On the Geometric Mechanics of Assignment Flows for Metric Data Labeling

TL;DR

This paper explores the geometric mechanics underlying assignment flows used for metric data labeling, establishing their relation to critical points of an action functional and characterizing when they are geodesics in a Riemannian metric.

Contribution

It generalizes previous results on uncoupled replicator equations, relates assignment flows to Lagrangian mechanics, and characterizes conditions for assignment flows to be critical points.

Findings

Assignment flows are related to critical points of an action functional.

Not all assignment flows are critical points; conditions are characterized.

Assignment flows are reparametrized geodesics of the Jacobi metric, except for measure-zero initial conditions.

Abstract

Metric data labeling refers to the task of assigning one of multiple predefined labels to every given datapoint based on the metric distance between label and data. This assignment of labels typically takes place in a spatial or spatio-temporal context. Assignment flows are a class of dynamical models for metric data labeling that evolve on a basic statistical manifold, the so called assignment manifold, governed by a system of coupled replicator equations. In this paper we generalize the result of a recent paper for uncoupled replicator equations and adopting the viewpoint of geometric mechanics, relate assignment flows to critical points of an action functional via the associated Euler-Lagrange equation. We also show that not every assignment flow is a critical point and characterize precisely the class of coupled replicator equations fulfilling this relation, a condition that has been missing in recent related work. Finally, some consequences of this connection to Lagrangian mechanics are investigated including the fact that assignment flows are, up to initial conditions of measure zero, reparametrized geodesics of the so-called Jacobi metric.