748 research outputs found
Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration
Computing optimal transport distances such as the earth mover's distance is a
fundamental problem in machine learning, statistics, and computer vision.
Despite the recent introduction of several algorithms with good empirical
performance, it is unknown whether general optimal transport distances can be
approximated in near-linear time. This paper demonstrates that this ambitious
goal is in fact achieved by Cuturi's Sinkhorn Distances. This result relies on
a new analysis of Sinkhorn iteration, which also directly suggests a new greedy
coordinate descent algorithm, Greenkhorn, with the same theoretical guarantees.
Numerical simulations illustrate that Greenkhorn significantly outperforms the
classical Sinkhorn algorithm in practice
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
Optimal Approximation Algorithms for Multi-agent Combinatorial Problems with Discounted Price Functions
Submodular functions are an important class of functions in combinatorial
optimization which satisfy the natural properties of decreasing marginal costs.
The study of these functions has led to strong structural properties with
applications in many areas. Recently, there has been significant interest in
extending the theory of algorithms for optimizing combinatorial problems (such
as network design problem of spanning tree) over submodular functions.
Unfortunately, the lower bounds under the general class of submodular functions
are known to be very high for many of the classical problems.
In this paper, we introduce and study an important subclass of submodular
functions, which we call discounted price functions. These functions are
succinctly representable and generalize linear cost functions. In this paper we
study the following fundamental combinatorial optimization problems: Edge
Cover, Spanning Tree, Perfect Matching and Shortest Path, and obtain tight
upper and lower bounds for these problems.
The main technical contribution of this paper is designing novel adaptive
greedy algorithms for the above problems. These algorithms greedily build the
solution whist rectifying mistakes made in the previous steps
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