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's leaf nodes to form Given the definition of our attention in Eq. 9 in the main text, the highest polynomial order is Before providing the proof of Theorem 4, we establish Lemma 1 as its foundation. We follow the principle of Y an et al's work [ Figure 1, we consider two kinds of value functions, i.e., In P AM-Net, we set the number of levels to 2. A grid search is performed over different configurations We conduct grid searches on the dropout rate over {0, 0.1, 0.2} and the initial

Theorem 6 is stated in terms of Gaussian complexity. Ben-David (2014) has a full proof. M (α)null is the linear class following the depth-K neural network. The second term relies on the Lipschitz constant of DNN, which we bound with the following lemma. Similar results are given by Scaman and Virmaux (2018); Fazlyab et al. (2019).

The implicit bias towards solutions with favorable properties is believed to be a key reason why neural networks trained by gradient-based optimization can generalize well. While the implicit bias of gradient flow has been widely studied for homogeneous neural networks (including ReLU and leaky ReLU networks), the implicit bias of gradient descent is currently only understood for smooth neural networks. Therefore, implicit bias in non-smooth neural networks trained by gradient descent remains an open question. In this paper, we aim to answer this question by studying the implicit bias of gradient descent for training two-layer fully connected (leaky) ReLU neural networks. We showed that when the training data are nearly-orthogonal, for leaky ReLU activation function, gradient descent will find a network with a stable rank that converges to $1$, whereas for ReLU activation function, gradient descent will find a neural network with a stable rank that is upper bounded by a constant. Additionally, we show that gradient descent will find a neural network such that all the training data points have the same normalized margin asymptotically.

Neural network training is usually accomplished by solving a non-convex optimization problem using stochastic gradient descent. Although one optimizes over the networks parameters, the main loss function generally only depends on the realization of the neural network, i.e. the function it computes. Studying the optimization problem over the space of realizations opens up new ways to understand neural network training. In particular, usual loss functions like mean squared error and categorical cross entropy are convex on spaces of neural network realizations, which themselves are non-convex. Approximation capabilities of neural networks can be used to deal with the latter non-convexity, which allows us to establish that for sufficiently large networks local minima of a regularized optimization problem on the realization space are almost optimal.

Laser cutting is a widely adopted technology in material processing across various industries, but it generates a significant amount of dust, smoke, and aerosols during operation, posing a risk to both the environment and workers' health. Speckle sensing has emerged as a promising method to monitor the cutting process and identify material types in real-time. This paper proposes a material classification technique using a speckle pattern of the material's surface based on deep learning to monitor and control the laser cutting process. The proposed method involves training a convolutional neural network (CNN) on a dataset of laser speckle patterns to recognize distinct material types for safe and efficient cutting. Previous methods for material classification using speckle sensing may face issues when the color of the laser used to produce the speckle pattern is changed. Experiments conducted in this study demonstrate that the proposed method achieves high accuracy in material classification, even when the laser color is changed. The model achieved an accuracy of 98.30 % on the training set and 96.88% on the validation set. Furthermore, the model was evaluated on a set of 3000 new images for 30 different materials, achieving an F1-score of 0.9643. The proposed method provides a robust and accurate solution for material-aware laser cutting using speckle sensing.