Convolutional neural networks (CNN) have been hugely successful recently with superior accuracy and performance in various imaging applications, such as classification, object detection, and segmentation. However, a highly accurate CNN model requires millions of parameters to be trained and utilized. Even to increase its performance slightly would require significantly more parameters due to adding more layers and/or increasing the number of filters per layer. Apparently, many of these weight parameters turn out to be redundant and extraneous, so the original, dense model can be replaced by its compressed version attained by imposing inter- and intra-group sparsity onto the layer weights during training. In this paper, we propose a nonconvex family of sparse group lasso that blends nonconvex regularization (e.g., transformed ${\ell }_1$, ${\ell }_1-{\ell }_2$, and ${\ell }_0$) that induces sparsity onto the individual weights and ${\ell }_{\mathrm{2,1}}$ regularization onto the output channels of a layer. We apply variable splitting onto the proposed regularization to develop an algorithm that consists of two steps per iteration: gradient descent and thresholding. Numerical experiments are demonstrated on various CNN architectures showcasing the effectiveness of the nonconvex family of sparse group lasso in network sparsification and test accuracy on par with the current state of the art.

Dealing with massive data is a challenging task for machine learning. An important aspect of machine learning is function approximation. In the context of massive data, some of the commonly used tools for this purpose are sparsity, divide-and-conquer, and distributed learning. In this paper, we develop a very general theory of approximation by networks, which we have called eignets, to achieve local, stratified approximation. The very massive nature of the data allows us to use these eignets to solve inverse problems, such as finding a good approximation to the probability law that governs the data and finding the local smoothness of the target function near different points in the domain. In fact, we develop a wavelet-like representation using our eignets. Our theory is applicable to approximation on a general locally compact metric measure space. Special examples include approximation by periodic basis functions on the torus, zonal function networks on a Euclidean sphere (including smooth ReLU networks), Gaussian networks, and approximation on manifolds. We construct pre-fabricated networks so that no data-based training is required for the approximation.