Dynamical Optimal Transport on Discrete Surfaces

We propose a technique for interpolating between probability distributions on discrete surfaces, based on the theory of optimal transport. Unlike previous attempts that use linear programming, our method is based on a dynamical formulation of quadratic optimal transport proposed for flat domains by Benamou and Brenier [2000], adapted to discrete surfaces. Our structure-preserving construction yields a Riemannian metric on the (finite-dimensional) space of probability distributions on a discrete surface, which translates the so-called Otto calculus to discrete language. From a practical perspective, our technique provides a smooth interpolation between distributions on discrete surfaces with less diffusion than state-of-the-art algorithms involving entropic regularization. Beyond interpolation, we show how our discrete notion of optimal transport extends to other tasks, such as distribution-valued Dirichlet problems and time integration of gradient flows

Stochastic Wasserstein Barycenters

We present a stochastic algorithm to compute the barycenter of a set of probability distributions under the Wasserstein metric from optimal transport. Unlike previous approaches, our method extends to continuous input distributions and allows the support of the barycenter to be adjusted in each iteration. We tackle the problem without regularization, allowing us to recover a sharp output whose support is contained within the support of the true barycenter. We give examples where our algorithm recovers a more meaningful barycenter than previous work. Our method is versatile and can be extended to applications such as generating super samples from a given distribution and recovering blue noise approximations.Comment: ICML 201

Dynamical optimal transport on discrete surfaces

We propose a technique for interpolating between probability distributions on discrete surfaces, based on the theory of optimal transport. Unlike previous attempts that use linear programming, our method is based on a dynamical formulation of quadratic optimal transport proposed for flat domains by Benamou and Brenier [2000], adapted to discrete surfaces. Our structure-preserving construction yields a Riemannian metric on the (finitedimensional) space of probability distributions on a discrete surface, which translates the so-called Otto calculus to discrete language. From a practical perspective, our technique provides a smooth interpolation between distributions on discrete surfaces with less diffusion than state-of-the-art algorithms involving entropic regularization. Beyond interpolation, we show how our discrete notion of optimal transport extends to other tasks, such as distribution-valued Dirichlet problems and time integration of gradient flows

Incorporating unlabeled data into distributionally-robust learning

We study a robust alternative to empirical risk minimization called distributionally robust learning (DRL), in which one learns to perform against an adversary who can choose the data distribution from a specified set of distributions. We illustrate a problem with current DRL formulations, which rely on an overly broad definition of allowed distributions for the adversary, leading to learned classifiers that are unable to predict with any confidence. We propose a solution that incorporates unlabeled data into the DRL problem to further constrain the adversary. We show that this new formulation is tractable for stochastic gradient-based optimization and yields a computable guarantee on the future performance of the learned classifier, analogous to—but tighter than—guarantees from conventional DRL. We examine the performance of this new formulation on 14 real data sets and find that it often yields effective classifiers with nontrivial performance guarantees in situations where conventional DRL produces neither. Inspired by these results, we extend our DRL formulation to active learning with a novel, distributionally-robust version of the standard model-change heuristic. Our active learning algorithm often achieves superior learning performance to the original heuristic on real data sets.Accepted manuscrip

Aggregation for modular robots in the pivoting cube model

Thesis: S.M., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2016.Cataloged from PDF version of thesis.Includes bibliographical references (pages 61-66).In this thesis, we present algorithms for self-aggregation and self-reconfiguration of modular robots in the pivoting cube model. First, we provide generic algorithms for aggregation of robots following integrator dynamics in arbitrary dimensional configuration spaces. We describe solutions to the problem under different assumptions on the capabilities of the robots, and the configuration space in which they travel. We also detail control strategies in cases where the robots are restricted to move on lower dimensional subspaces of the configuration space (such as being restricted to move on a 2D lattice). Second, we consider the problem of finding a distributed strategy for the aggregation of multiple modular robots into one connected structure. Our algorithm is designed for the pivoting cube model, a generalized model of motion for modular robots that has been effectively realized in hardware in the 3D M-Blocks. We use the intensity from a stimulus source as a input to a decentralized control algorithm that uses gradient information to drive the robots together. We give provable guarantees on convergence, and discuss experiments carried out in simulation and with a hardware platform of six 3D M-Blocks modules.by Sebastian Claici.S.M

Transportation tools for understanding data

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, May, 2020Cataloged from the official PDF of thesis.Includes bibliographical references (pages 169-187).The typical machine learning algorithms looks for a pattern in data, and makes an assumption that the signal to noise ratio of the pattern is high. This approach depends strongly on the quality of the datasets these algorithms operate on, and many complex algorithms fail in spectacular fashion on simple tasks by overfitting noise or outlier examples. These algorithms have training procedures that scale poorly in the size of the dataset, and their out-puts are difficult to intepret. This thesis proposes solutions to both problems by leveraging the theory of optimal transport and proposing efficient algorithms to solve problems in: (1) quantization, with extensions to the Wasserstein barycenter problem, and a link to the classical coreset problem; (2) natural language processing where the hierarchical structure of text allows us to compare documents efficiently;(3) Bayesian inference where we can impose a hierarchy on the label switching problem to resolve ambiguities.by Sebastian Claici.Ph. D.Ph.D. Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Scienc