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Organizing Committee
Tufts University
University of Massachusetts Amherst
Tufts University
UMass Amherst
Brown University
Abstract
This workshop will focus on the intersection of mathematics, statistics, machine learning, and computation, when viewed through the lens of optimal transport (OT). Mathematical topics will include low-dimensional models for OT, linearizations of OT, and the geometry of OT including gradient flows and gradient descent in the space of measures. Relevant statistical topics will include reliable and efficient estimation of OT plans in high dimensions, the role of regularization in computing OT distances and plans, with applications to robust statistics, uncertainty quantification, and overparameterized machine learning. Computation will be a recurring theme of the workshop, with emphasis on the development of fast algorithms and applications to computational biology, high energy physics, material science, spatio-temporal modeling, natural language processing, and image processing.
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Workshop Schedule
Monday, May 08, 2023
8:50 - 9:00 AM EDT
Welcome
11th Floor Lecture Hall
9:00 - 9:45 AM EDT
Network Analysis of High Dimensional Data
11th Floor Lecture Hall
Speaker
Allen Tannenbaum, Stony Brook University
Session Chair
James Murphy, Tufts University
Abstract
A major problem in data science is representation of data so that the variables driving key functions can be uncovered and explored. Correlation analysis is widely used to simplify networks of feature variables by reducing redundancies, but makes limited use of the network topology, relying on comparison of direct neighbor variables. The proposed method incorporates relational or functional profiles of neighboring variables along multiple common neighbors, which are fitted with Gaussian mixture models and compared using a data metric based on a version of optimal mass transport tailored to Gaussian mixtures. Hierarchical interactive visualization of the result leads to effective unbiased hypothesis generation. We will discuss several applications to medical imaging and cancer networks.
10:00 - 10:30 AM EDT
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM EDT
Signed Cumulative Distribution Transform for Machine Learning
11th Floor Lecture Hall
Speaker
Sumati Thareja, Vanderbilt University
Session Chair
James Murphy, Tufts University
Abstract
Classification and estimation problems are at the core of machine learning. In this talk we will see a new mathematical signal transform that renders data easy to classify or estimate, based on a very old theory of transportation that was started by Monge. We will learn about the existing Cumulative Distribution Transform and then extend to a more general measure theoretic framework, to define the new transform (Signed Cumulative Distribution Transform). We will look at both forward (analysis) and inverse (synthesis) formulas for the transform, and describe several of its properties including translation, scaling, convexity and isometry. Finally, we will demonstrate two applications of the transform in classifying (detecting) signals under random displacements and estimation of signal parameters under such displacements.
11:30 AM - 12:15 PM EDT
Towards a Mathematical Theory of Development
11th Floor Lecture Hall
Virtual Speaker
Geoffrey Schiebinger, University of British Columbia
Session Chair
James Murphy, Tufts University
Abstract
This talk introduces a mathematical theory of developmental biology, based on optimal transport. While, in principle, organisms are made of molecules whose motions are described by the Schödinger equation, there are simply too many molecules for this to be useful. Optimal transport—a fascinating topic in its own right, at the intersection of probability, statistics and optimization—provides a set of equations that describe development at the level of cells. Biology has entered a new era of precision measurement and massive datasets. Techniques like single-cell RNA sequencing (scRNA-seq) and single-cell ATAC-seq have emerged as powerful tools to profile cell states at unprecedented molecular resolution. One of the most exciting prospects associated with this new trove of data is the possibility of studying temporal processes, such as differentiation and development. If we could understand the genetic forces that control embryonic development, then we would have a better idea of how cell types are stabilized throughout adult life and how they destabilize with age or in diseases like cancer. This would be within reach if we could analyze the dynamic changes in gene expression, as populations develop and subpopulations differentiate. However, this is not directly possible with current measurement technologies because they are destructive (e.g. cells must be lysed to measure expression profiles). Therefore, we cannot directly observe the waves of transcriptional patterns that dictate changes in cell type.
This talk introduces a rigorous framework for understanding the developmental trajectories of cells in a dynamically changing, heterogeneous population based on static snapshots along a time-course. The framework is based on a simple hypothesis: over short time-scales cells can only change their expression profile by small amounts. We formulate this in precise mathematical terms using a classical tool called optimal transport (OT), and we propose that this optimal transport hypothesis is a fundamental mathematical principle of developmental biology.
12:30 - 2:00 PM EDT
Lunch - Optimal transport: Junior and seniors
Working Lunch
2:00 - 2:45 PM EDT
Multivariate Distribution-free testing using Optimal Transport
11th Floor Lecture Hall
Speaker
Bodhisattva Sen, Columbia University
Session Chair
James Murphy, Tufts University
Abstract
We propose a general framework for distribution-free nonparametric testing in multi-dimensions, based on a notion of multivariate ranks defined using the theory of optimal transport (see e.g., Villani (2003)). We demonstrate the applicability of this approach by constructing exactly distribution-free tests for two classical nonparametric problems: (i) testing for the equality of two multivariate distributions, and (ii) testing for mutual independence between two random vectors. In particular, we propose (multivariate) rank versions of Hotelling T^2 and kernel two-sample tests (e.g., Gretton et al. (2012), Szekely and Rizzo (2013)), and kernel tests for independence (e.g., Gretton et al. (2007), Szekely et al. (2007)) for scenarios (i) and (ii) respectively. We investigate the consistency and asymptotic distributions of these tests, both under the null and local contiguous alternatives. We also study the local power and asymptotic (Pitman) efficiency of these multivariate tests (based on optimal transport), and show that a subclass of these tests achieve attractive efficiency lower bounds that mimic the remarkable efficiency results of Hodges and Lehmann (1956) and Chernoff and Savage (1958) (for the Wilcoxon-rank sum test). To the best of our knowledge, these are the first collection of multivariate, nonparametric, exactly distribution-free tests that provably achieve such attractive efficiency lower bounds. We also study the rates of convergence of the rank maps (aka optimal transport maps).
3:00 - 3:30 PM EDT
Coffee Break
11th Floor Collaborative Space
3:30 - 4:15 PM EDT
Optimal transport for estimating generalization properties of machine learning models
11th Floor Lecture Hall
Speaker
Stefanie Jegelka, MIT
Session Chair
James Murphy, Tufts University
Abstract
One important challenge in practical machine learning is to estimate the generalization properties of a trained model, i.e., judging how well it will perform on unseen data. In this talk, we will discuss two examples of how optimal transport can help with this challenge.
First, we address the problem of estimating generalization of deep neural networks. A critical factor is the geometry of the data in the latent embedding space. We analyze this data arrangement via an optimal-transport-based generalization of variance, and show its theoretical and empirical relevance via generalization bounds that are also empirically predictive.
Second, we study the stability of neural networks for graph inputs, i.e., graph neural networks (GNNs), under shifts of the data distribution. In particular, to derive stability bounds, we need a suitable metric in the input space of graphs. We derive such a (pseudo)metric targeted to GNNs via a recursive optimal transport based distance between sets of trees. Our metric correlates better than state of the art with the behavior of GNNs under data distribution shifts.
4:30 - 6:30 PM EDT
Reception
11th Floor Collaborative Space
Tuesday, May 09, 2023
9:00 - 9:45 AM EDT
On the Convergence Rate of Sinkhorn’s Algorithm
11th Floor Lecture Hall
Speaker
Marcel Nutz, Columbia University
Session Chair
Shuchin Aeron, Tufts University
Abstract
We study Sinkhorn's algorithm for solving the entropically regularized optimal transport problem. Its iterate π_t is shown to satisfy H(π_t|π∗)+H(π∗|π_t)=O(1/t) where H denotes relative entropy and π∗ the optimal coupling. This holds for a large class of cost functions and marginals, including quadratic cost with subgaussian marginals. We also obtain the rate O(1/t) for the dual suboptimality and O(1/t^2) for the marginal entropies. More precisely, we derive non-asymptotic bounds, and in contrast to previous results on linear convergence that are limited to bounded costs, our estimates do not deteriorate exponentially with the regularization parameter. We also obtain a stability result for π∗ as a function of the marginals, quantified in relative entropy.
10:00 - 10:30 AM EDT
Coffee Break
11th Floor Collaborative Space