  The Mixup method (Zhang et al. 2018), which uses linearly interpolated data,
has emerged as an effective data augmentation tool to improve generalization
performance and the robustness to adversarial examples. The motivation is to
curtail undesirable oscillations by its implicit model constraint to behave
linearly at in-between observed data points and promote smoothness. In this
work, we formally investigate this premise, propose a way to explicitly impose
smoothness constraints, and extend it to incorporate with implicit model
constraints. First, we derive a new function class composed of
kernel-convoluted models (KCM) where the smoothness constraint is directly
imposed by locally averaging the original functions with a kernel function.
Second, we propose to incorporate the Mixup method into KCM to expand the
domains of smoothness. In both cases of KCM and the KCM adapted with the Mixup,
we provide risk analysis, respectively, under some conditions for kernels. We
show that the upper bound of the excess risk is not slower than that of the
original function class. The upper bound of the KCM with the Mixup remains
dominated by that of the KCM if the perturbation of the Mixup vanishes faster
than \(O(n^{-1/2})\) where \(n\) is a sample size. Using CIFAR-10 and CIFAR-100
datasets, our experiments demonstrate that the KCM with the Mixup outperforms
the Mixup method in terms of generalization and robustness to adversarial
examples.
