  Robust scatter estimation is a fundamental task in statistics. The recent
discovery on the connection between robust estimation and generative
adversarial nets (GANs) by Gao et al. (2018) suggests that it is possible to
compute depth-like robust estimators using similar techniques that optimize
GANs. In this paper, we introduce a general learning via classification
framework based on the notion of proper scoring rules. This framework allows us
to understand both matrix depth function and various GANs through the lens of
variational approximations of $f$-divergences induced by proper scoring rules.
We then propose a new class of robust scatter estimators in this framework by
carefully constructing discriminators with appropriate neural network
structures. These estimators are proved to achieve the minimax rate of scatter
estimation under Huber's contamination model. Our numerical results demonstrate
its good performance under various settings against competitors in the
literature.
