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1. Norm-Based Generalization Bounds for Compositionally Sparse Neural Networks

   Galanti, Tomer; Xu, Mengjia; Galanti, Liane; Poggio, Tomaso (February 2023, Center for Brains, Minds and Machines (CBMM))

   In this paper, we investigate the Rademacher complexity of deep sparse neural networks, where each neuron receives a small number of inputs. We prove generalization bounds for multilayered sparse ReLU neural networks, including convolutional neural networks. These bounds differ from previous ones, as they consider the norms of the convolutional filters instead of the norms of the associated Toeplitz matrices, independently of weight sharing between neurons. As we show theoretically, these bounds may be orders of magnitude better than standard norm-based generalization bounds and empirically, they are almost non-vacuous in estimating generalization in various simple classification problems. Taken together, these results suggest that compositional sparsity of the underlying target function is critical to the success of deep neural networks. 

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2. Dynamics in Deep Classifiers Trained with the Square Loss: Normalization, Low Rank, Neural Collapse, and Generalization Bounds

   https://doi.org/10.34133/research.0024  

   Xu, Mengjia; Rangamani, Akshay; Liao, Qianli; Galanti, Tomer; Poggio, Tomaso (January 2023, Research)

   We overview several properties—old and new—of training overparameterized deep networks under the square loss. We first consider a model of the dynamics of gradient flow under the square loss in deep homogeneous rectified linear unit networks. We study the convergence to a solution with the absolute minimumρ, which is the product of the Frobenius norms of each layer weight matrix, when normalization by Lagrange multipliers is used together with weight decay under different forms of gradient descent. A main property of the minimizers that bound their expected error for a specific network architecture isρ. In particular, we derive novel norm-based bounds for convolutional layers that are orders of magnitude better than classical bounds for dense networks. Next, we prove that quasi-interpolating solutions obtained by stochastic gradient descent in the presence of weight decay have a bias toward low-rank weight matrices, which should improve generalization. The same analysis predicts the existence of an inherent stochastic gradient descent noise for deep networks. In both cases, we verify our predictions experimentally. We then predict neural collapse and its properties without any specific assumption—unlike other published proofs. Our analysis supports the idea that the advantage of deep networks relative to other classifiers is greater for problems that are appropriate for sparse deep architectures such as convolutional neural networks. The reason is that compositionally sparse target functions can be approximated well by “sparse” deep networks without incurring in the curse of dimensionality. 

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3. Deep Classifiers trained with the Square Loss

   Xu, Mengjia; Rangamani, Akshay; Banburski, Andrzej; Liao, Qianli; Galanti, Tomer; Poggio, Tomaso (October 2022, Center for Brains, Minds and Machines (CBMM))

   We overview several properties -- old and new -- of training overparametrized deep networks under the square loss. We first consider a model of the dynamics of gradient flow under the square loss in deep homogeneous ReLU networks. We study the convergence to a solution with the absolute minimum $$\rho$$, which is the product of the Frobenius norms of each layer weight matrix, when normalization by Lagrange multipliers (LM) is used together with Weight Decay (WD) under different forms of gradient descent. A main property of the minimizers that bounds their expected error {\it for a specific network architecture} is $$\rho$$. In particular, we derive novel norm-based bounds for convolutional layers that are orders of magnitude better than classical bounds for dense networks. Next we prove that quasi-interpolating solutions obtained by Stochastic Gradient Descent (SGD) in the presence of WD have a bias towards low rank weight matrices -- that, as we also explain, should improve generalization. The same analysis predicts the existence of an inherent SGD noise for deep networks. In both cases, we verify our predictions experimentally. We then predict Neural Collapse and its properties without any specific assumption -- unlike other published proofs. Our analysis supports the idea that the advantage of deep networks relative to other classifiers is greater for the problems that are appropriate for sparse deep architectures such as CNNs. The deep reason compositionally sparse target functions can be approximated well by ``sparse'' deep networks without incurring in the curse of dimensionality. 

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4. On Generalization Bounds for Neural Networks with Low Rank Layers

   Pinto, Andrea; Rangamani, Akshay; Poggio, Tomaso (December 2024, Center for Brains, Minds and Machines (CBMM))

   While previous optimization results have suggested that deep neural networks tend to favour low-rank weight matrices, the implications of this inductive bias on generalization bounds remain underexplored. In this paper, we apply a chain rule for Gaussian complexity (Maurer, 2016a) to analyze how low-rank layers in deep networks can prevent the accumulation of rank and dimensionality factors that typically multiply across layers. This approach yields generalization bounds for rank and spectral norm constrained networks. We compare our results to prior generalization bounds for deep networks, highlighting how deep networks with low-rank layers can achieve better generalization than those with full-rank layers. Additionally, we discuss how this framework provides new perspectives on the generalization capabilities of deep networks exhibiting neural collapse. Keywords: Gaussian complexity, Generalization bounds, Neural collapse, Low rank layers 

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5. Manufacturing Feature Recognition With a Sparse Voxel-Based Convolutional Neural Network

   https://doi.org/10.1115/1.4067334  

   Vatandoust, Farzad; Yan, Xiaoliang; Rosen, David; Melkote, Shreyes N (March 2025, Journal of Computing and Information Science in Engineering)

   Abstract Automated manufacturing feature recognition is a crucial link between computer-aided design and manufacturing, facilitating process selection and other downstream tasks in computer-aided process planning. While various methods such as graph-based, rule-based, and neural networks have been proposed for automatic feature recognition, they suffer from poor scalability or computational inefficiency. Recently, voxel-based convolutional neural networks have shown promise in solving these challenges but incur a tradeoff between computational cost and feature resolution. This paper investigates a computationally efficient sparse voxel-based convolutional neural network for manufacturing feature recognition, specifically, an octree-based sparse voxel convolutional neural network. This model is trained on a large-scale manufacturing feature dataset, and its performance is compared to a voxel-based feature recognition model (FeatureNet). The results indicate that the octree-based model yields higher feature recognition accuracy (99.5% on the test dataset) with 44% lower graphics processing unit (GPU) memory consumption than a voxel-based model of comparable resolution. In addition, increasing the resolution of the octree-based model enables recognition of finer manufacturing features. These results indicate that a sparse voxel-based convolutional neural network is a computationally efficient deep learning model for manufacturing feature recognition to enable process planning automation. Moreover, the sparse voxel-based neural network demonstrated comparable performance to a boundary representation-based feature recognition neural network, achieving similar accuracy in single-feature recognition without having access to the exact 3D shape descriptors. 

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