Invariant Representations via Wasserstein Correlation Maximization

Abstract: This work investigates the use of Wasserstein correlation -- a normalized measure of statistical dependence based on the Wasserstein distance between a joint distribution and the product of its marginals -- for unsupervised representation learning. Unlike, for example, contrastive methods, which naturally cluster classes in the latent space, we find that an (auto)encoder trained to maximize Wasserstein correlation between the input and encoded distributions instead acts as a compressor, reducing dimensionality while approximately preserving the topological and geometric properties of the input distribution. More strikingly, we show that Wasserstein correlation maximization can be used to arrive at an (auto)encoder -- either trained from scratch, or else one that extends a frozen, pretrained model -- that is approximately invariant to a chosen augmentation, or collection of augmentations, and that still approximately preserves the structural properties of the non-augmented input distribution. To do this, we first define the notion of an augmented encoder using the machinery of Markov-Wasserstein kernels. When the maximization objective is then applied to the augmented encoder, as opposed to the underlying, deterministic encoder, the resulting model exhibits the desired invariance properties. Finally, besides our experimental results, which show that even simple feedforward networks can be imbued with invariants or can, alternatively, be used to impart invariants to pretrained models under this training process, we additionally establish various theoretical results for optimal transport-based dependence measures. Code is available at https://github.com/keenan-eikenberry/wasserstein_correlation_maximization .