat” and others labeled “dog,” say. Can you find a low-rank projection such that after projection, cats and dogs remain separated? If you can implement such a projection as a sensor, then that sensor collects enough information to classify cats versus dogs. This is the main idea behind compressive classification.

A neat application of the polynomial method

Two years ago, Boris Alexeev emailed me a problem:

Let .  Suppose you have distinct numbers in some field.  Is it necessarily possible to arrange the numbers into an  matrix of full rank?

Boris’s problem was originally inspired by a linear algebra exam problem at Princeton: Is it possible arrange four distinct prime numbers in a rank-deficient matrix? (The answer depends on whether you consider to be prime.) Recently, Boris reminded me of his email, and I finally bothered to solve it. His hint: Apply the combinatorial nullstellensatz. The solve was rather satisfying, and if you’re reading this, I highly recommend that you stop reading here and enjoy the solve yourself.

Polymath16, eleventh thread: Chromatic numbers of planar sets

This is the eleventh “research” thread of the Polymath16 project to make progress on the Hadwiger–Nelson problem, continuing this post. This project is a follow-up to Aubrey de Grey’s breakthrough result that the chromatic number of the plane is at least 5. Discussion of the project of a non-research nature should continue in the Polymath proposal page. We will summarize progress on the Polymath wiki page.

Here’s a brief summary of the progress made in the previous thread:

– Let w(k) denote the supremum of w such that is k-colorable. Then of course and for every . Furthermore,

Colorings that produce these lower bounds are depicted here. The upper bound for k=3 is given here.

– The largest known k-colorable disks for k=2,3,4,5 are depicted here.

Presumably, we can obtain descent upper bounds on w(4) by restricting (a finite subset of) the ring to an infinite strip.

Foundations of Data Science Boot Camp, V

This is the fifth (and final) entry to summarize talks in the “boot camp” week of the program on Foundations of Data Science at the Simons Institute for the Theory of Computing, continuing this post. On Friday, we heard talks from Ilya Razenshteyn and Michael Kapralov. Below, I link videos and provide brief summaries of their talks.

Ilya Razenshteyn — Nearest Neighbor Methods

Foundations of Data Science Boot Camp, IV

This is the fourth entry to summarize talks in the “boot camp” week of the program on Foundations of Data Science at the Simons Institute for the Theory of Computing, continuing this post. On Thursday, we heard talks from Santosh Vempala and Ilias Diakonikolas. Below, I link videos and provide brief summaries of their talks.

Santosh Vempala — High Dimensional Geometry and Concentration