# Beyond John’s ellipsoid

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Here is the video of my talk from the Learning and Optimization workshop, which was a part of Microsoft TechFest.

This covers (parts of) our very fresh work (joint with Alex Andoni, Assaf Naor, Sasho Nikolov and Erik Waingarten), which is currently under submission. In short, we show the first Approximate Nearest Neighbor (ANN) search1 algorithm for general $d$-dimensional normed spaces with approximation $d^{o(1)}$. The only previously known bound is the trivial $O(\sqrt{d})$, which immediately follows from John’s theorem and any good ANN data structure for the Euclidean distance.

A couple of minor glitches:

1. The Wasserstein-$p$ distance is not a norm for $p > 1$;
2. In the last couple of slides, I started writing $D(\varepsilon)$ instead of $R(\varepsilon)$, but it is meant to be the same.

1 For some flavor of the problem, see this classic, or – in case you are more adventurous – my PhD thesis.

2018