Geometric Numerical Integration of the Assignment Flow
Provides efficient numerical methods for assignment flows, enabling broader machine learning applications beyond supervised labeling.
The paper introduces the linear assignment flow and develops geometric numerical integration schemes (embedded Runge-Kutta-Munthe-Kaas, adaptive Runge-Kutta, exponential integrators) that are parameter-free except for tolerance or Krylov dimension. These schemes enable application of assignment flows to unsupervised labeling and learning.
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.