Quadratically Regularized Optimal Transport on Graphs

Abstract

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 minimumcost 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.Transport over graphs has a rich history and has been considered in mathematics, operations research, computer science, and many other disciplines. At the broadest level, this problem is a slight generalization of the linear assignment problem and can be solved using several classical techniques; see e.g. [6,4] for discussion.Without regularization, the particular problem we study would be an instance of minimum-cost flow without edge capacities.[30] discusses classical algorithms for this linear programming problem, such as the cycle canceling [20], network simplex [28], and Ford-Fulkerson algorithms [15]. Classical methods such as these often are not accompanied with systematic "tie-breaking" strategies in case the network flow is non-unique, indicating the potential application of a strictly convex variation of the problem such as ours.The theoretical computer science community recently has reconsidered this class of problems from an optimization perspective. In the undirected, capacity-free case, [36] proposes a preconditioner for approximate minimum-cost flow problems that achieves nearly-linear runtimes.[10] considers the more general case of minimum-cost flow in directed graphs with unit capacities, extending a framework proposed in [26] for approaching graph-based problems using the interior point method. These algorithms are primarily of theoretical interest but do employ interior point-style methods, possibly implying a systematic choice of flows in the case of multiple optima.In the continuum, the theory of optimal transport [42] classifies problems structured similarly to minimum-cost flow as the 1-Wasserstein distance or Beckmann problem [1]. See [33] for analysis and [42,17,32] for theoretical discussion. Even over general spaces, solutions of the 1-Wasserstein problem generally are nonunique and include some degenerate optima [42, §2.4.6]. Numerical algorithms for these problems include [39], which uses finite element methods for a vector field ver...

An LP-based, strongly-polynomial 2-approximation algorithm for sparse Wasserstein barycenters

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Computation of optimal transport on discrete metric measure spaces

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Topological Metric Detects Hidden Order in Disordered Media

“…To tackle these problems, we introduce here a topological earth mover's (TEM) distance by combining ideas from statistical topology [13][14][15]18,19] and optimal transport theory [20,21] with nonequilibrium statistical mechanics [22]. The TEM metric compares two discrete material structures by quantifying the statistical differences in the local network topology of their Delaunay triangulations (Fig.…”

“…Armed with this intuition, we can now define the TEM distance between the Delaunay triangulations of two materials A and B in a natural manner as the earth mover's or, equivalently, Wasserstein distance [20] between their neighborhood distributions P A and P B over the flip graph: If P A ðiÞ is the probability of neighborhood i occurring in material A, and P B ðjÞ is the probability of neighborhood j occurring in material B, then a transport map, γ ij ≥ 0, from A to B satisfies P j γ ij ¼ P A ðiÞ, P where dði; jÞ is the distance between the neighborhoods i and j on the flip graph, and the minimum is taken over all possible transport maps γ ¼ ðγ ij Þ. We emphasize that, in contrast to widely used entropic distance measures between distributions [14], the definition of TEM uses the physically relevant information encoded in the metric structure dði; jÞ of the underlying observable space, which in our case reflects the typical energy cost of a T1 transition between network motifs.…”

“…After computation of the Delaunay tessellation, which can be done with high parallel efficiency [42], we utilize a modification of the Weinberg algorithm [23,26] to determine the flip-graph distances dði; jÞ of N observed neighborhood motifs in OðNÞ steps. Given dði; jÞ, the minimization over the transport maps fγ ij g can be recast as an minimum cost flow problem [20,23], which is efficiently solved with linear programming [43]. To demonstrate the broad applicability of our TEM framework, we focus in the remainder of this Letter on applications relevant to current major research areas: colloidal packings, collective far-from-equilibrium dynamics, tissue development, and spatiotemporally heterogeneous multicellular systems.…”