Research map / Learning theory and algorithms / Optimal transport
Energy-Guided Continuous Entropic Barycenter Estimation for General Costs
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Proposes an energy-guided neural solver for continuous entropic OT barycenters with arbitrary costs, using the weak-OT dual with a congruence parameterization and ULA-sampled energy-based training.
Problem
Computing continuous entropic optimal transport (EOT) barycenters with general (non-quadratic) cost functions, with out-of-sample conditional plans, where existing continuous solvers are mostly limited to the quadratic cost or need fixed priors.
Approach
Reformulates the EOT barycenter via the weak OT dual combined with a congruence constraint; parameterizes potentials as g_k minus a weighted average so the constraint holds automatically, and optimizes them like energy-based models with Langevin MCMC sampling of the conditional plans. Also learns barycenters on a StyleGAN image manifold via latent-space costs.
Key results
- On Ave, celeba!, the proposed solver reaches FID 8.4/8.7/10.2 for the three input plans versus 49-59 for SCWB and WIN.
- Theory: the suboptimality L* - L(f) equals epsilon times a weighted KL between EOT plans and recovered plans, and generalization error decomposes into an O(sum N_k^{-1/2}) estimation term plus an approximation term that vanishes for MLPs.
- On the 2D twister example with epsilon=1e-2, the learned EOT barycenter qualitatively matches the analytically known unregularized barycenter.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use Ours instead of WIN | Ave, celeba! k=1 barycenter plan · Ave, celeba! k=1 | FID: 49.3 -> 8.4 (.3) | Helped |
| use Ours instead of WIN | Ave, celeba! k=2 barycenter plan · Ave, celeba! k=2 | FID: 46.9 -> 8.7 (.3) | Helped |
| use Ours instead of SCWB | Ave, celeba! k=3 barycenter plan · Ave, celeba! k=3 | FID: 58.8 -> 10.2 (.7) | Helped |
Limitations
- MCMC (ULA) sampling is time-consuming and may converge poorly to the target conditional plans.
- Entropic barycenters in data space contain noisy images; manifold constraint is needed to remove noise.
- Theoretical analysis ignores optimization and MCMC errors.
Experimental setup
Datasets: MNIST CelebA Ave, celeba!
Benchmarks: Ave, celeba! MNIST 0/1 barycenter 2D twister toy sphere von Mises
Metrics: FID KL divergence bound (theory)
Baselines: SCWB WIN
Builds on: weak optimal transport (Gushchin et al.) energy-based models Schrödinger barycenter unadjusted Langevin algorithm
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