  The pursuit algorithms integrated in multi-layer convolutional sparse coding
(ML-CSC) can interpret the convolutional neural networks (CNNs). However, many
current state-of-art (SOTA) pursuit algorithms require multiple iterations to
optimize the solution of ML-CSC, which limits their applications to deeper CNNs
due to high computational cost and large number of resources for getting very
tiny gain of performance. In this study, we focus on the 0th iteration in
pursuit algorithm by introducing an effective initialization strategy for each
layer, by which the solution for ML-CSC can be improved. Specifically, we first
propose a novel width-depth synchronous extension-based basis pursuit (WSEBP)
algorithm which solves the ML-CSC problem without the limitation of the number
of iterations compared to the SOTA algorithms and maximizes the performance by
an effective initialization in each layer. Then, we propose a simple and
unified ML-CSC-based classification network (ML-CSC-Net) which consists of an
ML-CSC-based feature encoder and a fully-connected layer to validate the
performance of WSEBP on image classification task. The experimental results
show that our proposed WSEBP outperforms SOTA algorithms in terms of accuracy
and consumption resources. In addition, the WSEBP integrated in CNNs can
improve the performance of deeper CNNs and make them interpretable. Finally,
taking VGG as an example, we propose WSEBP-VGG13 to enhance the performance of
VGG13, which achieves competitive results on four public datasets, i.e., 87.79%
vs. 86.83% on Cifar-10 dataset, 58.01% vs. 54.60% on Cifar-100 dataset, 91.52%
vs. 89.58% on COVID-19 dataset, and 99.88% vs. 99.78% on Crack dataset,
respectively. The results show the effectiveness of the proposed WSEBP, the
improved performance of ML-CSC with WSEBP, and interpretation of the CNNs or
deeper CNNs.
