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1. Structured Sparsity of Convolutional Neural Networks via Nonconvex Sparse Group Regularization

   https://doi.org/doi: 10.3389/fams.2020.529564  

   Bui, Kevin; Park, Fredrick; Zhang, Shuai; Qi, Yingyong; Xin, Jack (February 2021, Frontiers in applied mathematics and statistics)

   Tabacu, Lucia (Ed.)

   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 ℓ1, ℓ1 − ℓ2, and ℓ0) that induces sparsity onto the individual weights and ℓ2,1 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. 

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2. Convergence of a Relaxed Variable Splitting Coarse Gradient Descent Method for Learning Sparse Weight Binarized Activation Neural Network

   https://doi.org/DOI:10.3389/fams.2020.00013  

   Dinh, Thu; Xin, Jack (May 2020, Frontiers in applied mathematics and statistics)

   Sparsification of neural networks is one of the effective complexity reduction methods to improve efficiency and generalizability. Binarized activation offers an additional computational saving for inference. Due to vanishing gradient issue in training networks with binarized activation, coarse gradient (a.k.a. straight through estimator) is adopted in practice. In this paper, we study the problem of coarse gradient descent (CGD) learning of a one hidden layer convolutional neural network (CNN) with binarized activation function and sparse weights. It is known that when the input data is Gaussian distributed, no-overlap one hidden layer CNN with ReLU activation and general weight can be learned by GD in polynomial time at high probability in regression problems with ground truth. We propose a relaxed variable splitting method integrating thresholding and coarse gradient descent. The sparsity in network weight is realized through thresholding during the CGD training process. We prove that under thresholding of L1, L0, and transformed-L1 penalties, no-overlap binary activation CNN can be learned with high probability, and the iterative weights converge to a global limit which is a transformation of the true weight under a novel sparsifying operation. We found explicit error estimates of sparse weights from the true weights. 

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3. On sparse regression, L _p ‐regularization, and automated model discovery

   https://doi.org/10.1002/nme.7481  

   McCulloch, Jeremy A; St_Pierre, Skyler R; Linka, Kevin; Kuhl, Ellen (July 2024, International Journal for Numerical Methods in Engineering)

   Sparse regression and feature extraction are the cornerstones of knowledge discovery from massive data. Their goal is to discover interpretable and predictive models that provide simple relationships among scientific variables. While the statistical tools for model discovery are well established in the context of linear regression, their generalization to nonlinear regression in material modeling is highly problem‐specific and insufficiently understood. Here we explore the potential of neural networks for automatic model discovery and induce sparsity by a hybrid approach that combines two strategies: regularization and physical constraints. We integrate the concept of Lp regularization for subset selection with constitutive neural networks that leverage our domain knowledge in kinematics and thermodynamics. We train our networks with both, synthetic and real data, and perform several thousand discovery runs to infer common guidelines and trends: L2 regularization or ridge regression is unsuitable for model discovery; L1 regularization or lasso promotes sparsity, but induces strong bias that may aggressively change the results; only L0 regularization allows us to transparently fine‐tune the trade‐off between interpretability and predictability, simplicity and accuracy, and bias and variance. With these insights, we demonstrate that Lp regularized constitutive neural networks can simultaneously discover both, interpretable models and physically meaningful parameters. We anticipate that our findings will generalize to alternative discovery techniques such as sparse and symbolic regression, and to other domains such as biology, chemistry, or medicine. Our ability to automatically discover material models from data could have tremendous applications in generative material design and open new opportunities to manipulate matter, alter properties of existing materials, and discover new materials with user‐defined properties. 

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4. Global Minimizers of ℓp-Regularized Objectives Yield the Sparsest ReLU Neural Networks

   Nakhleh, J; Nowak, R (December 2025, The Thirty-Ninth Annual Conference on Neural Information Processing Systems (NeurIPS 2025))

   Overparameterized neural networks can interpolate a given dataset in many different ways, prompting the fundamental question: which among these solutions should we prefer, and what explicit regularization strategies will provably yield these solutions? This paper addresses the challenge of finding the sparsest interpolating ReLU network—i.e., the network with the fewest nonzero parameters or neurons—a goal with wide-ranging implications for efficiency, generalization, interpretability, theory, and model compression. Unlike post hoc pruning approaches, we propose a continuous, almost-everywhere differentiable training objective whose global minima are guaranteed to correspond to the sparsest single-hidden-layer ReLU networks that fit the data. This result marks a conceptual advance: it recasts the combinatorial problem of sparse interpolation as a smooth optimization task, potentially enabling the use of gradient-based training methods. Our objective is based on minimizing ℓp quasinorms of the weights for 0 < p < 1, a classical sparsity-promoting strategy in finite-dimensional settings. However, applying these ideas to neural networks presents new challenges: the function class is infinite-dimensional, and the weights are learned using a highly nonconvex objective. We prove that, under our formulation, global minimizers correspond exactly to sparsest solutions. Our work lays a foundation for understanding when and how continuous sparsity-inducing objectives can be leveraged to recover sparse networks through training. 

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5. PENNI: Pruned Kernel Sharing for Efficient CNN Inference

   LI, Shiyu; Hanson, Edward; Li, Hai Li; Chen, Yiran (July 2020, International Conference on Machine Learning)

   Although state-of-the-art (SOTA) CNNs achieve outstanding performance on various tasks, their high computation demand and massive number of parameters make it difficult to deploy these SOTA CNNs onto resource-constrained devices. Previous works on CNN acceleration utilize low-rank approximation of the original convolution layers to reduce computation cost. However, these methods are very difficult to conduct upon sparse models, which limits execution speedup since redundancies within the CNN model are not fully exploited. We argue that kernel granularity decomposition can be conducted with low-rank assumption while exploiting the redundancy within the remaining compact coefficients. Based on this observation, we propose PENNI, a CNN model compression framework that is able to achieve model compactness and hardware efficiency simultaneously by (1) implementing kernel sharing in convolution layers via a small number of basis kernels and (2) alternately adjusting bases and coefficients with sparse constraints. Experiments show that we can prune 97% parameters and 92% FLOPs on ResNet18 CIFAR10 with no accuracy loss, and achieve 44% reduction in run-time memory consumption and a 53% reduction in inference latency. 

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