Georgia Institute of Technology College of Sciences

We study the classic matrix cross approximation based on the maximal volume submatrices. Our main results consist of an improvement of the classic estimate for matrix cross approximation and a greedy approach for finding the maximal volume submatrices. More precisely, we present a new proof of the classic estimate of the inequality with an improved constant. Also, we present a family of greedy maximal volume algorithms to improve the computational efficiency of matrix cross approximation. The proposed algorithms are shown to have theoretical guarantees of convergence. Finally, we present two applications: image compression and the least squares approximation of continuous functions. Our numerical results demonstrate the effective performance of our approach.

In 2017 Moseley, Proudfoot, and Young conjectured that the reduced Orlik-Terao algebra of the braid matroid was isomorphic as a symmetric group representation to the cohomology of a certain configuration space. This was proved by Pagaria in 2022. We generalize Pagaria's result from the braid arrangement to arbitrary hyperplane arrangements and recover a new proof in the case of the braid arrangement. Along the way, we give formulas for several other invariants of a hyperplane arrangement. Joint with Nick Proudfoot.

How to achieve the optimal control for general stochastic nonlinear is notoriously difficult, which becomes even more difficult by involving learning and exploration for unknown dynamics in reinforcement learning setting. In this talk, I will present our recent work on exploiting the power of representation in RL to bypass these difficulties. Specifically, we designed practical algorithms for extracting useful representations, with the goal of improving statistical and computational efficiency in exploration vs. exploitation tradeoff and empirical performance in RL. We provide rigorous theoretical analysis of our algorithm, and demonstrate the practical superior performance over the existing state-of-the-art empirical algorithms on several benchmarks. 

In enumerative combinatorics, one is often asked to count the number of combinatorial objects.  But the inverse problem is even more interesting: given some numbers, do they have a combinatorial interpretation?  In the main part of the talk I will give a broad survey of this problem, formalize the question in the language of computational complexity, and describe some connections to deep results and open problems in algebraic and probabilistic combinatorics.  In the last part of the talk, I will discuss our recent results on the defect and equality cases of Stanley inequalities for the numbers of bases of matroids and for the numbers of linear extensions of posets (joint work with Swee Hong Chan).  The talk is aimed at the general audience.