Jan 30 (What is...)

Ian McPherson

• Title: What is Riemannian Optimization?

• Abstract: Within continuous optimization we make use of smoothness assumptions on our cost functions to construct well-principled algorithms to algorithmically search for an optimal solution. We often maintain that our search space is Euclidean, namely R^d. However, we can loosen this assumption on our search space and still leverage the same ideas fleshed out in most optimization courses. This talk will attempt to highlight that if we maintain analogous smoothness assumptions, we can still undergo first-order optimization routines in more general spaces. In words, we can carry out Gradient Descent algorithms not only on common manifolds (curved surfaces such as balls, matrix manifolds, etc.) but also can import these ideas to even more wild spaces (e.g.  the space of probability measures under the 2-Wasserstein distance, where I use these ideas). We look to highlight the key ideas that allow us to extend first-order optimization methods to these more general settings, so that we can all roughly say how Riemannian Gradient Descent works after this talk!

Feb 6

Sichen Yang (11am-12pm)

• Title: An overview of my work: Nonlinear model reduction of stochastic systems, Multilevel reinforcement learning, Applications at the intersection of machine learning and biology

• Abstract: First, we introduce a nonlinear stochastic model reduction technique for high-dimensional stochastic dynamical systems having a low-dimensional invariant effective manifold with slow dynamics, and high-dimensional, large fast modes. Given only access to short bursts of simulation, we design an algorithm that outputs an estimate of the invariant manifold, a process of the effective stochastic dynamics on it, which has averaged out the fast modes, and a simulator thereof. The algorithm and the estimation can be performed on-the-fly, leading to consistent and efficient exploration of the effective state space. This construction enables fast, efficient simulation of the effective dynamics, plus accurate estimation of crucial features of such dynamics, including the stationary distribution, identification of metastable states, and residence times and transition rates between them. Theoretical analysis demonstrates favorable sampling complexity scaling linearly in the high dimension of the state space, indicating that our approach overcomes the curse of dimensionality. Extensions of the technique motivated by some averaging and linear approximation theory address the nonlinearity of fast modes.

Second, we introduce a new fast multiscale procedure for repeatedly compressing Markov decision processes by using parametric families of policies to abstract sub-problems at finer scales. The multiscale representation yields substantial improvements in convergence rates both locally within sub-problems and globally across sub-problems. These multiscale decompositions also yield new transfer opportunities across different levels and different problems, by summarizing useful skills and higher-order functions from previously learned policies, which further enable systematic curriculum learning. In addition, we provide additional features like virtual policies and recursion. We demonstrate all the features above in a collection of illustrative domains.

Finally, we discuss several collaborative medical projects. One of the projects is a scientific machine learning project on digital cardiology in collaboration with Prof. Natalia A. Trayanova's lab, Prof. Yannis G. Kevrekidis' group, Prof. George Karniadakis' group, and Prof. Lu Lu's group, where we try to ease the prediction of cardiac reentry. Mathematically, we hope to use model reduction to speed up the simulation of the systems containing a reaction-diffusion equation associated to high-dimensional systems of ODEs with a timescale of separation between them. Using some homogenization of the solution, plus a selection of multiscale features contained in local spatial-temporal triangular boxes motivated by harmonic analysis, we could learn an operator mapping from the homogenized solution at the current time to the homogenized solution at the next time step accurately. Then, by iteratively applying this learned operator, we could simulate the system fast, stably, and accurately across many time steps in unseen future time intervals during the testing phase.

Feb 13

Hamilton Sawczuk, slides

• Title: Symbolic Listings as Computation

• Abstract: This talk is based on a forthcoming archive preprint written in collaboration with Professor Gnang. We begin with a review of symbolic polynomial listings and functional graph theory and discuss how these can be used to implement arbitrary Boolean functions via "differential computers." Next we introduce the Chow rank of a polynomial, which appears as a natural measure of complexity for "differential computers." We conclude by showing that in a specific sense it is sufficient to study functional graphs when upper-bounding the Chow ranks of general graph families.

Feb 20

Matt Santana

• Title: Total Variation Bounds for Sequential Monte Carlo on Multimodal Distributions

• Abstract: This talk will be structured around understanding the main result of a paper written in collaboration with Professor Lee. We will cover how basic sampling theory is tied to the convergence rates of Markov Chains and discuss how sampling from multimodal distributions via basic Markov Chain Monte Carlo methods (MCMC) has poor time complexity. To remedy the poor time complexity of simple MCMC methods we introduce Sequential Monte Carlo (SMC) and present our main result on total variation bounds for SMC on multimodal distributions which depend only on local Markov chain mixing dynamics.

Feb 27 (What is...)

Kaleigh Rudge

• Title: What are... Numerical Methods and Who are some mathematicians that use these?

• Abstract: This edition of the "What is..." series will focus on numerical methods for solving PDEs while highlighting the life and work of some under-represented mathematicians who have used and contributed to the development of this field. We will start by motivating why we care about numerical methods with some applications. Next we will introduce Euler's Method for parabolic PDEs. Then we will continue onto iterative methods with schedule relaxation for eliptic PDEs. Finally, we will conclude with a brief overview of multigrid methods.

Mar 5

Aranyak Acharyya

• Title: Convergence guarantees for response prediction in latent structure networks on unknown one-dimensional manifolds

• Abstract: TBA

Mar 12

Dapeng Yao, slides

• Title: Bayesian Sparse Gaussian Mixture Model for Clustering in High Dimensions

• Abstract: We study the sparse high-dimensional Gaussian mixture model when the number of clusters is allowed to grow with the sample size. A minimax lower bound for parameter estimation is established, and we show that a constrained maximum likelihood estimator achieves the minimax lower bound. However, this optimization-based estimator is computationally intractable because the objective function is highly nonconvex and the feasible set involves discrete structures. To address the computational challenge, we propose a computationally tractable Bayesian approach to estimate high-dimensional Gaussian mixtures whose cluster centers exhibit sparsity using a continuous spike-and-slab prior. We further prove that the posterior contraction rate of the proposed Bayesian method is minimax optimal. The mis- clustering rate is obtained as a by-product using tools from matrix perturbation theory. The proposed Bayesian sparse Gaussian mixture model does not require pre-specifying the number of clusters, which can be adaptively estimated. The validity and usefulness of the proposed method is demonstrated through simulation studies and the analysis of a real-world single-cell RNA sequencing data set.

Mar 19

Spring break.

Mar 26 (What is...)

Yuxin Ma, slides

• Title: What is Topological Data Analysis?

• Abstract: This talks serves as an introduction to Topological Data Analysis (TDA) – a method for analyzing the shape and structure of data using concepts from topology. TDA merges algebraic topology and other mathematical tools with real-world statistical analysis, offering both elegant mathematical theory and practical applications. The focus of this talk will be on Persistent Homology, widely regarded as the cornerstone technique in TDA.

We'll begin by exploring "why should we care about the topology or shape of data, and in what contexts is TDA valuable?" through examples such as brain artery tree data and prostate cancer histopathology images. Then, we'll provide a gentle introduction to key concepts in homology theory (within algebraic topology) and demonstrate how these ideas evolve into the concept of persistent homology. Finally, we will examine how we could perform statistical analysis on persistent homology - through vectorization.

If you are interested in understanding more about the topic, you could check out Vidit Nanda's lecture notes or (shameless plug here) an essay I wrote during my Master's program. 

Apr 2

Zhiyue Zhang

• Title: Attentive Joint Model of Longitudinal Data, Survival, and Recurrent Events with Concurrent Latent Structure

• Abstract: In this talk, I will introduce the concept of joint modeling of multivariate longitudinal data and survival data while capturing their interdependency. We will begin with a standard statistical joint model based on the linear mixed effects model and Cox proportional hazards model. Then, we will incorporate neural networks to make each submodel of the joint model more flexible. Finally, I will go over our novel deep joint model based on the transformer architecture, which further jointly models recurrent events.

Apr 9

Benjamin Brindle

• Title: Time Series Clustering

• Abstract: We examine how mixtures of Linear Gaussian State Space Models can be used to tackle unsupervised clustering problems for time series. We study the case of unequally sampled time series and motivate the discussion with several examples. We review important underlying concepts such as the Expectation Maximization Algorithm and Kalman Filter. To broaden the scope, we will also discuss how I came to work on this particular problem as well as the future directions I am currently working on.

Apr 16

Zan Ahmad

• Title: Neural Operators For Learning PDEs on Geometrically Varying Data

• Abstract: In this talk I'll introduce the concept of neural operators for learning mappings between infinite dimensional function spaces. I will briefly discuss some of the powerful neural network architectures for learning partial differential operators under this umbrella and discuss two new types of frameworks we have developed in the lab: 'Diffeomorphic Mapping Operator Learning and Graph Fourier Neural Kernels'. I will discuss their formulations and then show some experiments on pedagogical examples and on real patient data for speeding up cardiac electrophysiology simulations. Lastly, I will discuss some future directions in which we aim to extend our work to learn vector and tensor valued operators, towards the goal of learning fluid and solid mechanics of the heart.

Apr 23

Phillip Kerger

• Title: Explorations and Insights in Convex Geometry and Optimization 

• Abstract: This talk will cover three different very geometric ideas across convex optimization, less focused on research / published results and more on the fun geometry arising in the discussed topics. First, I will show that d^2 bits of first-order information are needed to solve the separation problem in d dimensions. Second, I will illustrate interesting behavior of regions in which gradients of smooth functions can be guaranteed to lie when iteratively optimizing a smooth function. This won't be computationally useful at all, but is fun. Third, I will consider an apparent mismatch between dimension-dependent upper- and lower- bounds for convex optimization, which is rectified once one considers the norms being used. If there is time, I might briefly say a couple words about my experience in the academic job search.