560 research outputs found
Regularized Wasserstein Means for Aligning Distributional Data
We propose to align distributional data from the perspective of Wasserstein
means. We raise the problem of regularizing Wasserstein means and propose
several terms tailored to tackle different problems. Our formulation is based
on the variational transportation to distribute a sparse discrete measure into
the target domain. The resulting sparse representation well captures the
desired property of the domain while reducing the mapping cost. We demonstrate
the scalability and robustness of our method with examples in domain
adaptation, point set registration, and skeleton layout
Scaling Algorithms for Unbalanced Transport Problems
This article introduces a new class of fast algorithms to approximate
variational problems involving unbalanced optimal transport. While classical
optimal transport considers only normalized probability distributions, it is
important for many applications to be able to compute some sort of relaxed
transportation between arbitrary positive measures. A generic class of such
"unbalanced" optimal transport problems has been recently proposed by several
authors. In this paper, we show how to extend the, now classical, entropic
regularization scheme to these unbalanced problems. This gives rise to fast,
highly parallelizable algorithms that operate by performing only diagonal
scaling (i.e. pointwise multiplications) of the transportation couplings. They
are generalizations of the celebrated Sinkhorn algorithm. We show how these
methods can be used to solve unbalanced transport, unbalanced gradient flows,
and to compute unbalanced barycenters. We showcase applications to 2-D shape
modification, color transfer, and growth models
Optimizing Nondecomposable Data Dependent Regularizers via Lagrangian Reparameterization offers Significant Performance and Efficiency Gains
Data dependent regularization is known to benefit a wide variety of problems
in machine learning. Often, these regularizers cannot be easily decomposed into
a sum over a finite number of terms, e.g., a sum over individual example-wise
terms. The measure, Area under the ROC curve (AUCROC) and Precision
at a fixed recall (P@R) are some prominent examples that are used in many
applications. We find that for most medium to large sized datasets, scalability
issues severely limit our ability in leveraging the benefits of such
regularizers. Importantly, the key technical impediment despite some recent
progress is that, such objectives remain difficult to optimize via
backpropapagation procedures. While an efficient general-purpose strategy for
this problem still remains elusive, in this paper, we show that for many
data-dependent nondecomposable regularizers that are relevant in applications,
sizable gains in efficiency are possible with minimal code-level changes; in
other words, no specialized tools or numerical schemes are needed. Our
procedure involves a reparameterization followed by a partial dualization --
this leads to a formulation that has provably cheap projection operators. We
present a detailed analysis of runtime and convergence properties of our
algorithm. On the experimental side, we show that a direct use of our scheme
significantly improves the state of the art IOU measures reported for MSCOCO
Stuff segmentation dataset
Learning and inference with Wasserstein metrics
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Brain and Cognitive Sciences, 2018.Cataloged from PDF version of thesis.Includes bibliographical references (pages 131-143).This thesis develops new approaches for three problems in machine learning, using tools from the study of optimal transport (or Wasserstein) distances between probability distributions. Optimal transport distances capture an intuitive notion of similarity between distributions, by incorporating the underlying geometry of the domain of the distributions. Despite their intuitive appeal, optimal transport distances are often difficult to apply in practice, as computing them requires solving a costly optimization problem. In each setting studied here, we describe a numerical method that overcomes this computational bottleneck and enables scaling to real data. In the first part, we consider the problem of multi-output learning in the presence of a metric on the output domain. We develop a loss function that measures the Wasserstein distance between the prediction and ground truth, and describe an efficient learning algorithm based on entropic regularization of the optimal transport problem. We additionally propose a novel extension of the Wasserstein distance from probability measures to unnormalized measures, which is applicable in settings where the ground truth is not naturally expressed as a probability distribution. We show statistical learning bounds for both the Wasserstein loss and its unnormalized counterpart. The Wasserstein loss can encourage smoothness of the predictions with respect to a chosen metric on the output space. We demonstrate this property on a real-data image tagging problem, outperforming a baseline that doesn't use the metric. In the second part, we consider the probabilistic inference problem for diffusion processes. Such processes model a variety of stochastic phenomena and appear often in continuous-time state space models. Exact inference for diffusion processes is generally intractable. In this work, we describe a novel approximate inference method, which is based on a characterization of the diffusion as following a gradient flow in a space of probability densities endowed with a Wasserstein metric. Existing methods for computing this Wasserstein gradient flow rely on discretizing the underlying domain of the diffusion, prohibiting their application to problems in more than several dimensions. In the current work, we propose a novel algorithm for computing a Wasserstein gradient flow that operates directly in a space of continuous functions, free of any underlying mesh. We apply our approximate gradient flow to the problem of filtering a diffusion, showing superior performance where standard filters struggle. Finally, we study the ecological inference problem, which is that of reasoning from aggregate measurements of a population to inferences about the individual behaviors of its members. This problem arises often when dealing with data from economics and political sciences, such as when attempting to infer the demographic breakdown of votes for each political party, given only the aggregate demographic and vote counts separately. Ecological inference is generally ill-posed, and requires prior information to distinguish a unique solution. We propose a novel, general framework for ecological inference that allows for a variety of priors and enables efficient computation of the most probable solution. Unlike previous methods, which rely on Monte Carlo estimates of the posterior, our inference procedure uses an efficient fixed point iteration that is linearly convergent. Given suitable prior information, our method can achieve more accurate inferences than existing methods. We additionally explore a sampling algorithm for estimating credible regions.by Charles Frogner.Ph. D
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