We study the problem of unsupervised heteroscedastic covariance estimation,
where the goal is to learn the multivariate target distribution $\mathcal{N}(y,
\Sigma_y | x )$ given an observation $x$. This problem is particularly
challenging as $\Sigma_{y}$ varies for different samples (heteroscedastic) and
no annotation for the covariance is available (unsupervised). Typically,
state-of-the-art methods predict the mean $f_{\theta}(x)$ and covariance
$\textrm{Cov}(f_{\theta}(x))$ of the target distribution through two neural
networks trained using the negative log-likelihood. This raises two questions:
(1) Does the predicted covariance truly capture the randomness of the predicted
mean? (2) In the absence of ground-truth annotation, how can we quantify the
performance of covariance estimation? We address (1) by deriving TIC: Taylor
Induced Covariance, which captures the randomness of the multivariate
$f_{\theta}(x)$ by incorporating its gradient and curvature around $x$ through
the second order Taylor polynomial. Furthermore, we tackle (2) by introducing
TAC: Task Agnostic Correlations, a metric which leverages conditioning of the
normal distribution to evaluate the covariance. We verify the effectiveness of
TIC through multiple experiments spanning synthetic (univariate, multivariate)
and real-world datasets (UCI Regression, LSP, and MPII Human Pose Estimation).
Our experiments show that TIC outperforms state-of-the-art in accurately
learning the covariance, as quantified through TAC.

我们研究了无监督异方差协方差估计的问题，其中的目标是学习给定观察值 x 的多元目标分布 N (y, Σy|x)。我们通过推导 TIC：Taylor Induced Covariance 来解决预测协方差是否能真正捕捉到预测均值的随机性的问题，并通过引入 TAC：Task Agnostic Correlations 度量来解决在没有基准注释的情况下如何评估协方差估计性能的问题。我们的实验证明 TIC 在准确学习协方差方面优于现有技术，并通过 TAC 进行了量化。