Later this month, Hans Parshall will participate in a summer school on “Sphere Packings and Optimal Configurations.” In preparation for this event, Hans was assigned the task of writing lecture notes that summarize the main results of the following paper:
P. Delsarte, J. M. Goethals, J. J. Seidel,
Spherical codes and designs,
Geometriae Dedicata 6 (1977) 363–388.
I found Hans’ notes to be particularly helpful, so I’m posting them here with his permission. I’ve lightly edited his notes for formatting and hyperlinks.
Without further ado:
Continue reading Spherical codes and designs
Emily King recently launched an online competition to find the best packings of points in complex projective space. The so-called Game of Sloanes is concerned with packing points in for and for . John Jasper, Emily King and I collaborated to make the baseline for this competition by curating various packings from the literature and then numerically optimizing sub-optimal packings. See our paper for more information:
J. Jasper, E. J. King, D. G. Mixon, Game of Sloanes: Best known packings in complex projective space
If you have a packing that improves upon the current leader board, you can submit your packing to the following email address:
asongofvectorsandangles [at] gmail [dot] com
In this competition, you can win money if you find a new packing that achieves equality in the Welch bound; see this paper for a survey of these so-called equiangular tight frames (ETFs).
Continue reading Game of Sloanes
Last week, the SIAM Conference on Applied Algebraic Geometry hosted a session on “Algebra, geometry, and combinatorics of subspace packings,” organized by Emily King and myself. Sadly, I wasn’t able to attend, but thankfully, most of the speakers gave me permission to post their slides on my blog. Here’s the lineup:
Emily King — Algebra, Geometry, and Combinatorics of Subspace Packings
Romanos Malikiosis — Group frames, full spark, and other topics
John Jasper — Equiangular tight frames from nonabelian groups
Gene Kopp — SIC-POVM existence and the Stark conjectures
Continue reading Algebra, geometry, and combinatorics of subspace packings
Last week, I co-organized (with Joey Iverson and John Jasper) a special session on “Recent Advances in Packing” for the AMS Spring Central Sectional Meeting at the Ohio State University. All told, our session had 13 talks that covered various aspects of packing, such as sphere packing, packing points in projective space, applications to quantum physics, and connections with combinatorics. It was a great time! And after the talks, we learned how to throw axes!
What follows is the list of speakers and links to their slides. (I anticipate referencing these slides quite a bit in the near future.) Thanks to all who participated!
Continue reading Recent Advances in Packing
I just returned from an amazing workshop in New Zealand organized by Shayne Waldron. The talks and activities were both phenomenal! Here’s a photo by Emily King that accurately conveys the juxtaposition:
A few of the talks gave me a lot to think about, and I wanted to take a moment to record some of these ideas.
Continue reading Tight Frames and Approximation 2018
Joey Iverson recently posted our latest paper with John Jasper on the arXiv. This paper can be viewed as a sequel of sorts to our previous paper, in which we introduced the idea of hunting for Gram matrices of equiangular tight frames (ETFs) in the adjacency algebras of association schemes, specifically group schemes. In this new paper, we focus on the so-called Schurian schemes. This proved to be a particularly fruitful restriction: We found an alternate construction of Hoggar’s lines, we found an explicit representation of the “elusive” packing from the real packings paper (based on a private tip from Henry Cohn), we found an packing involving the Mathieu group (this one beating the corresponding packing in Sloane’s database), we found some low-dimensional mutually unbiased bases, and we recovered nearly all small sized ETFs. In addition, we constructed the first known infinite family of ETFs with Heisenberg symmetry; while these aren’t SIC-POVMs, we suspect they are related to the objects of interest in Zauner’s conjecture (as in this paper, for example). This blog entry briefly describes the main ideas in the paper.
Continue reading Optimal line packings from finite group actions
This summer, I participated in several interesting conferences. This entry documents my slides and describes a few of my favorite talks from the summer. Here are links to my talks:
UPDATE: SIAM AG17 just posted a video of my talk.
Now for my favorite talks from FoCM, ILAS, SIAM AG17 and SPIE:
Ben Recht — Understanding deep learning requires rethinking generalization
In machine learning, you hope to fit a model so as to be good at prediction. To do this, you fit to a training set and then evaluate with a test set. In general, if a simple model fits a large training set pretty well, you can expect the fit to generalize, meaning it will also fit the test set. By conventional wisdom, if the model happens to fit the training set exactly, then your model is probably not simple enough, meaning it will not fit the test set very well. According to Ben, this conventional wisdom is wrong. He demonstrates this by presenting some observations he made while training neural nets. In particular, he allowed the number of parameters to far exceed the size of the training set, and in doing so, he fit the training set exactly, and yet he still managed to fit the test set well. He suggested that generalization was successful here because stochastic gradient descent implicitly regularizes. For reference, in the linear case, stochastic gradient descent (aka the randomized Kaczmarz method) finds the solution of minimal 2-norm, and it converges faster when the optimal solution has smaller 2-norm. Along these lines, Ben has some work to demonstrate that even in the nonlinear case, fast convergence implies generalization.
Continue reading Talks from the Summer of ’17
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:
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.”
Continue reading Notes on Zauner’s conjecture
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 . 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 ACFW17 — available here — appear in red):
Continue reading Zauner’s conjecture is true in dimensions 18, 20, 21, 30, 31, 37, 39 and 43