
Equivalence of approximation by convolutional neural networks and fullyconnected networks
Convolutional neural networks are the most widely used type of neural ne...
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On the Number of Linear Regions of Convolutional Neural Networks
One fundamental problem in deep learning is understanding the outstandin...
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On the Learning Property of Logistic and Softmax Losses for Deep Neural Networks
Deep convolutional neural networks (CNNs) trained with logistic and soft...
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Asymptotic Risk of Overparameterized Likelihood Models: Double Descent Theory for Deep Neural Networks
We investigate the asymptotic risk of a general class of overparameteriz...
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ButterflyNet: Optimal Function Representation Based on Convolutional Neural Networks
Deep networks, especially Convolutional Neural Networks (CNNs), have bee...
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The information of attribute uncertainties: what convolutional neural networks can learn about errors in input data
Errors in measurements are key to weighting the value of data, but are o...
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Approximation and Nonparametric Estimation of ResNettype Convolutional Neural Networks
Convolutional neural networks (CNNs) have been shown to achieve optimal ...
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Nonasymptotic Excess Risk Bounds for Classification with Deep Convolutional Neural Networks
In this paper, we consider the problem of binary classification with a class of general deep convolutional neural networks, which includes fullyconnected neural networks and fully convolutional neural networks as special cases. We establish nonasymptotic excess risk bounds for a class of convex surrogate losses and target functions with different modulus of continuity. An important feature of our results is that we clearly define the prefactors of the risk bounds in terms of the input data dimension and other model parameters and show that they depend polynomially on the dimensionality in some important models. We also show that the classification methods with CNNs can circumvent the curse of dimensionality if the input data is supported on an approximate lowdimensional manifold. To establish these results, we derive an upper bound for the covering number for the class of general convolutional neural networks with a bias term in each convolutional layer, and derive new results on the approximation power of CNNs for any uniformlycontinuous target functions. These results provide further insights into the complexity and the approximation power of general convolutional neural networks, which are of independent interest and may have other applications. Finally, we apply our general results to analyze the nonasymptotic excess risk bounds for four widely used methods with different loss functions using CNNs, including the least squares, the logistic, the exponential and the SVM hinge losses.
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