Monday, January 12, 2015

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.

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