Georgia Institute of Technology College of Sciences

Krylov subspace methods (KSMs) are among the most widely used algorithms for a number of core linear algebra tasks. However, despite their ubiquity throughout the computational sciences, there are many open questions regarding the remarkable convergence of commonly used KSMs. Moreover, there is still potential for the development of new methods, particularly through the incorporation of randomness as an algorithmic tool. This talk will survey some recent work on the analysis of the well-known Lanczos method for matrix functions and the design of new KSMs for low-rank approximation of matrix functions and approximating partial traces and reduced density matrices. 

Abstract: Fundamental to our understanding of Teichm\"uller space T(S) of a closed oriented genus $g \geq 2$ surface S are two different perspectives: one as connected  component in the  PSL(2,\R) character variety  \chi(\pi_1S, PSL(2,\R)) and one as the moduli space of marked hyperbolic structures on S. The latter can be thought of as a moduli space of (PSL(2,\R), \H^2) -structures. The G-Hitchin component, denoted Hit(S,G), for G a split real simple Lie group, is a connected component in \chi(\pi_1S, G) that is a higher rank generalization of T(S). In this talk, we discuss a new geometric structures (i.e., (G,X)-structures) interpretation of Hit(S, G_2'), where G_2' is the split real form of the exceptional complex simple Lie group G_2.

After discussing the motivation and background, we will present some of the main ideas of the theorem, including a family of almost-complex curves
that serve as bridge between the geometric structures and representations.

Part I (SAM as an Optimal Relaxation of Bayes) Dr. Thomas Moellenhoff

Sharpness-aware minimization (SAM) and related adversarial deep-learning methods can drastically improve generalization, but their underlying mechanisms are not yet fully understood. In this talk, I will show how SAM can be interpreted as optimizing a relaxation of the Bayes objective where the expected negative-loss is replaced by the optimal convex lower bound, obtained by using the so-called Fenchel biconjugate. The connection enables a new Adam-like extension of SAM to automatically obtain reasonable uncertainty estimates, while sometimes also improving its accuracy.

Part II (Lie Group updates for Learning Distributions on Machine Learning Parameters) Dr. Eren Mehmet Kıral

I will talk about our recent paper https://arxiv.org/abs/2303.04397 with Thomas Möllenhoff and Emtiyaz Khan, and other related results. Bayesian Learning learns a distribution over the model parameters, allowing for different descriptions of the same data. This is (contrary to classical learning which "bets-it-all" on a single set of parameters in describing a given dataset and making predictions. We focus on classes of distributions which have a transitive Lie group action on them given by pushforwards of an action on the parameter space. I will also specialize to a few concrete Lie groups and show distinct learning behavior.