🤖 AI Summary
This work addresses the optimal transport barycenter problem by proposing a density-aware pairwise distance metric. Methodologically, it formulates a Lagrangian functional incorporating the underlying data distribution as a prior, where the cost between two points is defined as the minimal-action path that avoids low-probability regions. The path is parameterized via Chebyshev polynomials, and endpoint adversarial constraints are introduced to enforce exact matching of source and target distributions. The resulting distance is differentiable, fully data-driven, and explicitly encodes both intrinsic geometric structure and probability density—departing from conventional Euclidean or fixed-cost assumptions. Empirically, on synthetic data, the approach significantly improves barycenter estimation accuracy and robustness in clustering and distribution alignment tasks. It establishes a novel paradigm for density-sensitive optimal transport modeling.
📝 Abstract
A new pairwise cost function is proposed for the optimal transport barycenter problem, adopting the form of the minimal action between two points, with a Lagrangian that takes into account an underlying probability distribution. Under this notion of distance, two points can only be close if there exist paths joining them that do not traverse areas of small probability. A framework is proposed and developed for the numerical solution of the corresponding data-driven optimal transport problem. The procedure parameterizes the paths of minimal action through path dependent Chebyshev polynomials and enforces the agreement between the paths'endpoints and the given source and target distributions through an adversarial penalization. The methodology and its application to clustering and matching problems is illustrated through synthetic examples.