Coupled Optimal Transport with Landmark Constraints
Abstract Overview
This paper addresses a limitation of standard optimal transport: minimizing transport cost alone may fail to identify a geometrically meaningful transformation between distributions. It proposes a coupled optimal transport framework that jointly optimizes a transport plan and a continuous deformation field, using a small set of annotated landmarks to guide the deformation while enforcing mutual consistency between the deformation and the transport plan. The authors establish the well-definedness and existence of solutions in a general variational setting, analyze limiting parameter regimes, and develop finite-element-based alternating numerical algorithms with theoretical convergence guarantees. Experiments on synthetic fish shapes and real handwritten digit matching demonstrate that the method recovers more spatially coherent deformations than transport-only barycentric projection or landmark-only fitting, especially under sparse landmark supervision.
Novelty
The primary novelty is a unified optimal transport formulation that explicitly couples cost-driven distribution matching with a landmark-guided deformation field through a mutual-consistency penalty, rather than treating transport and geometric registration as separate tasks. The work establishes a variational connection between landmark-based registration and optimal transport, supported by a finite-element computational realization and rigorous convergence theory.
Results
The authors prove the existence of minimizers under mild conditions, characterize the limiting behavior as the coupling weight approaches 0 and 1, and establish convergence guarantees for alternating discrete algorithms, including full-sequence convergence for an entropy-regularized variant. On synthetic benchmarks, the coupled approach achieves lower deformation, held-out landmark, and density alignment errors compared to OT barycentric projection and landmark-only baselines, with the largest gains observed when landmarks are sparse. Qualitative evaluations on MNIST digit pairs confirm that coupling transport with sparse landmarks produces spatially coherent deformation fields across the entire shape support.
Key Points
- The proposed framework jointly estimates a transport plan and a smooth deformation field, constrained by sparse landmark observations and a deformation-consistency penalty.
- Theoretical analysis establishes the existence of minimizers, variational interpretations under squared Euclidean cost, limiting behaviors, and convergence for discrete alternating optimization schemes.
- Empirical evaluations on synthetic shapes and MNIST digit matching show superior deformation recovery and spatial coherence compared to transport-only and landmark-only baselines.