Assignment flows for data labeling on graphs: convergence and stability

Abstract

The assignment flow recently introduced in the J. Math. Imaging and Vision 58/2 (2017) constitutes a high-dimensional dynamical system that evolves on a statistical product manifold and performs contextual labeling (classification) of data given in a metric space. Vertices of an underlying corresponding graph index the data points and define a system of neighborhoods. These neighborhoods together with nonnegative weight parameters define the regularization of the evolution of label assignments to data points, through geometric averaging induced by the affine e-connection of information geometry. From the point of view of evolutionary game dynamics, the assignment flow may be characterized as a large system of replicator equations that are coupled by geometric averaging. This paper establishes conditions on the weight parameters that guarantee convergence of the continuous-time assignment flow to integral assignments (labelings), up to a negligible subset of situations that will not be encountered when working with real data in practice. Furthermore, we classify attractors of the flow and quantify corresponding basins of attraction. This provides convergence guarantees for the assignment flow which are extended to the discrete-time assignment flow that results from applying a Runge–Kutta–Munthe–Kaas scheme for the numerical geometric integration of the assignment flow. Several counter-examples illustrate that violating the conditions may entail unfavorable behavior of the assignment flow regarding contextual data classification.

Assignment Flow for Order-Constrained OCT Segmentation

“…The solution W (t) ∈ W is numerically computed by geometric integration [ZSPS20] and determines a labeling W (T ) ∈ W * for sufficiently large T after a trivial rounding operation. Convergence and stability of the assignment flow have been studied by [ZZS20].…”

A Nonlocal Graph-PDE and Higher-Order Geometric Integration for Image Labeling

“…Integrating the system (2.35) numerically [ZSPS20] yields integral assignment vectors W (t, x), x ∈ V, for t → ∞, that uniquely assign a label from the set X * to each data point X(x) [ZZS21].…”

“…As argued in [ZZS21] by a range of counterexamples, using nonsymmetric parameter matrices Ω compromises convergence of the assignment flow (2.38a) to integral solutions (labelings) and is therefore not considered. The study of more general parameter matrices left for future work, see Section 8.…”

“…(5.2) with stepsize h = 1 γ < 1 due to γ > |λ min (Ω)|. Recent work [ZZS21] on the convergence of (2.38a) showed that, up to negligible situations that cannot occur when working with real data, limit points S * = lim t→∞ S(t) of (2.38a) are integral assignments S * ∈ W. Proposition 5.1 says that for small step sizes h < 1 the geometric integration step (5.1) yields a descent direction for moving S(t) ∈ W to S(t + h) ∈ W and therefore sufficiently approximates the integral curve corresponding (2.38a) at time t + h. We conclude that the fixed point determined by Algorithm 1 listed below solves the nonlocal PDE (3.8).…”

“…Stability and convergence to integral solutions of assignment flows hold under mild conditions [ZZS21]. A wide range of numerical schemes exist [ZSPS20] for integrating geometrically assignment flows with GPU-conforming operations.…”