Georgia Institute of Technology College of Sciences

Each point x in Gr(r, n) corresponds to an r × n matrix Ax which gives rise to a matroid Mx on its columns. Gel’fand, Goresky, MacPherson, and Serganova showed that the sets {y ∈ Gr(r, n)|My = Mx} form a stratification of Gr(r, n) with many beautiful properties. However, results of Mnëv and Sturmfels show that these strata can be quite complicated, and in particular may have arbitrary singularities. We study the ideals Ix of matroid varieties, the Zariski closures of these strata. We construct several classes of examples based on theorems from projective geometry and describe how the Grassmann-Cayley algebra may be used to derive non-trivial elements of Ix geometrically when the combinatorics of the matroid is sufficiently rich.

Each point x in Gr(r,n) corresponds to an r×n matrix A_x which gives rise to a matroid M_x on its columns. Gel'fand, Goresky, MacPherson, and Serganova showed that the sets {y∈Gr(r,n)|M_y=M_x} form a stratification of Gr(r,n) with many beautiful properties. However, results of Mnëv and Sturmfels show that these strata can be quite complicated, and in particular may have arbitrary singularities. We study the ideals I_x of matroid varieties, the Zariski closures of these strata. We construct several classes of examples based on theorems from projective geometry and describe how the Grassmann-Cayley algebra may be used to derive non-trivial elements of I_x geometrically when the combinatorics of the matroid is sufficiently rich. 

In 1995, Mitsumatsu constructed a large family of Liouville domains whose topology obstructs the existence of a Weinstein structure.  Stabilizing these domains yields Liouville domains for which the topological obstruction is no longer in effect, and in 2019 Huang asked whether Mitsumatsu's Liouville domains were stably homotopic to Weinstein domains.  We answer this question in the affirmative.  This is joint work-in-progress with J. Breen.

In this talk, I will introduce some recent advances in designing stochastic primal-dual methods for bilinear saddle point problems, in the form of min_x max_y y^TAx under different geometries of x and y. These problems are prominent in economics, linear programming, machine learning and reinforcement learning. Specifically, our methods apply to Markov decision processes (MDPs), linear regression, and computational geometry tasks. 

In our work, we propose a variance-reduced framework for solving convex-concave saddle-point problems, given a gradient estimator satisfying some local properties. Further, we show how to design such gradient estimators for bilinear objectives under different geometry including simplex (l_2), Euclidean ball (l_1) or box (l_inf) domains. For matrix A with larger dimension n, nonzero entries nnz and accuracy epsilon, our proposed variance-reduced primal dual methods obtain a runtime complexity of nnz+\sqrt{nnz*n}/epsilon, improving over the exact gradient methods and fully stochastic methods in the accuracy and/or the sparse regime (when epsilon < n/nnz). For finite-sum saddle-point problems sum_{k=1}^K f_k(x,y) where each f is 1-smooth, we show how to obtain an epsilon-optimal saddle point within gradient query complexity of K+\sqrt{K}/epsilon.

Moreover, we also provide a class of coordinate methods for solving bilinear saddle-point problems. These algorithms use either O(1)-sparse gradient estimators to obtain improved sublinear complexity over fully stochastic methods, or their variance-reduced counterparts for improved nearly-linear complexity, for sparse and numerically sparse instances A. 

This talk is based on several joint works with Yair Carmon, Aaron Sidford and Kevin Tian, with links of papers below:

Variance Reduction for Matrix Games

Coordinate Methods for Matrix Games

Efficiently Solving MDPs using Stochastic Mirror Descent

Bio of the speaker: Yujia Jin is a fourth-year Ph.D. student in Department of Management Science and Engineering, Stanford University, working with Aaron Sidford. She is interested in designing efficient continuous optimization methods, which often run in nearly linear / sublinear time and find vast applications in machine learning, data analysis, reinforcement learning, and graph problems.