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Register a new userApproximation Properties of Deep ReLU CNNs
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This paper is devoted to establishing $L^2$ approximation properties for deep ReLU convolutional neural networks (CNNs) on two-dimensional space. The analysis is based on a decomposition theorem for convolutional kernels with large spatial size and multi-channel. Given that decomposition and the property of the ReLU activation function, a universal approximation theorem of deep ReLU CNNs with classic structure is obtained by showing its connection with ReLU deep neural networks (DNNs) with one hidden layer. Furthermore, approximation properties are also obtained for neural networks with ResNet, pre-act ResNet, and MgNet architecture based on connections between these networks.
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This article is concerned with the approximation and expressive powers of deep neural networks. This is an active research area currently producing many interesting papers. The results most commonly found in the literature prove that neural networks approximate functions with classical smoothness to the same accuracy as classical linear methods of approximation, e.g. approximation by polynomials or by piecewise polynomials on prescribed partitions. However, approximation by neural networks depending on n parameters is a form of nonlinear approximation and as such should be compared with other nonlinear methods such as variable knot splines or n-term approximation from dictionaries. The performance of neural networks in targeted applications such as machine learning indicate that they actually possess even greater approximation power than these traditional methods of nonlinear approximation. The main results of this article prove that this is indeed the case. This is done by exhibiting large classes of functions which can be efficiently captured by neural networks where classical nonlinear methods fall short of the task. The present article purposefully limits itself to studying the approximation of univariate functions by ReLU networks. Many generalizations to functions of several variables and other activation functions can be envisioned. However, even in this simplest of settings considered here, a theory that completely quantifies the approximation power of neural networks is still lacking.
It has been widely assumed that a neural network cannot be recovered from its outputs, as the network depends on its parameters in a highly nonlinear way. Here, we prove that in fact it is often possible to identify the architecture, weights, and biases of an unknown deep ReLU network by observing only its output. Every ReLU network defines a piecewise linear function, where the boundaries between linear regions correspond to inputs for which some neuron in the network switches between inactive and active ReLU states. By dissecting the set of region boundaries into components associated with particular neurons, we show both theoretically and empirically that it is possible to recover the weights of neurons and their arrangement within the network, up to isomorphism.
Real world data often exhibit low-dimensional geometric structures, and can be viewed as samples near a low-dimensional manifold. This paper studies nonparametric regression of H{o}lder functions on low-dimensional manifolds using deep ReLU networks. Suppose $n$ training data are sampled from a H{o}lder function in $mathcal{H}^{s,alpha}$ supported on a $d$-dimensional Riemannian manifold isometrically embedded in $mathbb{R}^D$, with sub-gaussian noise. A deep ReLU network architecture is designed to estimate the underlying function from the training data. The mean squared error of the empirical estimator is proved to converge in the order of $n^{-frac{2(s+alpha)}{2(s+alpha) + d}}log^3 n$. This result shows that deep ReLU networks give rise to a fast convergence rate depending on the data intrinsic dimension $d$, which is usually much smaller than the ambient dimension $D$. It therefore demonstrates the adaptivity of deep ReLU networks to low-dimensional geometric structures of data, and partially explains the power of deep ReLU networks in tackling high-dimensional data with low-dimensional geometric structures.
We study the expressive power of deep ReLU neural networks for approximating functions in dilated shift-invariant spaces, which are widely used in signal processing, image processing, communications and so on. Approximation error bounds are estimated with respect to the width and depth of neural networks. The network construction is based on the bit extraction and data-fitting capacity of deep neural networks. As applications of our main results, the approximation rates of classical function spaces such as Sobolev spaces and Besov spaces are obtained. We also give lower bounds of the $L^p (1le p le infty)$ approximation error for Sobolev spaces, which show that our construction of neural network is asymptotically optimal up to a logarithmic factor.
Deep convolutional neural networks have achieved great success in various applications. However, training an effective DNN model for a specific task is rather challenging because it requires a prior knowledge or experience to design the network architecture, repeated trial-and-error process to tune the parameters, and a large set of labeled data to train the model. In this paper, we propose to overcome these challenges by actively adapting a pre-trained model to a new task with less labeled examples. Specifically, the pre-trained model is iteratively fine tuned based on the most useful examples. The examples are actively selected based on a novel criterion, which jointly estimates the potential contribution of an instance on optimizing the feature representation as well as improving the classification model for the target task. On one hand, the pre-trained model brings plentiful information from its original task, avoiding redesign of the network architecture or training from scratch; and on the other hand, the labeling cost can be significantly reduced by active label querying. Experiments on multiple datasets and different pre-trained models demonstrate that the proposed approach can achieve cost-effective training of DNNs.
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