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Continuous-domain assignment flows

Part of: Game theory Inference from stochastic processes Miscellaneous topics - Partial differential equations Existence theories

Published online by Cambridge University Press:  01 September 2020

F. SAVARINO Open the ORCID record for F. SAVARINO [Opens in a new window]  and
C. SCHNÖRR
F. SAVARINO
Affiliation:
Image and Pattern Analysis Group, Heidelberg University, Heidelberg, Germany emails: [email protected]; [email protected], URL: https://ipa.math.uni-heidelberg.de
C. SCHNÖRR
Affiliation:
Image and Pattern Analysis Group, Heidelberg University, Heidelberg, Germany emails: [email protected]; [email protected], URL: https://ipa.math.uni-heidelberg.de
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Abstract

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.

Keywords

Image labellingimage segmentationinformation geometryreplicator equationevolutionary dynamicsassignment flow

MSC classification

Primary: 35R02: Partial differential equations on graphs and networks (ramified or polygonal spaces) 62M45: Neural nets and related approaches 68U10: Image processing
Secondary: 49J40: Variational methods including variational inequalities 91A22: Evolutionary games
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Special Issue 3: Connections between Deep learning and Partial Differential Equations , June 2021 , pp. 570 - 597
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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