Exponential Integration of the Linear Assignment Flow

We introduce the linear assignment flow as an approximation of the full nonlinear assignment flow, which is a method for contextual data labeling on arbitrary graphs. The linear assignment flow is a dynamical system evolving on the tangent space of a statistical manifold. It is numerically determined using exponential integrators and Krylov subspace approximation, for which we provide error estimates. The approximation property of the linear assignment flow is illustrated by a numerical experiment. This work is supplemented by two papers on variational modeling and unsupervised labeling [1].

We introduce the linear assignment flow as an approximation of the full nonlinear assignment flow, which is a method for contextual data labeling on arbitrary graphs. The linear assignment flow is a dynamical system evolving on the tangent space of a statistical manifold. It is numerically determined using exponential integrators and Krylov subspace approximation, for which we provide error estimates. The approximation property of the linear assignment flow is illustrated by a numerical experiment. This work is supplemented by two papers on variational modeling and unsupervised labeling [1].

Motivation and Preliminaries
Recently, the paper [2] introduced the assignment flow which provides a novel approach to the image labeling problem, i.e. the task of finding a function (called labeling) which maps each pixel i from a set of pixels I to a label j from an applicationdependent prespecified set of labels J. For most applications in the field of image analysis and beyond, a good labeling is an adequate compromise between adhering to the image data and being spatially coherent.
The assignment flow is defined as a dynamical system evolving on the assignment manifold W = {W ∈ R |I|×|J| : W ij > 0, W 1 = 1}, which is the set of row-stochastic matrices with full support. Endowed with the Fisher-Rao (information) metric on the tangent space T 0 , it becomes a Riemannian manifold. The replicator operator R W and the similarity matrix S(W ), both introduced in [2], define the assignment flow aṡ After following the trajectory for some time, we round the current assignment to a vertex of the (closure of the) assignment manifold which corresponds to a labeling.

Linear Assignment Flow
By means of the exponential map corresponding to the geodesics of the affine e-connection from information geometry, the assignment flow can be represented by an ordinary differential equation (ODE) on the tangent space T 0 at the barycenter of W. Linearizing this ODE yields the linear assignment flow which was introduced in [3]: where A ∈ R |I||J|×|I||J| is a suitably chosen matrix that relates neighboring pixels in the image to each other. The vector a ∈ R |I||J| incorporates the image data.

Exponential Integration
Among various integration methods discussed in [3], we consider here exponential integration of the linear assignment flow. The solution of the autonomous linear ODE (2) can be written as (see [4]) where ϕ 0 denotes the matrix exponential. Even for small images the matrix A is too large to make the computation of ϕ 1 (tA) feasible. However, we only need the action of ϕ 1 (tA) on the vector a. This action can be approximated with Krylov subspaces For the m × m matrix H m we can compute ϕ 1 (tH m ) using the matrix exponential [6] of the (m + 1) × (m + 1) matrix [7] ϕ 0 tH m e 1 0 0 = ϕ 0 (tH m ) tϕ 1 (tH m )e 1 0 1 .
As the spectral radius of A satisfies ρ(A) ≤ 1 2 , we have the following error estimate [8] for the Krylov approximation (4) For common evaluation times t and vector sizes a this estimate ensures small errors already for Krylov orders m |I||J|. Thus the exponential integration with Krylov approximation produces a labeling which is close to the labeling returned by the linear assignment flow when using an alternative accurate (and more costly) numerical integration method [3]. In addition, an increase of the image size only has a small effect on a . Therefore, a constant Krylov order m can be used for a broad range of images sizes. Due to the final rounding from assignments to labelings, the error does not need to be equal or close to zero to produce a labeling that is close to the labeling returned by the full nonlinear assignment flow. In addition, the exponential integration of the linear assignment flow can be carried out approximately two orders of magnitude faster than the iterative integration of the nonlinear assignment flow. Possible future work includes the application of the linear assignment flow to the evaluation of graphical models [9] and to unsupervised labeling [10], and methods for representing and learning the spatial context of classes of images directly from data in terms of parameters that define the matrix A of the linear assignment flow.