Binary classification with linear classifiers is a fundamental problem in machine learning with broad applications. Traditionally, this problem focuses on using labeled data to learn a linear classifier, but data is often expensive to label or difficult to acquire. Instead of labeling all data points in a data set, we consider labeling a specially chosen subset of points and ask how well we can accomplish our learning task. To answer this question, we will reformulate this problem into the context of Gaussian spherical tessellations and study geometric properties of such tessellations. This work is joint with Rayan Saab and Eric Lybrand.

Topological and geometric data analysis (TGDA) is a powerful framework for quantitative description and simplification of datasets & shapes. It is especially suitable for modern biological data that are intrinsically complex and high-dimensional. Traditional topological data analysis considers the geometric features of a dataset, while in practice, there could be both geometric and non-geometric features. In this talk, I will introduce a persistent cohomology based method, enriched barcode to embed the non-geometric features in the topological invariants. I will then talk about a geometric method, unnormalized optimal transport for integrating heterogeneous datasets which is crucial for generating a comprehensive topological perspective for the system of interest. Scientific data often have limited size and high complexity, and a straightforward application of machine learning to raw data could result in suboptimal performances. To tackle this challenge, we integrate the TGDA method designed for biological data with deep learning. This topology-based deep learning strategy achieves top performance on standard benchmarks and D3R Grand Challenges, a worldwide competition series in computer-aided drug design. I will also show several applications of our geometric method to the analysis and integration of single-cell omics data. Finally, I will discuss future directions on data-driven modeling using topology, geometry, and machine learning-based approaches.

Single-cell RNA sequencing (scRNA-seq) enables the collection of rich phenotypic information of individual cells. During this sequencing process, each cell is destroyed, which poses challenges in the analysis of time-course data of heterogeneous populations since cell trajectories need to be computationally inferred. Standard trajectory inference methods approximate the transcriptional spectrum of all time points combined by a graph, not leveraging information about the time-points at which the profiles were captured. The recently proposed Waddington-OT algorithm instead appeals to the theory of optimal transport to infer probabilistic couplings between different time points. First, as an example of the benefits and drawbacks of these methods, I will show how we applied them to a problem in immunology, revealing previously unknown effector state potential of tissue-resident innate lymphoid cells. Second, I will discuss issues with the sample efficiency of optimal transport methods in high-dimensions and two approaches to overcome this problem. The first approach relies on the notion of the transport rank of a probabilistic coupling and I will provide empirical and theoretical evidence that it can be used to significantly improve the rates of estimation of optimal transport distances and plans. The second approach relies on a wavelet regularization and admits near minimax optimal rates for the estimation of smooth optimal transport maps.

Systems with multiple timescales, in which the dynamics separate into slow and fast components, occur ubiquitously in models of physical, biological and ecological processes. In this talk I will focus on two example applications: the dynamics of vegetation patterns in water-limited regions and the propagation of impulses along nerve fibers. These processes can be modeled by reaction diffusion PDEs in which the patterns arise as traveling wave solutions. The slow/fast timescale separation induces a geometry on the underlying equations which can be exploited to gain insight into the structure and stability of these patterns.

Michael Cranston is inviting you to a scheduled Zoom meeting.

Topic: Carter Colloquium
Time: Oct 9, 2020 03:30 PM Pacific Time (US and Canada)

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Meeting ID: 946 1422 4176
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Elliptic curves are simple-looking polynomial equations in two variables whose solutions are still a mystery. The Birch and Swinnerton-Dyer Conjecture (a millennium problem) relates these solutions to a complex function. The conjecture is deep because it connects algebra with analysis. After explaining the conjecture, we discuss some recent results towards it, along with strategies of proving it one prime at a time.