We present a greedy-based approach to construct an efficient single hidden layer neural network with the ReLU activation that approximates a target function. In our approach we obtain a shallow network by utilizing a greedy algorithm with the prescribed dictionary provided by the available training data and a set of possible inner weights. To facilitate the greedy selection process we employ an integral representation of the network, based on the ridgelet transform, that significantly reduces the cardinality of the dictionary and hence promotes feasibility of the greedy selection. Our approach allows for the construction of efficient architectures which can be treated either as improved initializations to be used in place of random-based alternatives, or as fully-trained networks in certain cases, thus potentially nullifying the need for backpropagation training. Numerical experiments demonstrate the tenability of the proposed concept and its advantages compared to the conventional techniques for selecting architectures and initializations for neural networks.
@cite_0 the authors suggest to use the values of the ridgelet transform to derive a probability distribution from which the inner weights are randomly selected. Then the resulted network is used as an initialization for training the network. In their approach the number of points on the sphere where the ridgelet transform needs to be computed, to derive the distribution can be potentially larger than what we require. And we replace the random selection step with a greedy selection step which, even though being more expensive as a selection method, results in a better approximation with a much fewer number of nodes.
@cite_0
2673818281 2913010492 2757143157 2941057241
In this paper, we analyze the spatial information of deep features, and propose two complementary regressions for robust visual tracking. First, we propose a kernelized ridge regression model wherein the kernel value is defined as the weighted sum of similarity scores of all pairs of patches between two samples. We show that this model can be formulated as a neural network and thus can be efficiently solved. Second, we propose a fully convolutional neural network with spatially regularized kernels, through which the filter kernel corresponding to each output channel is forced to focus on a specific region of the target. Distance transform pooling is further exploited to determine the effectiveness of each output channel of the convolution layer. The outputs from the kernelized ridge regression model and the fully convolutional neural network are combined to obtain the ultimate response. Experimental results on two benchmark datasets validate the effectiveness of the proposed method. We consider learning two layer neural networks using stochastic gradient descent. The mean-field description of this learning dynamics approximates the evolution of the network weights by an evolution in the space of probability distributions in @math (where @math is the number of parameters associated to each neuron). This evolution can be defined through a partial differential equation or, equivalently, as the gradient flow in the Wasserstein space of probability distributions. Earlier work shows that (under some regularity assumptions), the mean field description is accurate as soon as the number of hidden units is much larger than the dimension @math . In this paper we establish stronger and more general approximation guarantees. First of all, we show that the number of hidden units only needs to be larger than a quantity dependent on the regularity properties of the data, and independent of the dimensions. Next, we generalize this analysis to the case of unbounded activation functions, which was not covered by earlier bounds. We extend our results to noisy stochastic gradient descent. Finally, we show that kernel ridge regression can be recovered as a special limit of the mean field analysis. Although deep Convolutional Neural Network (CNN) has shown better performance in various computer vision tasks, its application is restricted by a significant increase in storage and computation. Among CNN simplification techniques, parameter pruning is a promising approach which aims at reducing the number of weights of various layers without intensively reducing the original accuracy. In this paper, we propose a novel progressive parameter pruning method, named Structured Probabilistic Pruning (SPP), which effectively prunes weights of convolutional layers in a probabilistic manner. Specifically, unlike existing deterministic pruning approaches, where unimportant weights are permanently eliminated, SPP introduces a pruning probability for each weight, and pruning is guided by sampling from the pruning probabilities. A mechanism is designed to increase and decrease pruning probabilities based on importance criteria for the training process. Experiments show that, with 4x speedup, SPP can accelerate AlexNet with only 0.3 loss of top-5 accuracy and VGG-16 with 0.8 loss of top-5 accuracy in ImageNet classification. Moreover, SPP can be directly applied to accelerate multi-branch CNN networks, such as ResNet, without specific adaptations. Our 2x speedup ResNet-50 only suffers 0.8 loss of top-5 accuracy on ImageNet. We further prove the effectiveness of our method on transfer learning task on Flower-102 dataset with AlexNet. We consider the problem of learning an unknown function @math on the @math -dimensional sphere with respect to the square loss, given i.i.d. samples @math where @math is a feature vector uniformly distributed on the sphere and @math . We study two popular classes of models that can be regarded as linearizations of two-layers neural networks around a random initialization: (RF) The random feature model of Rahimi-Recht; (NT) The neural tangent kernel model of Jacot-Gabriel-Hongler. Both these approaches can also be regarded as randomized approximations of kernel ridge regression (with respect to different kernels), and hence enjoy universal approximation properties when the number of neurons @math diverges, for a fixed dimension @math . We prove that, if both @math and @math are large, the behavior of these models is instead remarkably simpler. If @math , then RF performs no better than linear regression with respect to the raw features @math , and NT performs no better than linear regression with respect to degree-one and two monomials in the @math . More generally, if @math then RF fits at most a degree- @math polynomial in the raw features, and NT fits at most a degree- @math polynomial.