The rapid advances in Vision Transformer (ViT) refresh the state-of-the-art performances in various vision tasks, overshadowing the conventional CNN-based models. This ignites a few recent striking-back research in the CNN world showing that pure CNN models can achieve as good performance as ViT models when carefully tuned. While encouraging, designing such high-performance CNN models is challenging, requiring non-trivial prior knowledge of network design. To this end, a novel framework termed Mathematical Architecture Design for Deep CNN (DeepMAD) is proposed to design high-performance CNN models in a principled way. In DeepMAD, a CNN network is modeled as an information processing system whose expressiveness and effectiveness can be analytically formulated by their structural parameters. Then a constrained mathematical programming (MP) problem is proposed to optimize these structural parameters. The MP problem can be easily solved by off-the-shelf MP solvers on CPUs with a small memory footprint. In addition, DeepMAD is a pure mathematical framework: no GPU or training data is required during network design. The superiority of DeepMAD is validated on multiple large-scale computer vision benchmark datasets. Notably on ImageNet-1k, only using conventional convolutional layers, DeepMAD achieves 0.7% and 1.5% higher top-1 accuracy than ConvNeXt and Swin on Tiny level, and 0.8% and 0.9% higher on Small level.

We study anomaly detection for the case when the normal class consists of more than one object category. This is an obvious generalization of the standard one-class anomaly detection problem. However, we show that jointly using multiple one-class anomaly detectors to solve this problem yields poorer results as compared to training a single one-class anomaly detector on all normal object categories together. We further develop a new anomaly detector called DeepMAD that learns compact distinguishing features by exploiting the multiple normal objects categories. This algorithm achieves higher AUC values for different datasets compared to two top performing one-class algorithms that either are trained on each normal object category or jointly trained on all normal object categories combined. In addition to theoretical results we present empirical results using the CIFAR-10, fMNIST, CIFAR-100, and a new dataset we developed called RECYCLE.