Events

Title: Asymptotic count of edge-bicolored graphs

 Abstract: I will talk about recent joint work with Chiara Meroni and Max Wiesmann, where we showed that specific exponential bivariate integrals serve as generating functions of labeled edge-bicolored graphs. Based on this, we prove an asymptotic formula for the number of regular edge-bicolored graphs with arbitrary weights assigned to different vertex structures.

The asymptotic behavior is governed by the critical points of a polynomial. An interesting application of this purely combinatorial work to mathematical physics is the Ising model on a random graph. I will explain how its phase transitions arise from our formula.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm,

Title:The Inexact Augmented Lagrangian Method: Optimal Complexity Bounds and Applications to Solving Huge SDPs

Abstract:In the first part of this talk, we present optimal and nearly-optimal parameter-free augmented Lagrangian (AL) methods for convex and strongly optimization with linear constraints. Our AL methods employ tractable inexact criteria for solving their inner subproblems, which accelerated methods can be shown to achieve in a finite number of iterations that depends on the target accuracy.

In the second part of this talk, we present a new inexact augmented Lagrangian method, namely, HALLaR, that employs a Burer-Monteiro style low-rank factorization for solving large-scale semidefinite programs (SDPs). The AL subproblems are solved by a hybrid low-rank method, which is based on a combination of a low-rank Frank-Wolfe method and a nonconvex accelerated inexact proximal point method. In contrast to the classical low-rank method by Burer and Monteiro, HALLaR finds a near-optimal solution (with provable complexity bounds) of SDP instances satisfying strong duality. Computational results comparing HALLaR to state-of-the-art solvers on several large SDP instances show that the former finds higher accurate solutions in substantially less CPU time than the latter ones. For example, in less than 20 minutes, HALLaR can solve (on a personal laptop) a maximum stable set SDP with 1 million vertices and 10 million edges within 1e-5 relative accuracy.

This talk is based on joint work with Saeed Ghadimi and Henry Wolkowicz from University of Waterloo and Diego Cifuentes and Renato Monteiro from Georgia Tech.

Title: Finite bivariate Tratnik functions

Abstract: In the context of algebraic combinatorics, P- and Q-polynomial association schemes are important objects and are closely related to distance-regular graphs. The polynomials appearing in these structures are classified by Leonard's theorem, and they belong to the discrete part of the (q-)Askey scheme. Relatively recently, the notions of P- and Q-polynomial association schemes as well as of distance-regular graphs have been generalized to the multivariate case. There is however no multivariate analog of Leonard's theorem. With the purpose of progressing in that direction, I will discuss ongoing work concerning certain finite families of bivariate functions, said of Tratnik type, which are expressed as an intricate product of univariate polynomials of the (q-)Askey scheme. The goal is to classify such functions which satisfy some generalized bispectral properties, that is, two recurrence relations and two (q-)difference equations of certain types.

Title: The quasisymmetric Macdonald polynomials are quasi-Schur positive at t = 0

 Abstract: The quasisymmetric Macdonald polynomials G_\gamma (X; q, t) are a quasisymmetric refinement of the symmetric Macdonald polynomials that specialize to the quasisymmetric Schur functions QS_\alpha (X). We study the t = 0 specialization G_\gamma (X; q,0), which can be described as a sum over weighted multiline queues. We show that G_\gamma (X; q, 0) expands positively in the quasisymmetric Schur basis and give a charge formula for the quasisymmetric Kostka-Foulkes polynomials K_{\gamma,\alpha}(q) in the expansion G_\gamma (X; q, 0) = \sum K_{\gamma,\alpha}(q) QS_\alpha(X). The proof relies heavily on crystal operators, and if you do not know what that means, come find out! This is joint work with Olya Mandelshtam.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm,

Title:: The training dynamics and local geometry of high-dimensional learning

Abstract:Many modern data science tasks can be expressed as optimizing a complex, random functions in high dimensions. The go-to methods for such problems are variations of stochastic gradient descent (SGD), which perform remarkably well—c.f. the success of modern neural networks. However, the rigorous analysis of SGD on natural, high-dimensional statistical models is in its infancy. In this talk, we study a general model that captures a broad range of learning tasks, from Matrix and Tensor PCA to training two-layer neural networks to classify mixture models. We show the evolution of natural summary statistics along training converge, in the high-dimensional limit, to a closed, finite-dimensional dynamical system called their effective dynamics. We then turn to understanding the landscape of training from the point-of-view of the algorithm. We show that in this limit, the spectrum of the Hessian and Information matrices admit an effective spectral theory: the limiting empirical spectral measure and outliers have explicit characterizations that depend only on these summary statistics. I will then illustrate how these techniques can be used to give rigorous demonstrations of phenomena observed in the machine learning literature such as the lottery ticket hypothesis and the "spectral alignment" phenomenona. This talk surveys a series of joint works with G. Ben Arous (NYU), R. Gheissari (Northwestern), and J. Huang (U Penn).

This talk is based on joint work with Saeed Ghadimi and Henry Wolkowicz from University of Waterloo and Diego Cifuentes and Renato Monteiro from Georgia Tech.

Title: New Strongly Regular Graphs from Finite Semifields and Finite Geometry

Abstract: Finite geometry often provides natural examples of highly structured combinatorial objects, many of which exhibit strong symmetry properties. 

In particular, many constructions of strongly regular graphs arise from classical geometric configurations. In this talk, we will present two new constructions of quasi-polar spaces, that give rise to two families of pairwise non-isomorphic strongly regular graphs, having the same non-trivial automorphism group.

Both constructions are related to a pair of commuting polarities in a projective space. Surprisingly, one of these constructions is connected to the algebraic structure of finite semifields and their tensor representation.

Title: Log-concavity of random Radon partitions

 Abstract: Over one hundred years ago, Radon proved that any set of d+2 points in R^d can be partitioned into two sets whose convex hulls intersect. I will talk about Radon partitions when the points are selected randomly. In particular, if the points are independent normal random vectors, let p_k be the probability that the Radon partition has size (k, d+2-k). Answering a conjecture of Kalai and White, we show that the sequence (p_k) is ultra log-concave and that, in fact, a balanced partition is the most likely. Joint work with Swee Hong Chan, Gil Kalai, Bhargav Narayanan, and Moshe White.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm,

Title: Exact algorithms for combinatorial interdiction problems

Abstract: Typical optimization paradigms involve a single decision maker that can control all variables involved. However, several practical situations involve multiple (potentially adversarial) decision makers. Bilevel optimization is a field that involves two decision makers. The key paradigm is that an upper-level decision maker acts first and after observing the upper-level decisions, a lower level decision maker optimizes their own objective. One particular important instance of such problems are so-called interdiction problems, where the upper-level decision is to try and forbid access to some decisions by the lower level and the goal of the upper level is to make the most impact into the lower-level decisions. These problems are, in general $\Sigma^P_2$-hard (likely harder than NP-hard). 

In this talk I will present recent work on improving significantly on state-of-the-art exact algorithms to obtain optimal solutions to some combinatorial interdiction problems. Despite the hardness results, our algorithm can solve instances consistently in a short amount of time. We also generalize our algorithms to propose a framework that could be applied to other similar problems, by deriving relaxations based on looking at these problems as games and performing operations on such games. 

This is joint work with Noah Weninger.