Geometric Numerical Integration of the Assignment Flow

TL;DR

This paper introduces geometric numerical integration methods for the assignment flow, a dynamical system used in data labeling on graphs, enabling more efficient and adaptable algorithms for machine learning applications.

Contribution

It develops and analyzes numerical schemes tailored for the nonlinear and linear assignment flows, facilitating their application in diverse machine learning tasks.

Findings

Embedded Runge-Kutta-Munthe-Kaas schemes effectively integrate nonlinear flows.

Adaptive Runge-Kutta and exponential integrators suit linear flows with parameter-free algorithms.

Algorithms support applications beyond supervised labeling, including unsupervised learning.

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.