# All Seminars

Title: The Extremal Number of Tight Cycles
Seminar: Combinatorics
Speaker: Istvan Tomon of ETH Zurich
Contact: Dr. Hao Huang, hao.huang@emory.edu
Date: 2020-10-02 at 10:00AM
Venue: https://emory.zoom.us/j/96323787117
Abstract:
A tight cycle in an $r$-uniform hypergraph $\mathcal{H}$ is a sequence of $\ell\geq r+1$ vertices $x_1,\dots,x_{\ell}$ such that all $r$-tuples $\{x_{i},x_{i+1},\dots,x_{i+r-1}\}$ (with subscripts modulo $\ell$) are edges of $\mathcal{H}$. An old problem of V. S\'os, also posed independently by J. Verstra\"ete, asks for the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices which has no tight cycle. Although this is a very basic question, until recently, no good upper bounds were known for this problem for $r\geq 3$. In my talk, I will present a brief outline of the proof of the upper bound $n^{r-1+o(1)}$, which is tight up to the $o(1)$ error term. This is based on a joint work with Benny Sudakov.
Title: Imputing Missing Data with the Gaussian Copula
Seminar: Numerical Analysis and Scientific Computing
Speaker: Madeleine Udell of Cornell University
Contact: James Nagy, jnagy@emory.edu
Date: 2020-10-02 at 2:40PM
Venue: https://emory.zoom.us/j/95900585494
Abstract:
Missing data imputation forms the first critical step of many data analysis pipelines. The challenge is greatest for mixed data sets, including real, Boolean, and ordinal data, where standard techniques for imputation fail basic sanity checks: for example, the imputed values may not follow the same distributions as the data. This talk introduces a new semiparametric algorithm to impute missing values, with no tuning parameters. The algorithm models mixed data as a Gaussian copula. This model can fit arbitrary marginals for continuous variables and can handle ordinal variables with many levels, including Boolean variables as a special case. We develop an efficient approximate EM algorithm to estimate copula parameters from incomplete mixed data, and low rank and online extensions of the method that can handle extremely large datasets. The resulting model reveals the statistical associations among variables. Experimental results on several synthetic and real datasets show the superiority of the proposed algorithm to state-of-the-art imputation algorithms for mixed data.
Title: Scientific Machine Learning: Learning from Small Data
Seminar: Numerical Analysis and Scientific Computing
Speaker: Dr. Lu Lu of Brown University
Contact: Yuanzhe Xi, yxi26@emory.edu
Date: 2020-04-24 at 2:00PM
Venue: https://emory.zoom.us/j/313230176
Abstract:
Deep learning has achieved remarkable success in diverse applications; however, its use in scientific applications has emerged only recently. I have developed multi-fidelity neural networks to extract mechanical properties of solid materials (including 3D printing materials) from instrumented indentation. I have improved the physics-informed neural networks (PINNs) and developed the library DeepXDE for solving forward and inverse problems for differential equations, including partial differential equations (PDEs), fractional PDEs, and stochastic PDEs. I have also developed the deep operator network (DeepONet) based on the universal approximation theorem of operators to learn nonlinear operators (e.g., dynamical systems) accurately and efficiently from a relatively small dataset. In addition, I will present my work on the deep learning theory of optimization and generalization.
Title: Recent Development of Multigrid Solvers in HYPRE on Modern Heterogeneous Computing Platforms
Seminar: Numerical Analysis and Scientific Computing
Speaker: Dr. Ruipeng Li of Lawrence Livermore National Lab
Contact: Yuanzhe Xi, yxi26@emory.edu
Date: 2020-04-17 at 2:00PM
Venue: https://emory.zoom.us/j/313230176
Abstract:
Modern many-core processors such as the graphics processing units (GPUs) are becoming an integral part of many high performance computing systems nowadays. These processors yield enormous raw processing power in the form of massive SIMD parallelism. Accelerating multigrid methods on GPUs has drawn a lot of research attention in recent years. For instance, in recent releases of the HYPRE package, the structured multigrid solvers (SMG, PFMG) have full GPU-support for both the setup and the solve phases, whereas the algebraic multigrid (AMG) solver, namely BoomerAMG, has only its solve phase been ported and the setup can still be computed on CPUs only. In this talk, we will provide an overview of the available GPU-acceleration in HYPRE and present our current work on the algorithms in the AMG setup that are suitable for GPUs including the parallel coarsening algorithms, the interpolation methods and the triple-matrix multiplications. The recent results as well as the future work will also be included.
Title: A discussion on the Log-Brunn-Minkowski Conjecture and Related Questions
Seminar: Analysis and Differential Geometry
Speaker: Professor Galyna Livshytz of Georgia Institute of Technology
Date: 2020-04-07 at 4:00PM
Venue: https://emory.zoom.us/j/352530072
Abstract:
We shall discuss the Log-Brunn-Minkowski conjecture, a conjectured strengthening of the Brunn-Minkowski inequality proposed by Boroczky, Lutwak, Yang and Zhang. The discussion will involve introduction and explanation of how the local version of the conjecture arises naturally, a collection of ‘’hands on’’ examples and elementary geometric tricks leading to various related partial results, statements of related questions as well as a discussion of more technically involved approaches and results. Based on work with Johannes Hosle and Alexander Kolesnikov, as well as on previous joint results with Colesanti, Marsiglietti, Nayar, Zvavitch.
Title: Applied differential geometry and harmonic analysis in deep learning regularization
Seminar: Numerical Analysis and Scientific Computing
Speaker: Dr. Wei Zhu of Duke University
Contact: Yuanzhe Xi, yxi26@emory.edu
Date: 2020-04-03 at 2:00PM
Venue: https://emory.zoom.us/j/313230176
Abstract:
With the explosive production of digital data and information, data-driven methods, deep neural networks (DNNs) in particular, have revolutionized machine learning and scientific computing by gradually outperforming traditional hand-craft model-based algorithms. While DNNs have proved very successful when large training sets are available, they typically have two shortcomings: First, when the training data are scarce, DNNs tend to suffer from overfitting. Second, the generalization ability of overparameterized DNNs still remains a mystery despite many recent efforts. In this talk, I will discuss two recent works to “inject” the “modeling” flavor back into deep learning to improve the generalization performance and interpretability of DNNs. This is accomplished by deep learning regularization through applied differential geometry and harmonic analysis. In the first part of the talk, I will explain how to improve the regularity of the DNN representation by imposing a “smoothness” inductive bias over the DNN model. This is achieved by solving a variational problem with a low-dimensionality constraint on the data-feature concatenation manifold. In the second part, I will discuss how to impose scale-equivariance in network representation by conducting joint convolutions across the space and the scaling group. The stability of the equivariant representation to nuisance input deformation is also proved under mild assumptions on the Fourier-Bessel norm of filter expansion coefficients
Title: A GAN-based Approach for High-Dimensional Stochastic Mean Field Games
Seminar: Numerical Analysis and Scientific Computing
Speaker: Samy Wu Fung of UCLA
Contact: Lars Ruthotto, lruthotto@emory.edu
Date: 2020-03-27 at 2:00PM
Venue: https://emory.zoom.us/j/313230176
Abstract:
We present APAC-Net, an alternating population and agent control neural network for solving stochastic mean field games (MFGs). Our algorithm is geared toward high-dimensional instances of MFGs that are beyond reach with existing methods. We achieve this in two steps. First, we take advantage of the underlying variational primal-dual structure that MFGs exhibit and phrase it as a convex-concave saddle point problem. Second, we parameterize the value and density functions by two neural networks, respectively. By phrasing the problem in this manner, solving the MFG can be interpreted as a special case of training a generative adversarial generative network (GAN). We show the potential of our method on up to 50-dimensional MFG problems.
Title: Increasing paths in edge-ordered hypergraphs
Defense: Dissertation
Speaker: Bradley Elliott of Emory University
Date: 2020-03-27 at 3:30PM
Venue: https://emory.zoom.us/j/345080312
Abstract:
Abstract: In this defense, we will see many variations on a classic problem of ordering the vertices or edges of a graph and determining the length of "increasing" paths in the graph. For finite graphs, the vertex-ordering problem is completely solved, and there has been recent progress on the edge-ordering problem. Here, we also provide upper bounds for the edge-ordering problem with respect to complete hypergraphs and Steiner triple systems. We also prove the hypergraph version of the vertex-ordering problem: every vertex-ordered hypergraph H has an increasing path of at least $chi(H)-1$ edges, where $chi(H)$ is the chromatic number of $H$.\\ \\ For countably infinite graphs, a similar problem is studied, asking which graphs contain an infinite increasing path regardless of how their vertices or edges are ordered. Here we answer such questions for hypergraphs. For example, we show that the condition that a hypergraph contains a subhypergraph with all infinite degrees is equivalent to the condition that any vertex-ordering permits an infinite increasing path. We prove a similar result for edge-orderings. In addition, we find an equivalent condition for a graph to have the property that any vertex-ordering permits a path of arbitrary finite length. Finally, we study related problems for orderings by all integers $Z$ (instead of just positive integers $N$). For example, we show that for every countable graph, there is an ordering of its edges by $Z$ that forbids infinite increasing paths.
Title: Local-global principles for norm one tori over semi-global fields.
Defense: Dissertation
Speaker: Sumit Chandra Mishra of Emory University
Contact: David Zureick-Brown, DAVID.M.BROWN.JR@GMAIL.COM
Date: 2020-03-24 at 4:00PM
Venue: https://emory.zoom.us/j/382949597
Abstract:
Let K be a complete discretely valued field with residue field k (e.g. k((t)) ). Let n be an integer coprime to char(k). Let F = K(x) be the rational function field in one variable over F and L/F be any Galois extension of degree n. Suppose that either k is algebraically closed or k is finite field containing a primitive nth root of unity. Then we show that an element in F? is a norm from the extension L/F if and only if it is a norm from the corresponding extensions over the completions of F at all discrete valuations of F. We also prove that such a local-global principle holds for product of norms from cyclic extensions of prime degree if k is algebraically closed.
Title: Deep Learning with Graph Structured Data: Methods, Theory, and Applications
Seminar: Numerical Analysis and Scientific Computing
Speaker: Jie Chen of MIT-IBM Watson AI Lab, IBM Research
Contact: Yuanzhe Xi, yxi26@emory.edu
Date: 2020-03-06 at 2:00PM
Venue: MSC W201
Abstract:
Graphs are universal representations of pairwise relationship. With the rise of deep learning that demonstrates promising parameterizations of functions on Euclidean and regularly structured data (e.g., images and sequences), natural interests seek extensions of neural networks for irregularly structured data, including notably, graphs. This talk aims at painting a global picture of the emerging research on graph deep learning and inspiring novel directions. The speaker will share his recent research on modeling, computation, and applications of graph neural networks. Whereas modeling network architectures under different learning settings draws major interests in the field, understanding the capacity and limits of these networks attracts increasing attention. Moreover, efficient training and inference with large graphs or large collections of graphs need to address challenges beyond those of usual neural networks with regularly structured data. Of separate interest is the learning of a hidden graph structure if objects or variables interact, the subject of which interfaces with causality in machine learning. Last but not least, graphs admit numerous interesting applications, among which the speaker touches drug design, cryptocurrency forensics, cybersecurity, and power systems.