Abstract
Assignment flows are a general class of dynamical models for context dependent data classification on graphs. These flows evolve on the product manifold of probability simplices, called assignment manifold, and are governed by a system of coupled replicator equations. In this paper, we adopt the general viewpoint of Lagrangian mechanics on manifolds and show that assignment flows satisfy the Euler-Lagrange equations associated with an action functional. Besides providing a novel interpretation of assignment flows, our result rectifies the analogous statement of a recent paper devoted to uncoupled replicator equations evolving on a single simplex, and generalizes it to coupled replicator equations and assignment flows.
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Acknowledgments
This work is supported by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC-2181/1 - 390900948 (the Heidelberg STRUCTURES Cluster of Excellence). PA was also supported by the Transregional Collaborative Research Center CRC/TRR 191 (281071066).
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Savarino, F., Albers, P., Schnörr, C. (2021). On the Geometric Mechanics of Assignment Flows for Metric Data Labeling. In: Elmoataz, A., Fadili, J., Quéau, Y., Rabin, J., Simon, L. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2021. Lecture Notes in Computer Science(), vol 12679. Springer, Cham. https://doi.org/10.1007/978-3-030-75549-2_32
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DOI: https://doi.org/10.1007/978-3-030-75549-2_32
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