Continuous-Domain Assignment Flows

TL;DR

This paper introduces a new parametrization of assignment flows for data labeling on graphs, revealing how information geometry guides regularization and decision enforcement, and develops a PDE-based algorithm for continuous-domain problems.

Contribution

It presents a novel parametrization that links information geometry with assignment flows and introduces a PDE-based algorithm for continuous-domain data labeling.

Findings

Assignment flows can be characterized as Riemannian gradient flows.

A new algorithm solves linear elliptic PDEs with convex constraints.

The approach enables future learning problem formulations using PDE control.

Abstract

Assignment flows denote a class of dynamical models for contextual data labeling (classification) on graphs. We derive a novel parametrization of assignment flows that reveals how the underlying information geometry induces two processes for assignment regularization and for gradually enforcing unambiguous decisions, respectively, that seamlessly interact when solving for the flow. Our result enables to characterize the dominant part of the assignment flow as a Riemannian gradient flow with respect to the underlying information geometry. We consider a continuous-domain formulation of the corresponding potential and develop a novel algorithm in terms of solving a sequence of linear elliptic PDEs subject to a simple convex constraint. Our result provides a basis for addressing learning problems by controlling such PDEs in future work.