Towards Self-Supervised Covariance Estimation in Deep Heteroscedastic Regression

The challenge arises from heteroscedasticity, which implies that the covariance is sample dependent and is often unknown. Consequently, recent methods learn the covariance through unsupervised frameworks, which unfortunately yield a trade-off between computational complexity and accuracy. While this trade-off could be alleviated through supervision, obtaining labels for the covariance is non-trivial. Here, we study self-supervised covariance estimation in deep heteroscedastic regression. We address two questions: (1) How should we supervise the covariance assuming ground truth is available? We address (1) by analysing two popular measures: the KL Divergence and the 2-Wasserstein distance. Subsequently, we derive an upper bound on the 2-Wasserstein distance between normal distributions with non-commutative covariances that is stable to optimize. We address (2) through a simple neighborhood based heuristic algorithm which results in surprisingly effective pseudo-labels for the covariance. Our experiments over a wide range of synthetic and real datasets demonstrate that the proposed 2-Wasserstein bound coupled with pseudo-label annotations results in a computationally cheaper yet accurate deep heteroscedastic regression. The target distribution is typically used for downstream tasks such as uncertainty estimation, correlation analysis, sampling, and in bayesian frameworks. The key challenge in deep heteroscedastic regression lies in estimating heteroscedasticity, which implies that the variance of the target is input dependent and variable. Moreover, unlike the mean, the covariance lacks direct supervision and needs to be inferred. The standard approach without the ground-truth covariance relies on optimizing the negative loglikelihood to jointly learn the mean and covariance (Dorta et al., 2018).