Paper

Geometric Numerical Integration of the Assignment Flow

Savarino, Fabrizio, Zeilmann, Alexander, Stefania Petra, Schnörr, Christoph

2018

Overview

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.

• Publisher: Cornell University Library, arXiv.org
• Subject: Algorithms / Integrators / Labeling / Machine learning / Manifolds (mathematics) / Mathematical models / Numerical integration / Runge-Kutta method