You are provided with the abstract of a scientific paper and the abstracts of the papers it references. Your task is to write a Related Work section for the main paper. This section should concisely summarize and integrate the key points from the cited abstracts, reflecting how they relate to the main paper's contributions and context. In your summary, use numerical citation markers like [1], [2], etc., to refer to the corresponding papers.

Main Abstract:
In this paper, we propose training very deep neural networks (DNNs) for supervised learning of hash codes. Existing methods in this context train relatively "shallow" networks limited by the issues arising in back propagation (e.e. vanishing gradients) as well as computational efficiency. We propose a novel and efficient training algorithm inspired by alternating direction method of multipliers (ADMM) that overcomes some of these limitations. Our method decomposes the training process into independent layer-wise local updates through auxiliary variables. Empirically we observe that our training algorithm always converges and its computational complexity is linearly proportional to the number of edges in the networks. Empirically we manage to train DNNs with 64 hidden layers and 1024 nodes per layer for supervised hashing in about 3 hours using a single GPU. Our proposed very deep supervised hashing (VDSH) method significantly outperforms the state-of-the-art on several benchmark datasets.

Cited Abstract(s):
[1]: In this work we investigate the effect of the convolutional network depth on its accuracy in the large-scale image recognition setting. Our main contribution is a thorough evaluation of networks of increasing depth using an architecture with very small (3x3) convolution filters, which shows that a significant improvement on the prior-art configurations can be achieved by pushing the depth to 16-19 weight layers. These findings were the basis of our ImageNet Challenge 2014 submission, where our team secured the first and the second places in the localisation and classification tracks respectively. We also show that our representations generalise well to other datasets, where they achieve state-of-the-art results. We have made our two best-performing ConvNet models publicly available to facilitate further research on the use of deep visual representations in computer vision.

[2]: We introduce DropConnect, a generalization of Dropout (, 2012), for regularizing large fully-connected layers within neural networks. When training with Dropout, a randomly selected subset of activations are set to zero within each layer. DropConnect instead sets a randomly selected subset of weights within the network to zero. Each unit thus receives input from a random subset of units in the previous layer. We derive a bound on the generalization performance of both Dropout and DropConnect. We then evaluate DropConnect on a range of datasets, comparing to Dropout, and show state-of-the-art results on several image recognition benchmarks by aggregating multiple DropConnect-trained models.

[3]: Training Deep Neural Networks is complicated by the fact that the distribution of each layer's inputs changes during training, as the parameters of the previous layers change. This slows down the training by requiring lower learning rates and careful parameter initialization, and makes it notoriously hard to train models with saturating nonlinearities. We refer to this phenomenon as internal covariate shift, and address the problem by normalizing layer inputs. Our method draws its strength from making normalization a part of the model architecture and performing the normalization for each training mini-batch. Batch Normalization allows us to use much higher learning rates and be less careful about initialization. It also acts as a regularizer, in some cases eliminating the need for Dropout. Applied to a state-of-the-art image classification model, Batch Normalization achieves the same accuracy with 14 times fewer training steps, and beats the original model by a significant margin. Using an ensemble of batch-normalized networks, we improve upon the best published result on ImageNet classification: reaching 4.9 top-5 validation error (and 4.8 test error), exceeding the accuracy of human raters.

[4]: 

[5]: We explore the redundancy of parameters in deep neural networks by replacing the conventional linear projection in fully-connected layers with the circulant projection. The circulant structure substantially reduces memory footprint and enables the use of the Fast Fourier Transform to speed up the computation. Considering a fully-connected neural network layer with d input nodes, and d output nodes, this method improves the time complexity from O(d2) to O(dlogd) and space complexity from O(d2) to O(d). The space savings are particularly important for modern deep convolutional neural network architectures, where fully-connected layers typically contain more than 90 of the network parameters. We further show that the gradient computation and optimization of the circulant projections can be performed very efficiently. Our experiments on three standard datasets show that the proposed approach achieves this significant gain in storage and efficiency with minimal increase in error rate compared to neural networks with unstructured projections.

[6]: Deep neural networks have achieved remarkable performance in both image classification and object detection problems, at the cost of a large number of parameters and computational complexity. In this work, we show how to reduce the redundancy in these parameters using a sparse decomposition. Maximum sparsity is obtained by exploiting both inter-channel and intra-channel redundancy, with a fine-tuning step that minimize the recognition loss caused by maximizing sparsity. This procedure zeros out more than 90 of parameters, with a drop of accuracy that is less than 1 on the ILSVRC2012 dataset. We also propose an efficient sparse matrix multiplication algorithm on CPU for Sparse Convolutional Neural Networks (SCNN) models. Our CPU implementation demonstrates much higher efficiency than the off-the-shelf sparse matrix libraries, with a significant speedup realized over the original dense network. In addition, we apply the SCNN model to the object detection problem, in conjunction with a cascade model and sparse fully connected layers, to achieve significant speedups.

[7]: We trained a large, deep convolutional neural network to classify the 1.2 million high-resolution images in the ImageNet LSVRC-2010 contest into the 1000 different classes. On the test data, we achieved top-1 and top-5 error rates of 37.5 and 17.0 , respectively, which is considerably better than the previous state-of-the-art. The neural network, which has 60 million parameters and 650,000 neurons, consists of five convolutional layers, some of which are followed by max-pooling layers, and three fully connected layers with a final 1000-way softmax. To make training faster, we used non-saturating neurons and a very efficient GPU implementation of the convolution operation. To reduce overfitting in the fully connected layers we employed a recently developed regularization method called "dropout" that proved to be very effective. We also entered a variant of this model in the ILSVRC-2012 competition and achieved a winning top-5 test error rate of 15.3 , compared to 26.2 achieved by the second-best entry.

[8]: Deep neural nets with a large number of parameters are very powerful machine learning systems. However, overfitting is a serious problem in such networks. Large networks are also slow to use, making it difficult to deal with overfitting by combining the predictions of many different large neural nets at test time. Dropout is a technique for addressing this problem. The key idea is to randomly drop units (along with their connections) from the neural network during training. This prevents units from co-adapting too much. During training, dropout samples from an exponential number of different "thinned" networks. At test time, it is easy to approximate the effect of averaging the predictions of all these thinned networks by simply using a single unthinned network that has smaller weights. This significantly reduces overfitting and gives major improvements over other regularization methods. We show that dropout improves the performance of neural networks on supervised learning tasks in vision, speech recognition, document classification and computational biology, obtaining state-of-the-art results on many benchmark data sets.

[9]: We propose a deep convolutional neural network architecture codenamed "Inception", which was responsible for setting the new state of the art for classification and detection in the ImageNet Large-Scale Visual Recognition Challenge 2014 (ILSVRC 2014). The main hallmark of this architecture is the improved utilization of the computing resources inside the network. This was achieved by a carefully crafted design that allows for increasing the depth and width of the network while keeping the computational budget constant. To optimize quality, the architectural decisions were based on the Hebbian principle and the intuition of multi-scale processing. One particular incarnation used in our submission for ILSVRC 2014 is called GoogLeNet, a 22 layers deep network, the quality of which is assessed in the context of classification and detection.

[10]: Recent work in unsupervised feature learning and deep learning has shown that being able to train large models can dramatically improve performance. In this paper, we consider the problem of training a deep network with billions of parameters using tens of thousands of CPU cores. We have developed a software framework called DistBelief that can utilize computing clusters with thousands of machines to train large models. Within this framework, we have developed two algorithms for large-scale distributed training: (i) Downpour SGD, an asynchronous stochastic gradient descent procedure supporting a large number of model replicas, and (ii) Sandblaster, a framework that supports a variety of distributed batch optimization procedures, including a distributed implementation of L-BFGS. Downpour SGD and Sandblaster L-BFGS both increase the scale and speed of deep network training. We have successfully used our system to train a deep network 30x larger than previously reported in the literature, and achieves state-of-the-art performance on ImageNet, a visual object recognition task with 16 million images and 21k categories. We show that these same techniques dramatically accelerate the training of a more modestly- sized deep network for a commercial speech recognition service. Although we focus on and report performance of these methods as applied to training large neural networks, the underlying algorithms are applicable to any gradient-based machine learning algorithm.

[11]: While Deep Neural Networks (DNNs) have achieved tremendous success for large vocabulary continuous speech recognition (LVCSR) tasks, training of these networks is slow. One reason is that DNNs are trained with a large number of training parameters (i.e., 10-50 million). Because networks are trained with a large number of output targets to achieve good performance, the majority of these parameters are in the final weight layer. In this paper, we propose a low-rank matrix factorization of the final weight layer. We apply this low-rank technique to DNNs for both acoustic modeling and language modeling. We show on three different LVCSR tasks ranging between 50-400 hrs, that a low-rank factorization reduces the number of parameters of the network by 30-50 . This results in roughly an equivalent reduction in training time, without a significant loss in final recognition accuracy, compared to a full-rank representation.
( 2 ) DNNs: In the literature, many different DNN architectures ( LeNet [4] , AlexNet [7] , GoogLeNet [9] and VGG-VD [1] ) and weighting structures ( sparse network [5] , circulant structure [6] , low-rank approximation [11] ) have been proposed. Several techniques have been proposed to improve the generalization of networks such as dropout [8] and dropconnet [2] , which can be viewed as better regularization. Some techniques for speeding-up the training have been proposed as well such as distributed training [10] and batch normalization [3] . These architectures and methods, however, are trained using backprop, suffering from the same issues such as vanishing gradients.