Continuous-domain assignment flows

• Published in: European journal of applied mathematics Vol. 32; no. 3; pp. 570 - 597
• Main Authors: SAVARINO, F., SCHNÖRR, C.
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.06.2021
• Subjects: Algorithms; Applied mathematics; Domains; Dynamic models; Elliptic functions; Gradient flow; Parameterization; Partial differential equations; Regularization; 68U10; 62M45; 91A22; information geometry; image segmentation; 49J40; evolutionary dynamics; replicator equation; 35R02; Image labelling; assignment flow
• ISSN: 0956-7925; 1469-4425
• DOI: 10.1017/S0956792520000273

Assignment flows denote a class of dynamical models for contextual data labelling (classification) on graphs. We derive a novel parametrisation of assignment flows that reveals how the underlying information geometry induces two processes for assignment regularisation and for gradually enforcing unambiguous decisions, respectively, that seamlessly interact when solving for the flow. Our result enables to characterise 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 partial differential equations (PDEs) subject to a simple convex constraint. Our result provides a basis for addressing learning problems by controlling such PDEs in future work.