Multi-Anchor LOT: from Local Transport Maps to Global Wasserstein Geometry

Linearized optimal transport (LOT) provides Euclidean representations of distributions whose coordinate distances approximate the Wasserstein distance. However, linearization at a single reference cannot generally preserve the nonflat geometry of Wasserstein space. On regular Wasserstein models, we quantify this limitation through a local distortion bound governed by curvature and squared reference distance. This analysis motivates Multi-Anchor Linearized Optimal Transport (MALOT) for recovering global Wasserstein geometry from local transport maps. Starting from a multi-anchor Riemannian metric that gives greater weight to nearby references, we derive a symmetric two-point approximation with endpoint-dependent weights on squared LOT separations. Radial completion corrects each discrete coordinate block to preserve its reference radius. We prove that the resulting dissimilarities converge uniformly to the Wasserstein distance as the anchor covering radius and weight-localization error vanish, and establish coverage guarantees for farthest-point sampling. Simulations demonstrate improved recovery over LOT and decreasing error with finer anchor coverage; real-data studies evaluate distance recovery and downstream prediction.