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).

**The Jasper Prize.** John Jasper will pay US$100 to the first person who submits an ETF of vectors in such that

(i) , , and furthermore

(ii) no ETF of vectors in is currently known to exist.

To be explicit, there are 22 pairs in the “Game of Sloanes range” for which an ETF is currently known to exist:

Note: Fickus and Jasper conjecture that ETFs exist for both and ; see Conjecture 5.1 in this paper. Can you find either of these ETFs?

The goal of this competition is to find nice packings that generate new conjectures for the community to eventually prove (akin to Neil Sloane’s table of packings in real Grassmannian spaces). As an example, when building up the baseline for this competition, we happened upon a particularly nice packing of 5 points in which is the subject of the following conjecture:

**Conjecture.** The columns of the following matrix span optimally packed points in :

(This is Conjecture 8 in the paper.) If you need motivation to prove this conjecture, consider the following:

**King’s Coffee Prize.** Emily King will buy a coffee for the first person to prove the above conjecture.

(FYI – Emily is a coffee snob, so the coffee you get from her will be worth, like, two or three normal coffees.)

We look forward to your contributions to the Game of Sloanes!

]]>The purpose of this blog entry is to discuss recent observations regarding the Paley graph.

**— New bound on the clique number —**

There are several proofs that . In 2006, Maistrelli and Penman proved , and in 2013, Bachoc, Matolcsi and Ruzsa proved that holds for most . This contrasts with the best-known lower bound of . Considering this disparity between upper and lower bounds, I was excited to see progress on the upper bound this year:

**Theorem **(Hanson–Petridis 2019)**.**

The proof uses Stepanov’s method of auxiliary polynomials: Given such that , Hanson and Petridis construct a polynomial of degree with coefficients in such that every member of is a root of of multiplicity at least . It follows that .

Considering when is an even power of an odd prime, the proof necessarily makes use of the fact that is prime. We highlight how in the following proof sketch:

Fix , , and , and let denote the unique solution to the linear system

(Indeed, this Vandermonde system is invertible since the ‘s are distinct.) Define

By our choice of ‘s, an application of the binomial theorem reveals that has degree . (Here, it’s important that is a prime rather than a prime power, since otherwise the coefficient of might be zero.) Since , we have for every . This identity allows us to relate to the polynomial

which equals 1 by the binomial theorem and our choice of ‘s. In particular, for every , it holds that

To show that each is a root of with multiplicity at least , we take derivatives of before evaluating at . For example,

(Notice that division by might not be possible if were a prime power rather than a prime.) Similarly, the th derivative of vanishes on for each . This gives the desired result.

**— 3-point bounds on are computationally fast —**

Recall that the Lovasz theta number provides a semidefinite relaxation of the clique number of the complement graph. To see this, let denote the indicator vector of any vertex subset of a graph . Defining

then one may write

One may relax this integer program by recording convex constraints that satisfies:

This is known as the Lovasz theta number of , and by virtue of the relaxation, it holds that . When Lovasz first introduced the theta function, he proved that

with equality when is vertex-transitive. Since is both vertex-transitive and self-complementary, it follows that .

For a sharper bound on the clique number, one might consider a higher-order member of an appropriate SDP hierarchy. For example, Gvozdenovic, Laurent and Vallentin proposed such a hierarchy in 2009, and they evaluated its performance on the Paley graph. In this setting, it has become popular to refer to the classical Lovasz program as a 2-point bound, while higher-order programs produce -point bounds for . As an application, Gvozdenovic, Laurent and Vallentin computed 3-point bounds for , denoting them by ; see their Table 2 for explicit estimates. Unfortunately, going up the SDP hierarchy comes at a substantial computational price. Case in point, it took 4.5 hours to compute on a 3GHz processor with 1GB of RAM. In joint work with Mark Magsino and Hans Parshall, we managed to exploit symmetries in the Paley graph to reduce this runtime to under 20 seconds. Thanks to this speedup, we were able to run extensive experiments that suggest that is always away from the Hanson–Petridis bound:

Above, blue dots correspond to the 3-point bound, whereas the red curve gives the closed-form Hanson–Petridis bound. In particular, the 3-point bound improves upon the Hanson–Petridis bound in what appears to be a fraction of primes.

An easy interpretation of the 3-point bound is as follows: Since is vertex-transitive, we know that resides in a maximum clique. As such, we can restrict our search to the local graph , i.e., the subgraph of induced by the neighborhood of . In particular, it holds that

Next, the vertices of form a cyclic multiplicative group, and is circulant over this group. Since is circulant, the Lovasz SDP is invariant under conjugation by circulant permutations, and so we may average over these orbits to reduce to a linear program (this last move is common practice). While our numerical experiments solved this linear program for all primes less than 3000, we suspect that it’s possible to compute with in the millions by exploiting a fast Fourier transform in the LP solver. It would be interesting if one could construct feasible points for the dual LP that improve upon the Hanson–Petridis bound infinitely often. Our experiments suggest that the improvement would be an additive constant of 1 infinitely often, much like Bachoc, Matolcsi and Ruzsa’s improvement over the “trivial” bound.

**— Random Paley subgraphs satisfy a Kesten–McKay law —**

Select , and consider the random graph obtained by deleting vertices of independently with probability . What can be said about the statistics of such a random graph?

Intuitively, when is small, this random graph resembles the Erdos–Renyi graph with edge probability 1/2. The adjacency matrix of the latter graph is given by

where is symmetric with zeros on the diagonal and independent Rademacher variables on the off-diagonal. Since is a Wigner matrix, its spectrum follows a semi-circle law, and so enjoys a similar spectrum with an additional spike corresponding to the all-ones contribution. To corroborate our intuition, consider the following histograms; one illustrates the bulk of the spectrum of a 998-vertex instance of with and , and the other corresponds to an instance of the Erdos–Renyi graph on 998 vertices with edge probability 1/2. The largest eigenvalue of both graphs is off the chart at about 498. (Can you tell which is which?)

When is larger, we expect to inherit more of the structure from its parent graph , and therefore lose the semicircular behavior. For example, the following is a histogram of the bulk of the spectrum of an instance of with and :

In general, it turns out that the limiting spectral distribution for (where we fix and send ) is the so-called Kesten–McKay distribution. In joint work with Mark Magsino and Hans Parshall, we prove this using the moment method.

This result corresponds to a recent observation by Haikin, Zamir and Gavish: If is a random submatrix of a equiangular tight frame, then the eigenvalues of appear to exhibit a Wachter distribution with parameters determined by and . This observation will likely have implications for compressed sensing, where it’s important to have control over the conditioning of submatrices. To see the connection with the Paley graph, recall that the Paley equiangular tight frame is a real matrix with such that

where is the Seidel adjacency matrix of the Paley graph ; that is, , if , and otherwise . Furthermore, in this special case where , the Wachter distribution reduces to the Kesten–McKay distribution.

We’re excited to tackle more of the Haikin–Zamir–Gavish observations in the coming months.

]]>The biggest development in the previous thread:

The method used for finding this graph is vaguely described here and here. It seems that the method is currently more of an art form than an algorithm. A next step might be to automate the art away, code up any computational speedups that are available, and then throw more computing power at the problem.

]]>**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

**Matt Fickus** — Equiangular Tight Frames from Group Divisible Designs

**Mark Magsino** — A Delsarte-Style Proof of the Bukh–Cox Bound

**Joey Iverson **— Doubly transitive lines: Symmetry implies optimality

**Gary Greaves** — Equiangular lines in and the characteristic polynomial of a Seidel matrix

**Fabricio Machado** — -point semidefinite programming bounds for equiangular lines

**Marcin Pawlowski** — Using quantum information techniques to find the number of mutually unbiased bases in any given dimension

**Frederic Matter** — Detection of Ambiguities in Linear Arrays in Signal Processing

**Clayton Shonkwiler** — Symplectic Geometry and Frame Theory

**Tom Needham** — Symplectic Geometry, Optimization and Applications to Frame Theory

**Hans Parshall **— The optimal packing of eight points in

**Bill Martin** — Spherical configurations with few angles

Interest in this project has spiked since approaching (and passing) our original deadline of April 15. For this reason, I propose we extend the deadline to October 15, 2019. We can discuss this in the Polymath proposal page.

Here are some recent developments:

- The fractional chromatic number of the plane is at least 3.98.
- The smallest known 5-chromatic unit distance graph has 529 vertices and 2630 edges.
- Dömötör has an exciting idea for finding a human-verifiable proof that the chromatic number of the plane is at least 5.

I’m interested to see if this last point has legs!

]]>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.

Dömötör Pálvölgyi:

**Finding 5-chromatic UD graph with few vertices.**The record 553-vertex graph has been published by Marijn, not much news.**Probabilistic formulation.**Here I believe we have quite interesting results, my student is planning to write his MSc thesis on it (due June).**Bichromatic origin.**Here again we have some nice small graphs (in the plane, still looking for them on the sphere), and mathematically intriguing conjectures, which would imply improved bounds for the chromatic number of the space (if we find that graph on the sphere). About an attack on a conjecture by similar copies of almost monochromatic sets with limited success (and known limitations) we will soon publish something with Nora Frankl and Tamas Hubai.**Finding bounds for the width of the largest disk/strip that can be k-colored.**We have some bounds.**Determining the chromatic number of the extended rational field Q[w1,w2,..] for certain wi’s.**I know there are some results here, but what?

Aubrey de Grey:

**Fractional CN.**Jaan Parts has improved on the published record, but no one has yet computed the FCN of a 5-chromatic graph; however I put Jaan in contact with Marijn, who has much better hardware, so that may change soon. Then the question will be to find how to get a FCN above 4; presumably it will need at least a few interlocking 5-chromatic subgraphs, and that might be a clue for a 6-chromatic strategy.**Small graphs in higher dimensions.**There has been a fair bit of interest in my 59-vertex 6-chromatic graph in R^3, which is unpublished, so it should probably be described here. For completeness it may also be worth mentioning my 7-chromatic 24-vertex one in R^4, since it’s the best that has been achieved by explicit spindling and rotating and such like. My gut feeling is that the current records for higher dimensions can be beaten by spindling and rotation if only we can come up with some kind of pan-dimensional modular toolkit with which to build larger graphs, so it may be worth laying out what is so far known as a way to motivate such work.**Upper bounds in higher dimensions.**Philip coded up a nice fast way to find the most efficient colouring of a permutahedron-based tiling in higher dimensions and provided answers for R^5 and R^6; the number is still growing at about 3^n, so there seems to be a strong case for finding better tilings, especially ones in which tiles have fewer too-near neighbours. We should probably include my tiling of the plane in which each tile has onlly 16 too-near neighbours (as against 18 for the hexagonal tiling) as a way to motivate such a search in higher dimensions.**Fraction of the plane that can be k-coloured.**We have improvements (due to Jaan) for k=5 and 6, while Croft’s solution for k=4 remains unimproved (and also solves k=1, 2, 3).**The general concept of clamping.**Three families of 5-chromatic graphs (mine, Marijn’s and Dan/Geoff’s) are constructed by identifying simpler graphs with particular properties of this or that subset of vertices in any 4-colouring, and combining them so as to preclude all such properties being simultaneously true. There is sure to be masses to discover here. Way back in the first thread we had a brief exchange about finding simple M’s at the expense of more complex L’s (using graph names from my paper) but it never really went anywhere. A promising approach could be to find hexagonally symmetric graphs in which there is a hexagon (of non-unit radius) of vertices in which all three colours different from the centre are represented in any 4-colouring; I bet there are human-provable examples of that. We already have ones in which a hexagon of radius 8/3 has to all be the same colour as the centre, and there might be human-provable versions of that too.**Measurable CNP.**Marijn found a graph with 5617 vertices for which (in his words) it was “computationally hard” to find a 5 coloring with the center vertex being bichromatic. Sounds like it’s worth keeping looking for one that can’t be done, which would give MCNP >= 6.**Siamese tilings.**It turned out to be not particularly obvious how to tile the plane with seven colours in such a way that no two points of the same colour are exactly 1 apart but yet there are, irreducibly, separate tiles less than 1 apart. We never did anything much with such tilings, but they do seem to me to be so qualitatively different from ones in which all pairs of tiles are >=1 apart that we should say something about this.**SAT-based approaches to proving the nonexistence of a 6-tiling.**Boris’s original approach couldn’t find unscaleable tilings, but my suggestion for a variation seems to be theoretically more promising; however, Boris didn’t make it work. I think we should document what we did achieve so that others have a starting-point.- On Sept 29th, Jaan optimised Philip’s brilliant 6-tiled disk from June 17th. It ended up making sense! (in terms of regular pentagons)
- Just to be sure we remember: our bounds on Euclidean dimensions of k-coloured things need to include both the largest k-tileable disks or strips and the smallest disks or strips containing a graph that is not k-colourable.

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.

**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.

At it’s heart, this problem concerns linear dimensionality reduction. For the sake of illustration, suppose we want to find an appropriate projection for this dataset:

The gut-instinct method of dimensionality reduction is PCA, but this delivers poor results:

Of course, PCA ignores labels. Instead, you could run PCA on the differences between points of different labels, but in this case, you’d still get a dominant z-component, so this doesn’t help. Alternatively, you could run LDA, which projects onto the difference between class centroids (times an inverse covariance matrix). This also produces poor results:

Intuitively, we want to thumb through all possible projections to find a good one. This is what **SqueezeFit** does:

In particular, SqueezeFit is an SDP relaxation of the problem “find the minimum-rank orthogonal projection that keeps points of different labels separated.” In the paper, we prove some performance guarantees before applying SqueezeFit to real data. Overall, SqueezeFit provides an improvement over the standard approaches for linear dimensionality reduction. We’re excited to apply variants of SqueezeFit to various important settings.

**2. Utility Ghost: Gamified redistricting with partisan symmetry.**

There has been a lot of effort lately to use mathematical tools to help detect partisan gerrymandering. However, any detection procedure requires a technical definition of “excessively favoring one party over the other.” Any choice of definition can be perceived as arbitrary, or even sociological gobbledygook.

Considering this difficulty of fighting partisan gerrymandering in the courts, one might instead prevent gerrymandering from happening in the first place. Almost half of the states in the country use some sort of redistricting commission to draw a new map after the decennial census. Gamified redistricting offers a protocol for a bipartisan redistricting commission that leads to provably beneficial results. For example, the I-cut-you-freeze protocol is a modification of the I-cut-you-choose solution to the fair cake-cutting problem that provides a beautiful votes–seats curve in the limit as the number of districts goes to infinity (see Figure 1 in the paper).

Sadly, for smaller numbers of districts (which we frequently encounter in the real world), I-cut-you-freeze gives significant advantage to the first player. As an alternative, we propose **Utility Ghost**, which is a modification of the word game Ghost in which players take turns assigning precincts to districts. In the paper, we prove that in an idealized setting, if both players have the same number of votes, then under optimal play, they end up with the same number of seats.

We also show that Utility Ghost performs well in real-world settings. For example, consider the case of New Hampshire, which is made up of 10 counties and two U.S. congressional districts. If we don’t split counties, there are seven ways to partition New Hampshire into two districts with roughly the same sized population:

Here, proto-districts are colored according to the 2016 presidential election returns, where in New Hampshire, Hillary Clinton received 47.62 percent of the vote and Donald Trump received 47.25 percent. Since there are only two districts, the I-cut-you-freeze protocol is not helpful: the first player becomes *de facto* map maker, while the second player has no say in the matter. In particular, if the Democrats play first, they get to select the map in which they win both seats. This seems unfair, considering half of the voters are Republican. Alternatively, Utility Ghost avoids this map under optimal play, regardless of who plays first. Time will tell whether such gamified redistricting will be incorporated in protocols for bipartisan commissions following Census 2020.

**3. Derandomizing compressed sensing with combinatorial design.**

In compressed sensing, we encode sparse signals with random measurements, and then reconstruct the signals using L1 minimization. Here, the number of random measurements scales roughly linearly with the sparsity level. There has been some work to replicate this encoding performance with deterministic measurements, but the best theory to date requires a number of measurements that scales almost quadratically with the sparsity level.

Instead, one might attempt to minimize the number of random bits needed to accomplish the desired linear scaling. To this end, a previous paper leveraged pseudorandom properties of the Legendre symbol to derandomize sensing matrices composed of entries. Our new paper provides a more general treatment of derandomization for compressed sensing. As a special instance of our result, we can accommodate entries by sampling rows from an orthogonal array. The resulting sensing matrix uses slightly fewer measurements and slightly more randomness than the Legendre symbol–based construction. Our methods also provide derandomization by sampling from mutually unbiased bases.

In practice, reconstruction performance is identical to the fully random counterparts:

Still, our sensing matrices require a number of random bits that fail to break the “Johnson–Lindenstrauss bottleneck” identified in this paper. Is this a fundamental barrier to derandomized compressed sensing?

]]>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.

**Theorem.** Pick any field , fix , and let be any subset of size . Then there exists a full rank matrix such that every member of appears as an entry of .

**Proof:** Consider the homogeneous polynomial defined by

where means that either or but . Notice that precisely when is invertible with all distinct entries. We will identify a term of such that and so that the result follows immediately from the combinatorial nullstellensatz.

First, we observe that is a Vandermonde polynomial, and so the Leibniz formula of the corresponding determinant gives

Next, consider the term of corresponding to the identity permutation and the term of corresponding to any permutation satisfying for every . The product of these terms has exponent , where denotes the matrix representation of (namely, the identity matrix).

We claim that with and only if and , meaning there exists such that is a term of . To see this, first note that is uniquely minimized at , thereby forcing . Next, the remainder of is uniquely minimized at , thereby forcing . Continuing in this way, we obtain for every . Since for every , this forces , meaning , and so , i.e., .

Finally, since , and so we are done.

]]>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.

]]>