Tuesday, April 24, 2018
04/24/2018 - 4:00pm to 5:00pm
Three fundamental factors determine the quality of a statistical learning algorithm: expressiveness, optimization and generalization. The classic strategy for handling these factors is relatively well understood. In contrast, the radically different approach of deep learning, which in the last few years has revolutionized the world of artificial intelligence, is shrouded by mystery. This talk will describe a series of works aimed at unraveling some of the mysteries behind expressiveness and optimization. I will begin by establishing an equivalence between convolutional networks - the most successful deep learning architecture to date, and hierarchical tensor decompositions. The equivalence will be used to answer various questions concerning the expressiveness of convolutional networks. I will then turn to discuss recent work analyzing optimization of deep linear networks. Surprisingly, in stark contrast with conventional wisdom, we find that depth, despite its non-convex nature, can accelerate optimization.
Works covered in this talk were in collaboration with Sanjeev Arora, Elad Hazan, Yoav Levine, Or Sharir, Amnon Shashua, Ronen Tamari and David Yakira.
04/24/2018 - 4:15pm to 5:15pm
A classical result by Bieberbach says that uniform lattices acting on Euclidean spaces by isometries are virtually free abelian. On the other hand, uniform lattices acting on trees are virtually free. This motivates the study of commensurability classification of uniform lattices acting on $CAT(0)$ cube complexes associated with right-angled Artin groups (RAAG complexes). These complexes can be thought as “interpolations” between Euclidean spaces and trees. Uniform lattices acting on the same RAAG complex may not belong to the same commensurability class, as there are irreducible lattices acting on products of trees. However, we show that the tree times tree obstruction is the only obstruction for commmensurability of label-preserving lattices acting on RAAG complexes. Some connection of this problem with Haglund and Wise’s work on special cube complexes will be explained. It time permits, I will also discuss some applications to quasi-isometric rigidity of RAAGs.
04/24/2018 - 4:30pm to 5:30pm
I will explain how to refine the statement of the denominator and evaluation conjectures for affine Macdonald polynomials proposed by Etingof-Kirillov Jr. and to prove the first non-trivial cases of these conjectures. Our method applies recent work of the speaker to relate these conjectures for quantum affine sl(2) to evaluations of certaintheta hypergeometric integrals defined by Felder-Varchenko. We then evaluate the resulting integrals, which may be of independent interest, by well-chosen applications of the elliptic beta integral of Spiridonov.These results are joint work with E. Rains and A. Varchenko.