Continuous Regularized Wasserstein Barycenters
This work addresses a computational bottleneck for researchers and practitioners in machine learning and statistics who need to aggregate continuous distributions geometrically, representing an incremental improvement over prior finite-support methods.
The paper tackled the problem of computing Wasserstein barycenters for continuous probability distributions, which was previously limited to finite supports, by introducing a stochastic algorithm that constructs a continuous approximation using a new dual formulation and stochastic gradient descent, achieving efficient online computation.
Wasserstein barycenters provide a geometrically meaningful way to aggregate probability distributions, built on the theory of optimal transport. They are difficult to compute in practice, however, leading previous work to restrict their supports to finite sets of points. Leveraging a new dual formulation for the regularized Wasserstein barycenter problem, we introduce a stochastic algorithm that constructs a continuous approximation of the barycenter. We establish strong duality and use the corresponding primal-dual relationship to parametrize the barycenter implicitly using the dual potentials of regularized transport problems. The resulting problem can be solved with stochastic gradient descent, which yields an efficient online algorithm to approximate the barycenter of continuous distributions given sample access. We demonstrate the effectiveness of our approach and compare against previous work on synthetic examples and real-world applications.