Geodesic convexity of the variance functional in the discrete transport geometry on graphs

Determine whether the variance functional J[ν] = ∑_{i=1}^p (λ_i/2) W^2(ν, ν_i) is geodesically convex on the manifold of probability measures on a finite graph endowed with the discrete optimal transport metric W defined via the Benamou–Brenier-style formulation of Maas (2011). In particular, ascertain whether J is convex along W-geodesics for arbitrary finite graphs and fixed reference measures ν_i and weights λ ∈ Δ^{p−1}.

Background

The paper develops synthesis and analysis algorithms for barycenters of probability measures supported on graphs by leveraging the discrete dynamic optimal transport metric W on the probability simplex. In this setting, barycenters are minimizers of a variance functional J[ν] = ∑_{i=1}^p (λ_i/2) W^2(ν, ν_i). Existence of minimizers follows from convexity with respect to linear interpolation and compactness.

In the classical Euclidean Wasserstein-2 setting, it is known that the analogous variance functional is not geodesically convex, though first-order methods can still converge under additional conditions. Whether geodesic convexity holds in the discrete transport geometry on graphs is explicitly identified as unknown, and resolving this would clarify theoretical guarantees for intrinsic gradient descent on J in this setting.