569 research outputs found
Quadratically-Regularized Optimal Transport on Graphs
Optimal transportation provides a means of lifting distances between points
on a geometric domain to distances between signals over the domain, expressed
as probability distributions. On a graph, transportation problems can be used
to express challenging tasks involving matching supply to demand with minimal
shipment expense; in discrete language, these become minimum-cost network flow
problems. Regularization typically is needed to ensure uniqueness for the
linear ground distance case and to improve optimization convergence;
state-of-the-art techniques employ entropic regularization on the
transportation matrix. In this paper, we explore a quadratic alternative to
entropic regularization for transport over a graph. We theoretically analyze
the behavior of quadratically-regularized graph transport, characterizing how
regularization affects the structure of flows in the regime of small but
nonzero regularization. We further exploit elegant second-order structure in
the dual of this problem to derive an easily-implemented Newton-type
optimization algorithm.Comment: 27 page
Gerrymandering and Compactness: Implementation Flexibility and Abuse
The shape of an electoral district may suggest whether it was drawn with
political motivations, or gerrymandered. For this reason, quantifying the shape
of districts, in particular their compactness, is a key task in politics and
civil rights. A growing body of literature suggests and analyzes compactness
measures mathematically, but little consideration has been given to how these
scores should be calculated in practice. Here, we consider the effects of a
number of decisions that must be made in interpreting and implementing a set of
popular compactness scores. We show that the choices made in quantifying
compactness may themselves become political tools, with seemingly innocuous
decisions leading to disparate scores. We show that when the full range of
implementation flexibility is used, it can be abused to make clearly
gerrymandered districts appear quantitatively reasonable. This complicates
using compactness as a legislative or judicial standard to counteract unfair
redistricting practices. This paper accompanies the release of packages in C++,
Python, and R which correctly, efficiently, and reproducibly calculate a
variety of compactness scores.Comment: 10 pages, 17 figures, 1 tabl
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Computational Optimal Transport
Optimal transport is the mathematical discipline of matching supply to demand while minimizing shipping costs. This matching problem becomes extremely challenging as the quantity of supply and demand points increases; modern applications must cope with thousands or millions of these at a time. Here, we introduce the computational optimal transport problem and summarize recent ideas for achieving new heights in efficiency and scalability
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
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