## The Lovasz number of the plane is about 3.48287

As seen in this comment on Polymath and explicated further in Fernando de Oliveira Filho’s thesis, section 4.4.

I actually spent much of today thinking about this so let me try to explain it in a down-to-earth way, because it involved me thinking about Bessel functions for the first time ever, surely a life event worthy of recording.

So here’s what we’re going to do.  As I mentioned last week, you can express this problem as follows:  suppose you have a map h: R^2 -> V, for some normed vector space V, which is a unit-distance embedding; that is, if |x-x’|_{R^2} = 1, then |h(x)-h(x’)|_V = 1.  (We don’t ask that h is an isometry, only that it preserves the distance-1 set.)

Then let t be the radius of the smallest hypersphere in V containing h(R^2).

Then any graph embeddable in R^2 with all edges of length 1 is sent to a unit-distance graph in V contained in the hyperplane of radius t; this turns out to be equivalent to saying the Lovasz number of G (ok, really I mean the Lovasz number of the complement of G) is at most 1/(1-2t).  So we want to show that t is bounded below 1, is the point.  Or rather:  we can find a V and a map from R^2 to V to make this the case.

So here’s one!  Let V be the space of L^2 functions on R^2 with the usual inner product.  Choose a square-integrable function F on R^2 — in fact let’s normalize to make F^2 integrate to 1 — and for each a in R^2 we let h(a) be the function F(x-a).

We want the distance between F(x-a) and F(x-b) to be the same for every pair of points at distance 1 from each other; the easiest way to arrange that is to insist that F(x) be a radially symmetric function F(x) = f(|x|); then it’s easy to see that the distance between F(x-a) and F(x-b) in V is a function G(a-b) which depends only on |a-b|.  We write

$g(r) = \int_{\mathbf{R}^2} F(x)F(x-r) dx$

so that the squared distance between F(x) and F(x-r) is

$\int F(x)^2 dx - 2 \int F(x)F(x-r) dx + \int F(x-r)^2 dx = 2(1-g(r))$.

In particular, if two points in R^2 are at distance 1, the squared distance between their images in V is 2(1-g(1)).  Note also that g(0) is the square integral of F, which is 1.

What kind of hypersphere encloses all the points F(x-a) in V?  We can just go ahead and take the “center” of our hypersphere to be 0; since |F| = 1, every point in h(R^2) lies in (indeed, lies on) the sphere of radius 1 around the origin.

Hey but remember:  we want to study a unit-distance embedding of R^2 in V.  Right now, h sends unit distances to the distance 2(1-g(1)), whatever that is.  We can fix that by scaling h by the square root of that number.  So now h sends unit distances to unit distances, and its image is enclosed in a hypersphere of radius

2(1-g(1))^{-1}

The more negative g(1) is, the smaller this sphere is, which means the more we can “fold” R^2 into a small space.  Remember, the relationship between hypersphere number and Lovasz theta is

$2t + 1/\theta = 1$

and plugging in the above bound for the hypersphere number, we find that the Lovasz theta number of R^2, and thus the Lovasz theta number of any unit-distance graph in R^2, is at most

1-1/g(1).

So the only question is — what is g(1)?

Well, that depends on what g is.

Which depends on what F is.

Which depends on what f is.

And of course we get to choose what f is, in order to make g(1) as negative as possible.

How do we do this?  Well, here’s the trick.  The function G is not arbitrary; if it were, we could make g(1) whatever we wanted.  It’s not hard to see that G is what’s called a positive definite function on R^2.  And moreover, if G is positive definite, there exists some f giving rise to it.  (Roughly speaking, this is the fact that a positive definite symmetric matrix has a square root.)  So we ask:  if G is a positive definite (radially symmetric) function on R^2, and g(0) = 1, how small can g(1) be?

And now there’s an old theorem of (Wisconsin’s own!) Isaac Schoenberg which helpfully classifies the positive definite functions on R^2; they are precisely the functions G(x) = g(|x|) where g is a mixture of scalings of the Bessel function $J_0$:

$g(r) = \int_0^\infty J_0(ur) A(u)$

for some everywhere nonnegative A(u).  (Actually it’s more correct to say that A is a distribution and we are integrating J_0(ur) against a non-decreasing measure.)

So g(1) can be no smaller than the minimum value of J_0 on [0,infty], and in fact can be exactly that small if you let A become narrowly supported around the minimum argument.  This is basically just taking g to be a rescaled version of J_0 which achieves its minimum at 1.  That minimum value is about -0.4, and so the Lovasz theta for any unit-distance subgraph on the plane is bounded above by a number that’s about 1 + 1/0.4 = 3.5.

To sum up:  I give you a set of points in the plane, I connect every pair that’s at distance 1, and I ask how you can embed that graph in a small hypersphere keeping all the distances 1.  And you say:  “Oh, I know what to do, just assign to each point a the radially symmetrized Bessel function J_0(|x-a|) on R^2, the embedding of your graph in the finite-dimensional space of functions spanned by those Bessel translates will do the trick!”

That is cool!

Remark: Oliveira’s thesis does this for Euclidean space of every dimension (it gets more complicated.)  And I think (using analysis I haven’t really tried to understand) he doesn’t just give an upper bound for the Lovasz number of the plane as I do in this post, he really computes that number on the nose.

Update:  DeCorte, Oliveira, and Vallentin just posted a relevant paper on the arXiv this morning!

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## Suriya Gunasekar, optimization geometry, loss minimization as dynamical system

Awesome SILO seminar this week by Suriya Gunasekar of TTI Chicago.  Here’s the idea, as I understand it.  In a classical optimization problem, like linear regression, you are trying to solve a problem which typically has no solution (draw a line that passes through every point in this cloud!) and the challenge is to find the best approximate solution.  Algebraically speaking:  you might be asked to solve

$Ax = b$

for x; but since x may not be in the image of the linear transformation A, you settle for minimizing

$||Ax-b||$

in whatever norm you like (L^2 for standard linear regression.)

In many modern optimization problems, on the other hand, the problem you’re trying to solve may have a lot more degrees of freedom.  Maybe you’re setting up an RNN with lots and lots and lots of parameters.  Or maybe, to bring this down to earth, you’re trying to pass a curve through lots of points but the curve is allowed to have very high degree.  This has the advantage that you can definitely find a curve that passes through all the points.  But it also has the disadvantage that you can definitely find a curve that passes through all the points.  You are likely to overfit!  Your wildly wiggly curve, engineered to exactly fit the data you trained on, is unlikely to generalize well to future data.

Everybody knows about this problem, everybody knows to worry about it.  But here’s the thing.  A lot of modern problems are of this form, and yet the optima we find on training data often do generalize pretty well to test data!  Why?

Make this more formal.  Let’s say for the sake of argument you’re trying to learn a real-valued function F, which you hypothesize is drawn from some giant space X.  (Not necessarily a vector space, just any old space.)  You have N training pairs (x_i, y_i), and a good choice for F might be one such that F(x_i) = y_i.  So you might try to find F such that

$F(x_i) = y_i$

for all i.  But if X is big enough, there will be a whole space of functions F which do the trick!  The solution set to

$F(\mathbf{x}) = \mathbf{y}$

will be some big subspace F_{x,y} of X.  How do you know which of these F’s to pick?

One popular way is to regularize; you decide that some elements of X are just better than others, and choose the point of F_{x,y} that optimizes that objective.  For instance, if you’re curve-fitting, you might try to find, among those curves passing through your N points, the least wiggly one (e.g. the one with the least total curvature.)  Or you might optimize for some combination of hitting the points and non-wiggliness, arriving at a compromise curve that wiggles only mildly and still passes near most of the points.  (The ultimate version of this strategy would be to retreat all the way back to linear regression.)

But it’s not obvious what regularization objective to choose, and maybe trying to optimize that objective is yet another hard computational problem, and so on and so on.  What’s really surprising is that something much simpler often works pretty well.  Namely:  how would you find F such that F(x) = y in the first place?  You would choose some random F in X, then do some version of gradient descent.  Find the direction in the tangent space to X at F that decreases $||F(\mathbf{x})-\mathbf{y}||$ most steeply, perturb F a bit in that direction, lather, rinse, repeat.

If this process converges, it ought to get you somewhere on the solution space F_{x,y}. But where?  And this is really what Gunasekar’s work is about.  Even if your starting F is distributed broadly, the distribution of the spot where gradient descent “lands” on F_{x,y} can be much more sharply focused.  In some cases, it’s concentrated on a single point!  The “likely targets of gradient descent” seem to generalize better to test data, and in some cases Gunasekar et al can prove gradient descent likes to find the points on F_{x,y} which optimize some regularizer.

I was really struck by this outlook.  I have tended to think of function learning as a problem of optimization; how can you effectively minimize the training loss ||F(x)  – y||?  But Gunasekar asks us instead to think about the much richer mathematical structure of the dynamical system of gradient descent on X guided by the loss function.  (Or I should say dynamical systems; gradient descent comes in many flavors.)

The dynamical system has a lot more stuff in it!  Think about iterating a function; knowing the fixed points is one thing, but knowing which fixed points are stable and which aren’t, and knowing which stable points have big basins of attraction, tells you way more.

What’s more, the dynamical system formulation is much more natural for learning problems as they are so often encountered in life, with streaming rather than static training data.  If you are constantly observing more pairs (x_i,y_i), you don’t want to have to start over every second and optimize a new loss function!  But if you take the primary object of study to be, not the loss function, but the dynamical system on the hypothesis space X, new data is no problem; your gradient is just a longer and longer sum with each timestep (or you exponentially deweight the older data, whatever you want my friend, the world is yours.)

Anyway.  Loved this talk.  Maybe this dynamical framework is the way other people are already accustomed to think of it but it was news to me.

Slides for a talk of Gunasekar’s similar to the one she gave here

“Characterizing Implicit Bias in terms of Optimization Geometry” (2018)

“Convergence of Gradient Descent on Separable Data” (2018)

A little googling for gradient descent and dynamical systems shows me that, unsurprisingly, Ben Recht is on this train.

## New SIAM journal on applied algebra

As you know I’m a fan of “applied pure math.”  You can be an applied mathematician and not do PDEs or numerical analysis now!  And to reflect this new reality, there’s a new journal, SIAM Journal on Applied Algebra and Geometry (SIAGA).  They’re accepting submissions starting March 23; I actually have a little something that might be finished by then that would be perfect!

There are a bunch of cool pictures on the SIAGA flyer:  UC-Berkeley grad student Anna Seigal is explaining them on her blog, one by one.  The first entry depicts the algebraic variety you encounter when you try to find a point minimizing the summed distances to k other points in the plane.  Handsome!

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