Matrix Multiplication Not Yet Doable In Quadratic Time

Last night, a paper appeared on the arXiv titled A \Theta(n^2) time matrix multiplication algorithm.

At first look the paper seemed to score zero on Scott Aaronson’s list of Ten Signs a Claimed Mathematical Breakthrough is Wrong, and it came from a researcher who has worked extensively on the related problem of all-pairs shortest path, so it looked exciting. Unfortunately, it actually scores one on Scott’s list: it contradicts known impossibility results. (Thanks to Yuval Filmus for the references.)

The first step in the algorithm is to show how to compute the inner product of two 3^n-dimensional vectors using n^{O(1)} \cdot 2^n multiplication. This contradicts a lower bound due to Pan (see here the proof of a more general result) that computing the inner product of two N-dimensional vectors require at least N multiplications.

In addition, Raz has proved an \Omega (n^2 \log n) lower bound for matrix multiplication arithmetic circuits, and, to the best of my understanding, the paper’s claims imply an O(n^2) size arithmetic circuit. (Raz’s lower bound assumes that the circuit is not given access to large constants; as far as I can see, the formulas in Han’s paper do not require large constants.)

Holiday Readings

Now that the Winter break is coming, what to read in between decorating, cooking, eating, drinking and being merry?

The most exciting theoretical computer science development of the year was the improved efficiency of matrix multiplication by Stother and Vassilevska-Williams. Virginia’s 72-page write-up will certainly keep many people occupied.

Terence Tao is teaching a class on expanders, and posting the lecture notes, of exceptional high quality. It is hard to imagine something that would a more awesome combination, to me, than Terry Tao writing about expanders. Maybe a biopic on Turing’s life, starring Joseph Gordon-Levitt, written by Dustin Lance Black and directed by Clint Eastwood.

Meanwhile, Ryan O’Donnell has been writing a book on analysis of boolean function, by “serializing” it on a blog platform.

Now that I have done my part, you do yours. If I wanted to read a couple of books (no math, no theory) during the winter, what would you recommend. Don’t recommend what you think I would like, recommend what you like best, and why.