Estimating Barycenters of Measures in High Dimensions
arXiv:2007.07105
Abstract
Barycentric averaging is a principled way of summarizing populations of measures. Existing algorithms for estimating barycenters typically parametrize them as weighted sums of Diracs and optimize their weights and/or locations. However, these approaches do not scale to high-dimensional settings due to the curse of dimensionality. In this paper, we propose a scalable and general algorithm for estimating barycenters of measures in high dimensions. The key idea is to turn the optimization over measures into an optimization over generative models, introducing inductive biases that allow the method to scale while still accurately estimating barycenters. We prove local convergence under mild assumptions on the discrepancy showing that the approach is well-posed. We demonstrate that our method is fast, achieves good performance on low-dimensional problems, and scales to high-dimensional settings. In particular, our approach is the first to be used to estimate barycenters in thousands of dimensions.
arXiv commentIn submission
References in corpus (14)
- Fast Computation of Wasserstein Barycenters
- MMD GAN: Towards Deeper Understanding of Moment Matching Network
- The Cramer Distance as a Solution to Biased Wasserstein Gradients
- Training generative neural networks via Maximum Mean Discrepancy optimization
- Sample Complexity of Sinkhorn divergences
- Differential Properties of Sinkhorn Approximation for Learning with Wasserstein Distance
- On the Convergence and Robustness of Training GANs with Regularized Optimal Transport
- On gradient regularizers for MMD GANs
- Stochastic Wasserstein Barycenters
- Stochastic subgradient method converges at the rate on weakly convex functions
- Entropic Optimal Transport between Unbalanced Gaussian Measures has a Closed Form
- Gradient descent algorithms for Bures-Wasserstein barycenters
- Wasserstein barycenters are NP-hard to compute
- Wasserstein Barycenter Model Ensembling