  Relying on the premise that the performance of a binary neural network can be
largely restored with eliminated quantization error between full-precision
weight vectors and their corresponding binary vectors, existing works of
network binarization frequently adopt the idea of model robustness to reach the
aforementioned objective. However, robustness remains to be an ill-defined
concept without solid theoretical support. In this work, we introduce the
Lipschitz continuity, a well-defined functional property, as the rigorous
criteria to define the model robustness for BNN. We then propose to retain the
Lipschitz continuity as a regularization term to improve the model robustness.
Particularly, while the popular Lipschitz-involved regularization methods often
collapse in BNN due to its extreme sparsity, we design the Retention Matrices
to approximate spectral norms of the targeted weight matrices, which can be
deployed as the approximation for the Lipschitz constant of BNNs without the
exact Lipschitz constant computation (NP-hard). Our experiments prove that our
BNN-specific regularization method can effectively strengthen the robustness of
BNN (testified on ImageNet-C), achieving state-of-the-art performance on CIFAR
and ImageNet.
