Static and Dynamic Approaches to Computing Barycenters of Probability Measures on Graphs

Abstract

The optimal transportation problem defines a geometry of probability measures which leads to a definition for weighted averages (barycenters) of measures, finding application in the machine learning and computer vision communities as a signal processing tool. Here, we implement a barycentric coding model for measures which are supported on a graph, a context in which the classical optimal transport geometry becomes degenerate, by leveraging a Riemannian structure on the simplex induced by a dynamic formulation of the optimal transport problem. We approximate the exponential mapping associated to the Riemannian structure, as well as its inverse, by utilizing past approaches which compute action minimizing curves in order to numerically approximate transport distances for measures supported on discrete spaces. Intrinsic gradient descent is then used to synthesize barycenters, wherein gradients of a variance functional are computed by approximating geodesic curves between the current iterate and the reference measures; iterates are then pushed forward via a discretization of the continuity equation. Analysis of measures with respect to given dictionary of references is performed by solving a quadratic program formed by computing geodesics between target and reference measures. We compare our novel approach to one based on entropic regularization of the static formulation of the optimal transport problem where the graph structure is encoded via graph distance functions, we present numerical experiments validating our approach, and we conclude that intrinsic gradient descent on the probability simplex provides a coherent framework for the synthesis and analysis of measures supported on graphs.

Figures (17)

• Figure 1: A schematic of the barycentric coding model for measures supported on a graph. At the extremal points of the simplex, which correspond to the vertices of the triangle, are the reference measures; away from the extremal points are barycentric interpolations of the reference measures for certain weights $\lambda$. The coordinates recovered by solving the analysis problem are displayed and labeled with $\hat{\lambda}$.
• Figure 2: Weighted center of mass on a non-Euclidean space; $\nu_\lambda$ is the synthesis of the points $\nu_1, \nu_2, \nu_3$ with weights $\lambda = (0.2,0.5,0.3)$
• Figure 3: To solve the analysis problem, we compute the tangent vectors associated to the unit time geodesic curves connecting the target measure to each reference measure, then form the associated Gram matrix.
• Figure 4: Comparison between a classic optimal transport plan and an entropically regularized one. Left: two samples of $100$ points each in the plane (normally distributed with different means and variances). Center: optimal assignment between points to minimize transport cost, obtained via computation of a linear program. Right: entropic transport plan for minimizing entropic transport cost. Notice the difference in sparsity between the two plans: entropic transport results in a diffusive effect.
• Figure 5: The Chambolle-Pock routine is used to approximate energy minimizing curves in $\mathcal{P}(\mathcal{X})$, ultimately producing a rectified curve encoded as a sequence of points and a collection of momentum vector fields. We leverage these vector fields to approximate the gradient of \ref{['GraphVF']}.

Theorems & Definitions (15)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1: brenierPolarFactorizationMonotone1991, Theorem 1.3
• Theorem 2: benamouComputationalFluidMechanics2000, Proposition 1.1
• Theorem 3
• Theorem 4: werenskiMeasureEstimationBarycentric2022, Proposition 1
• Theorem 5: erbarRicciCurvatureFinite2012, Proposition 2.11
• Corollary 1
• proof