Colloquia

DMS Colloquium: Dr. Dana Bartošová

Apr 28, 2023 04:00 PM

Speaker: Dana Bartošová, University of Florida 

 

Title: Model Theory in Ramsey Theory

Abstract: Classical finite Ramsey theorem states that for any natural number \(m\leq n\) and a number of colours \(k\geq 2\), there is a natural number \(N\) such that whenever we colour \(m\)-tuples of \(\{1,2,\ldots, N\}\) by \(k\) colours, there is an \(n\)-element subset \(Y\) of  \(\{1,2,\ldots, N\}\) whose all \(m\)-element subsets received the same colour. This theorem has generalizations in possibly every area of mathematics. In this talk, we will focus on structural Ramsey theory, which in place of finite sets speaks about finite structures in some first-order language, such as finite graphs, finite Boolean algebras, or finite vector spaces over finite fields. I will talk about recent research in model-theoretic transfer principles of Ramsey properties between classes of finite structure (joint work with Lynn Scow) and from classes of finite structures to their ultraproducts (AIMSQuaREs project with Mirna Dzamonja, Rehana Patel, and Lynn Scow).

 

Faculty host: Joseph Briggs

 

Short Bio: Dana Bartosova obtained her undergraduate and master’s degree from the Charles University in Prague, as well as a master’s degree from the Free University of Amsterdam. She went on to complete her Ph.D. at the University of Toronto in 2013. Dana had postdocs at a semester program at HIM institute in Bonn, University of Sao Paulo, and Carnegie-Mellon University, and since 2018, she has been an assistant professor at the University of Florida. Her research is supported by an individual NSF grant and an NSF CAREER grant. She is also the co-founder of Math Parents Coffee and the director of UF Math Circle.

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DMS Colloquium: Dr. Lu Zhang

Apr 21, 2023 04:00 PM

Speaker: Dr. Lu Zhang, University of Southern California

 

Title: Bayesian inference for high-dimensional latent spatial model—Why we should and how to avoid random walk in MCMC

Abstract: High-dimensional latent spatial process models can present a significant challenge when it comes to obtaining Bayesian inference. In this talk. we will explore the pathological geometry features of the posterior distribution of latent spatial process models that impede the efficiency of MCMC sampling. I will present our proposed solutions that use conjugate and conditional conjugate prior along with scalable spatial modeling techniques to facilitate posterior sampling and posterior prediction. Our approaches exploit distribution theory for the Matrix-Normal distribution, which we use to construct scalable versions of a hierarchical linear model of coregionalization (LMC) and spatial factor models that deliver inference over a high-dimensional parameter space including the latent spatial process. Additionally, we develop Bayesian predictive stacking of spatial process models for geostatistical applications, which extends the prediction performance of conjugate spatial models. Our findings will be demonstrated through simulation studies and analyses of a large vegetation index data set, highlighting the computational and inferential benefits of our approaches compared to competing methods.

 

Short Bio: Lu Zhang is an Assistant Professor in the Division of Biostatistics in the Department of Population and Public Health Sciences at Keck School of Medicine, University of Southern California (USC). Prior to joining USC, she worked as a postdoctoral researcher with Andrew Gelman in the Department of Statistics at Columbia University, and with Bob Carpenter from Flatiron Institute. In 2020, she received her doctoral degree in Biostatistics under the supervision of Sudipto Banerjee from the Department of Biostatistics at University of California, Los Angeles (UCLA). Her research interests include statistical modeling and analysis for geographically referenced data, Bayesian statistics (theory and methods), statistical computing, and related software development.

Faculty host: Haoran Li

 

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DMS Colloquium: Dr. Wanli Qiao

Apr 14, 2023 04:00 PM

Speaker: Dr. Wanli Qiao (George Mason University)

 

Title: Connection and Consistency of Modal Clustering Methods 

Abstract: In the 1970s, Fukunaga and Hostetler put forward the idea of clustering points in space based on the gradient flow of the underlying sampling density. Despite generating significant interest, the theoretical basis of their proposal has not been fully developed, even in recent years. Our contribution to this effort is twofold. Firstly, we demonstrate how this clustering definition aligns with Hartigan's well-known definition, which also emerged in the 1970s, based on density level sets. These two approaches to clustering can thus be jointly referred to as "modal clustering," a term that has been in use for some time. Secondly, we present new consistency results for the main modal clustering methods. 

 

Faculty host: Guanqun Cao

 

Short Bio

Wanli Qiao is an Associate Professor in the Department of Statistics at George Mason University, where he also serves as the graduate program director. His research focuses on statistical inference, probability theory and computational algorithms related to geometric data analysis, which are applied towards analyzing the geometric and topological structures of molecular energy landscapes. His research is supported by the National Science Foundation. 

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DMS Colloquium: Dr. Roberto Molinari

Apr 07, 2023 04:00 PM

Speaker: Dr. Roberto Molinari (Auburn University)

 

Title: Inferential Solutions for Stochastic Processes, Differential Privacy and Multi-Model Paradigms (and others)

Abstract: The task of statistical inference, which includes the need for interpretation and uncertainty quantification, has become increasingly challenging due to the scale and complexity of data structures and types that are being collected. In this talk I will discuss three areas of methodological research that aim to address some of these challenges: (i) scalable inference for (composite) stochastic processes with complex data features (e.g. contamination, missing data); (ii) increased utility and exact inference for differentially private outputs which aim at protecting individual privacy; and (iii) multi-model inference and decision-making with a new paradigm for interpretation. To conclude, I will also briefly discuss other projects that do not fall within these three areas and then highlight the future directions we plan to take.

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DMS Colloquium: Dr. Jungeum Kim

Mar 31, 2023 04:00 PM

Speaker: Dr. Jungeum Kim (University of Chicago)

 

Title: Computational Complexity of Bayesian CART

Abstract: The success of Bayesian inference with MCMC depends critically on Markov chains reaching the posterior distribution reasonably fast. A host of theoretical results exist showing inferential validity of posteriors for Bayesian non-parametrics procedures (such as Gaussian processes or BART). However, convergence properties of MCMC algorithms that simulate from such ideal inferential targets are not always available. This work focuses on the Bayesian CART algorithm which forms a building block of Bayesian Additive Regression Trees (BART). We derive upper bounds on mixing times for typical posteriors under various proposal distributions. Exploiting the wavelet representation of trees, we provide sufficient conditions for Bayesian CART to mix well (polynomially) under certain hierarchical connectivity restrictions on the signal. We also derive a negative result showing that Bayesian CART (based on simple grow and prune steps) cannot reach deep isolated signals in faster than exponential mixing time. To remediate myopic chain exploration, we propose Twiggy Bayesian CART which attaches/detaches entire twigs (not just single nodes) in the proposal distribution. We show polynomial mixing of Twiggy Bayesian CART without assuming that the signal is connected on a tree. This talk is based on a joint work with Veronika Rockova at the University of Chicago Booth School of Business

 

Short Bio: Jungeum Kim is a Principal Researcher working with Professor Rockova at the University of Chicago Booth School of Business. She received her Ph.D. degree at Purdue University under the supervision of Dr. Xiao Wang. She studied three important problems in modern deep learning in her dissertation: adversarial robustness, dimensional reduction for visualization (manifold learning), and partially monotonic function modeling. She has also worked on generative models, such as generative adversarial networks, variational autoencoder, and energy-based models. Before her Ph.D., she received her Master's and Bachelor's degree at Seoul National University under the supervision of Dr. Hee-seok Oh and worked on robust PCA.

 

Faculty host: Guanqun Cao

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DMS Colloquium: Dr. Chris Cox

Mar 24, 2023 04:00 PM

Speaker: Dr. Chris Cox (Iowa State University)

 

Title: Small projective codes and equiangular lines

Abstract: How can one arrange \(d+k\) many vectors in \(\mathbb{R}^d\) so that they are as close to orthogonal as possible? Such arrangements are known as projective codes (or antipodal spherical codes) and are a natural generalization of balanced error-correcting codes. In this talk, we will consider the case when \(k\) is constant and \(d\to\infty\), i.e., when there is a "small excess" of vectors. In this realm, we show that there is an intimate connection to the existence of systems of \({k+1\choose 2}\) equiangular lines in \(\mathbb{R}^k\) and use this to obtain tight bounds for \(k\in\{1,2,3,7,23\}\) and outperform the Welch bound otherwise. To expose this relationship, we show how to "dualize" the problem and instead discuss bounding the first moment of isotropic probability masses (a.k.a. probabilistic tight frames) on \(\mathbb{R}^k\), which may be of independent interest. While we will focus mainly on the real case in this talk, all of these results translate naturally to the complex case (and even to the quaternions), wherein the answer relates to Zauner's conjecture on the existence of systems of \(k^2\) equiangular lines in \(\mathbb{C}^k\), also known as SIC-POVM in physics literature.

 

Faculty host: Joseph Briggs

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