# School of Mathematics

## Learning from Multiple Biased Sources

## Is the variety of singular tuples of matrices a null cone?

The following multi-determinantal algebraic variety plays an important role in algebra and computational complexity theory: SING_{n,m}, consisting of all m-tuples of n x n complex matrices which span only singular matrices. In particular, an efficient deterministic algorithm testing membership in SING_{n,m} will imply super-polynomial circuit lower bounds, a holy grail of the theory of computation. <br>

## Observable events" and "typical trajectories" in finite and infinite dimensional dynamical systems

## Classification of n-component links with Khovanov homology of rank 2^n

Suppose L is a link with n components and the rank of Kh(L;Z/2) is 2^n, we show that L can be obtained by disjoint unions and connected sums of Hopf links and unknots. This result gives a positive answer to a question asked by Batson-Seed, and generalizes the unlink detection theorem of Khovanov homology by Hedden-Ni and Batson-Seed. The proof relies on a new excision formula for the singular instanton Floer homology introduced by Kronheimer and Mrowka.

This is joint work with Yi Xie.

## Direct and dual Information Bottleneck frameworks for Deep Learning

The Information Bottleneck (IB) is an information theoretic framework for optimal representation learning. It stems from the problem of finding minimal sufficient statistics in supervised learning, but has insightful implications for Deep Learning. In particular, its the only theory that gives concrete predictions on the different representations in each layer and their potential computational benefit. I will review the theory and its new version, the dual Information Bottleneck, related to the variational Information Bottleneck which is gaining practical popularity.

## Strong Average-Case Circuit Lower Bounds from Non-trivial Derandomization

We prove that, unconditionally, for all constants a, NQP = NTIME[n^polylog(n)] cannot be (1/2 + 2^(-log^a n) )-approximated by 2^(log^a n)-size ACC^0 circuits. Previously, it was even open whether E^NP can be (1/2+1/sqrt(n))-approximated by AC^0[2] circuits. As a straightforward application, we obtain an infinitely often non-deterministic pseudorandom generator for poly-size ACC^0 circuits with seed length 2^{log^eps n}, for all eps > 0.