More FOCS 2014-blogging

In the spirit of better late than never, some more updates from Amirali Abdullah from his sojourn at FOCS 2014. Previously, he had blogged about the higher-order Fourier analysis workshop at FOCS.

I'll discuss now the first official day of FOCS, with a quick digression into the food first: the reception was lovely, with some nice quality beverages, and delectable appetizers which I munched on to perhaps some slight excess. As for the lunches given to participants, I will think twice in future about selecting a kosher option under dietary restrictions. One hopes for a little better than a microwave instant meal at a catered lunch, with the clear plastic covering still awaiting being peeled off. In fairness to the organizers, once I decided to revert to the regular menu on the remaining days, the chicken and fish were perfectly tasty.

I will pick out a couple of the talks I was most interested in to summarize briefly. This is of course not necessarily a reflection of comparative quality or scientific value; just which talk titles caught my eye.

The first talk is "Discrepancy minimization for convex sets" by Thomas Rothvoss. The basic setup of a discrepany problem is this: consider a universe of $n$ elements, $[n]$ and a set system of $m$ sets ($m$ may also be infinite), $S = \{S_1, S_2, \ldots, S_m \}$, where $S_i \subset [n]$. Then we want to find a $2$-coloring $\chi : [n] \to \{-1, +1 \}$ such that each set is as evenly colored as possible. The discrepany then measures how unevenly colored some set $S_i \in S$ must be under the best possible coloring.

One fundamental result is that of Spencer, which shows there always exists a coloring of discrepancy $O(\sqrt{n})$. This shaves a logarithmic factor off of a simple random coloring, and the proof is non-constructive. This paper by Rothvoss gives the first algorithm that serves as a constructive proof of the theorem.

The first (well-known) step is that Spencer's theorem can be recast as a problem in convex geometry. Each set $S_i$ can be converted to a geometric constraint in $R^n$, namely define a region $x \in R^n : \{ \sum_{j \in S_i} | x_j | \leq 100 \sqrt{n} \}$. Now the intersection of these set of constraints define a polytope $K$, and iff $K$ contains a point of the hypercube $\{-1 , +1 \}^n$ then this corresponds to the valid low discrepancy coloring.

One can also of course do a partial coloring iteratively - if a constant fraction of the elements can be colored with low discrepancy, it suffices to repeat.

The algorithm is surprisingly simple and follows from the traditional idea of trying to solve a discrete problem from the relaxation. Take a point $y$ which is generated from the sphercial $n$-dimensional Gaussian with variance 1. Now find the point $x$ closest to $y$ that lies in the intersection of the constraint set $K$ with the continuous hypercube $[-1, +1]^n$. (For example, by using the polynomial time ellipsoid method.) It turns out some constant fraction of the coordinates of $x$ are actually tight(i.e, integer valued in $\{-1, +1 \}$) and so $x$ turns out to be a good partial coloring.

To prove this, the paper shows that with high probability all subsets of $[-1 +1]^n$ with very few tight coordinates are far from the starting point $y$. Whereas with high probability, the intersection of $K$ with some set having many tight coordinates is close to $y$. This boils down to showing the latter has sufficiently large Gaussian measure, and can be shown by standard tools in convex analysis and probabilitiy theory. Or to rephrase, the proof works by arguing about the isoperimetry of the concerned sets.

The other talk I'm going to mention from the first day is by Karl Bringmann on the hardness of computing the Frechet distance between two curves. The Frechet distance is a measure of curve similarity, and is often popularly described as follows: "if a man and a dog each walk along two curves, each with a designated start and finish point, what is the shortest length leash required?"

The problem is solvable in $O(n^2)$ time by simple dynamic programming, and has since been improved to $O(n^2 / \log n)$ by Agarwal, Avraham, Kaplan and Sharir. It has long been conjectured that there is no strongly subquadratic algorithm for the Frechet distance. (A strongly subquadratic algorithm being defined as $O(n^{2 -\delta})$ complexity for some constant $\delta$, as opposed to say  $O(n^2 / polylog(n))$.)

The work by Bringmann shows this conjecture to be true, assuming SETH (the Strongly Exponential Time Hypothesis), or more precisely that there is no $O*((2- \delta)^N)$ algorithm for CNF-SAT. The hardness result holds for both the discrete and continuous versions of the Frechet distance, as well as for any $1.001$ approximation.

The proof works on a high level by directly reducing an instance of CNF-SAT to two curves where the Frechet distance is smaller than $1$ iff the instance is satisfiable. Logically, one can imagine the set of variables are split into two halves, and assigned to each curve. Each curve consists of a collection of "clause and assignment" gadgets, which encode whether all clauses are satisfied by a particular partial assignment. A different such gadget is created for each possible partial assignment, so that there are $O*(2^{N/2})$ vertices in each curve. (This is why solving Frechet distance by a subquadratic algorithm would imply a violation of SETH.)

There are many technical and geometric details required in the gadgets which I won't go into here. I will note admiringly that the proof is surprisingly elementary. No involved machinery or complexity result is needed in the clever construction of the main result; mostly just explicit computations of the pairwise distances between the vertices of the gadgets.


I will have one more blog post in a few days about another couple of results I thought were interesting, and then comment on the Knuth Prize lecture by the distinguished Dick Lipton.

FOCS Workshop on higher-order Fourier analysis: A Review

This is a guest post by my student Amirali Abdullah. Amirali attended FOCS 2014 and has a number of interesting reflections on the conference.

2014 was my first experience of attending a FOCS conference, and finally seeing the faces* (attached to some of the cutting edge as well as classical results in theoretical computer science. Not only did I get to enjoy the gentle strolls between conference halls, and hearing about fields I'd never known existed, I had the pleasure of doing so while revisiting historic Philadelphia.

With the talk videos now available, I thought now would be a good time to revisit some of the talks and my thoughts on the event. The usual disclaimers apply - this will be by no means comprehensive, and I won't go into the technicalities in much depth.

I'll begin with the first day, where I chose to attend the workshop on Higher-Order Fourier analysis. The starting point is that for the study of a function $f$, it is standard to consider its correlations with the Fourier basis of exponential functions (i.e., of the form $e(x) = e^{2 \pi \iota x}$) (also called as linear phase functions). This basis is orthonormal and has many nice analytic properties.

A standard technique is to decompose a function $f$ into the heavy components of its Fourier basis, and then argue that the contribution of the lower weight components is negligible. This gives a decompositon $f= f_1+ f_2$, where $f_1$ has few non-zero Fourier coefficients, and $f_2$ has all Fourier coefficients close to 0. Another way to view this under certain perspectives is $f_1$ representing the structured part of $f$ and $f_2$ the pseudorandom part.

However for some applications, the analysis of correlation with quadratic (or even higher order) phase functions of the form $e(x) = e^{2 \pi \iota x^2}$ is more powerful and indeed required. (An example of such a problem is where given a function on the integers, one desires to study its behavior on arithmetic progressions of length four or more.)

A subtlety in the study of higher order Fourier analysis is that notions such as "tail" and "weight" now have to be redefined. For regular Fourier analysis (at least on the Hamming cube) the natural notion corresponds with the $\ell_2^2$ norm or the linear correlation of a function and a linear basis element. However, in higher order Fourier analysis one has to define norms known as the Gowers uniformity norms which capture higher order correlations with these higher degree functions. This yields a decomposition of $f = f_1 + f_2 + f_3$, where $f_1$ consists of few non-zero higher order phase functions, $f_2$ has small $\ell_2$ norm and $f_3$ has small \emph{Gower's norm} under the right notion.

There were several talks discussing the various subfields of higher order Fourier analysis. Madhur Tulsiani discussed some of the algorithmic questions, including computing such a decomposition of the function into higher order Fourier terms in an analog of the Goldreich-Levin algorithm. Yui Yoshida discussed applications to algebraic property testing.

Abhishek Bhowmick discussed his very interesting paper with Lovett showing that the list-decoding radius of the Reed-Muller code over finite prime fields equals (approximately) the minimum distance of the code. The application of Fourier order analysis here is essentially to decompose the input space of the code by high-degree polynomials so that any random distribution is well-spread over these partition atoms.

I thought the workshop was an interesting exposure to some deep mathematics I had not previously seen, and gave some good pointers to the literature/work I can consult if I ever want a richer understanding of the toolset in the field.

Note: Thanks to Terry Tao's book on the subject for some useful background and context. All mistakes in this post are entirely mine. For a much more comprehensive, mathematically correct and broad view on the subject, do check out his blog.

* One can of course claim that 'seeing faces' could also include the digital images on faculty and student websites snapped in haste or misleading poses, but I choose to ignore this subtlety.