Variational Inference Optimized Using the Curved Geometry of Coupled Free Energy

We introduce an optimization framework for variational inference based on the coupled free energy, extending variational inference techniques to account for the curved geometry of the coupled exponential family. This family includes important heavy-tailed distributions such as the generalized Pareto and the Student's t. By leveraging the coupled free energy, which is equal to the coupled evidence lower bound (ELBO) of the inverted probabilities, we improve the accuracy and robustness of the learned model. The coupled generalization of Fisher Information metric and the affine connection. The method is applied to the design of a coupled variational autoencoder (CVAE). By using the coupling for both the distributions and cost functions, the reconstruction metric is derived to still be the mean-square average loss with modified constants. The novelty comes from sampling the heavy-tailed latent distribution with its associated coupled probability, which has faster decaying tails. The result is the ability to train a model with high penalties in the tails, while assuring that the training samples have a reduced number of outliers. The Wasserstein-2 or Fr\'echet Inception Distance of the reconstructed CelebA images shows the CVAE has a 3\% improvement over the VAE after 5 epochs of training.