Geometric numerical integration of the assignment flow

• Verfasst von: Zeilmann, Alexander [VerfasserIn]
• Verfasst von: Savarino, Fabrizio [VerfasserIn]
• Verfasst von: Petra, Stefania [VerfasserIn]
• Verfasst von: Schnörr, Christoph [VerfasserIn]
• Titel: Geometric numerical integration of the assignment flow
• Verf.angabe: Alexander Zeilmann, Fabrizio Savarino, Stefania Petra and Christoph Schnörr
• E-Jahr: 2020
• Jahr: 20 February 2020
• Umfang: ? S.
• Teil: volume:36
• Teil: year:2020
• Teil: number:3
• Teil: elocationid:034003
• Teil: extent:?
• Fussnoten: Gesehen am 25.03.2021
• Titel Quelle: Enthalten in: Inverse problems
• Ort Quelle: Bristol [u.a.] : Inst., 1985
• Jahr Quelle: 2020
• Band/Heft Quelle: 36(2020), 3, Artikel-ID 034003
• ISSN Quelle: 1361-6420
• DOI: doi:10.1088/1361-6420/ab2772
• Datenträger: Online-Ressource
• Sprache: eng
• K10plus-PPN: 1752408136

Abstract

The assignment flow is a smooth dynamical system that evolves on an elementary statistical manifold and performs contextual data labeling on a graph. We derive and introduce the linear assignment flow that evolves nonlinearly on the manifold, but is governed by a linear ODE on the tangent space. Various numerical schemes adapted to the mathematical structure of these two models are designed and studied, for the geometric numerical integration of both flows: embedded Runge-Kutta-Munthe-Kaas schemes for the nonlinear flow, adaptive Runge-Kutta schemes and exponential integrators for the linear flow. All algorithms are parameter free, except for setting a tolerance value that specifies adaptive step size selection by monitoring the local integration error, or fixing the dimension of the Krylov subspace approximation. These algorithms provide a basis for applying the assignment flow to machine learning scenarios beyond supervised labeling, including unsupervised labeling and learning from controlled assignment flows.