In this paper, we present a novel and general network structure towards accelerating the inference process of convolutional neural networks, which is more complicated in network structure yet with less inference complexity. The core idea is to equip each original convolutional layer with another low-cost collaborative layer (LCCL), and the element-wise multiplication of the ReLU outputs of these two parallel layers produces the layer-wise output. The combined layer is potentially more discriminative than the original convolutional layer, and its inference is faster for two reasons: 1) the zero cells of the LCCL feature maps will remain zero after element-wise multiplication, and thus it is safe to skip the calculation of the corresponding high-cost convolution in the original convolutional layer, 2) LCCL is very fast if it is implemented as a 1*1 convolution or only a single filter shared by all channels. Extensive experiments on the CIFAR-10, CIFAR-100 and ILSCRC-2012 benchmarks show that our proposed network structure can accelerate the inference process by 32 on average with negligible performance drop.
Low Rank . Tensor decomposition with low-rank approximation-based methods are commonly used to accelerate deep convolutional networks. For example, in @cite_30 @cite_21 , the authors exploited the redundancy between convolutional filters and used low-rank approximation to compress convolutional weight tensors and fully connected weight matrices. Yang al @cite_16 use an adaptive fastfood transform was used to replace a fully connected layer with a series of simple matrix multiplications, rather than the original dense and large ones. Liu al @cite_26 propose a sparse decomposition to reduce the redundancy in convolutional parameters. @cite_34 @cite_8 , the authors used generalized singular vector decomposition (GSVD) to decompose an original layer to two approximated layers with reduced computation complexity.
@cite_30 @cite_26 @cite_8 @cite_21 @cite_16 @cite_34
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Deep compression refers to removing the redundancy of parameters and feature maps for deep learning models. Low-rank approximation and pruning for sparse structures play a vital role in many compression works. However, weight filters tend to be both low-rank and sparse. Neglecting either part of these structure information in previous methods results in iteratively retraining, compromising accuracy, and low compression rates. Here we propose a unified framework integrating the low-rank and sparse decomposition of weight matrices with the feature map reconstructions. Our model includes methods like pruning connections as special cases, and is optimized by a fast SVD-free algorithm. It has been theoretically proven that, with a small sample, due to its generalizability, our model can well reconstruct the feature maps on both training and test data, which results in less compromising accuracy prior to the subsequent retraining. With such a warm start to retrain, the compression method always possesses several merits: (a) higher compression rates, (b) little loss of accuracy, and (c) fewer rounds to compress deep models. The experimental results on several popular models such as AlexNet, VGG-16, and GoogLeNet show that our model can significantly reduce the parameters for both convolutional and fully-connected layers. As a result, our model reduces the size of VGG-16 by 15&#xd7;, better than other recent compression methods that use a single strategy. Convolutional neural networks (CNNs) have achieved remarkable performance in a wide range of computer vision tasks, typically at the cost of massive computational complexity. The low speed of these networks may hinder real-time applications especially when computational resources are limited. In this paper, an efficient and effective approach is proposed to accelerate the test-phase computation of CNNs based on low-rank and group sparse tensor decomposition. Specifically, for each convolutional layer, the kernel tensor is decomposed into the sum of a small number of low multilinear rank tensors. Then we replace the original kernel tensors in all layers with the approximate tensors and fine-tune the whole net with respect to the final classification task using standard backpropagation. Comprehensive experiments on ILSVRC-12 demonstrate significant reduction in computational complexity, at the cost of negligible loss in accuracy. For the widely used VGG-16 model, our approach obtains a 6.6 @math speed-up on PC and 5.91 @math speed-up on mobile device of the whole network with less than 1 increase on top-5 error. The success of CNNs in various applications is accompanied by a significant increase in the computation and parameter storage costs. Recent efforts toward reducing these overheads involve pruning and compressing the weights of various layers without hurting original accuracy. However, magnitude-based pruning of weights reduces a significant number of parameters from the fully connected layers and may not adequately reduce the computation costs in the convolutional layers due to irregular sparsity in the pruned networks. We present an acceleration method for CNNs, where we prune filters from CNNs that are identified as having a small effect on the output accuracy. By removing whole filters in the network together with their connecting feature maps, the computation costs are reduced significantly. In contrast to pruning weights, this approach does not result in sparse connectivity patterns. Hence, it does not need the support of sparse convolution libraries and can work with existing efficient BLAS libraries for dense matrix multiplications. We show that even simple filter pruning techniques can reduce inference costs for VGG-16 by up to 34 and ResNet-110 by up to 38 on CIFAR10 while regaining close to the original accuracy by retraining the networks. We propose a simple two-step approach for speeding up convolution layers within large convolutional neural networks based on tensor decomposition and discriminative fine-tuning. Given a layer, we use non-linear least squares to compute a low-rank CP-decomposition of the 4D convolution kernel tensor into a sum of a small number of rank-one tensors. At the second step, this decomposition is used to replace the original convolutional layer with a sequence of four convolutional layers with small kernels. After such replacement, the entire network is fine-tuned on the training data using standard backpropagation process. We evaluate this approach on two CNNs and show that it is competitive with previous approaches, leading to higher obtained CPU speedups at the cost of lower accuracy drops for the smaller of the two networks. Thus, for the 36-class character classification CNN, our approach obtains a 8.5x CPU speedup of the whole network with only minor accuracy drop (1 from 91 to 90 ). For the standard ImageNet architecture (AlexNet), the approach speeds up the second convolution layer by a factor of 4x at the cost of @math increase of the overall top-5 classification error. Abstract: We propose a simple two-step approach for speeding up convolution layers within large convolutional neural networks based on tensor decomposition and discriminative fine-tuning. Given a layer, we use non-linear least squares to compute a low-rank CP-decomposition of the 4D convolution kernel tensor into a sum of a small number of rank-one tensors. At the second step, this decomposition is used to replace the original convolutional layer with a sequence of four convolutional layers with small kernels. After such replacement, the entire network is fine-tuned on the training data using standard backpropagation process. We evaluate this approach on two CNNs and show that it is competitive with previous approaches, leading to higher obtained CPU speedups at the cost of lower accuracy drops for the smaller of the two networks. Thus, for the 36-class character classification CNN, our approach obtains a 8.5x CPU speedup of the whole network with only minor accuracy drop (1 from 91 to 90 ). For the standard ImageNet architecture (AlexNet), the approach speeds up the second convolution layer by a factor of 4x at the cost of @math increase of the overall top-5 classification error. Deep neural networks currently demonstrate state-of-the-art performance in several domains. At the same time, models of this class are very demanding in terms of computational resources. In particular, a large amount of memory is required by commonly used fully-connected layers, making it hard to use the models on low-end devices and stopping the further increase of the model size. In this paper we convert the dense weight matrices of the fully-connected layers to the Tensor Train [17] format such that the number of parameters is reduced by a huge factor and at the same time the expressive power of the layer is preserved. In particular, for the Very Deep VGG networks [21] we report the compression factor of the dense weight matrix of a fully-connected layer up to 200000 times leading to the compression factor of the whole network up to 7 times. In this paper we propose a novel decomposition method based on filter group approximation, which can significantly reduce the redundancy of deep convolutional neural networks (CNNs) while maintaining the majority of feature representation. Unlike other low-rank decomposition algorithms which operate on spatial or channel dimension of filters, our proposed method mainly focuses on exploiting the filter group structure for each layer. For several commonly used CNN models, including VGG and ResNet, our method can reduce over 80 floating-point operations (FLOPs) with less accuracy drop than state-of-the-art methods on various image classification datasets. Besides, experiments demonstrate that our method is conducive to alleviating degeneracy of the compressed network, which hurts the convergence and performance of the network. In recent years considerable research efforts have been devoted to compression techniques of convolutional neural networks (CNNs). Many works so far have focused on CNN connection pruning methods which produce sparse parameter tensors in convolutional or fully-connected layers. It has been demonstrated in several studies that even simple methods can effectively eliminate connections of a CNN. However, since these methods make parameter tensors just sparser but no smaller, the compression may not transfer directly to acceleration without support from specially designed hardware. In this paper, we propose an iterative approach named Auto-balanced Filter Pruning, where we pre-train the network in an innovative auto-balanced way to transfer the representational capacity of its convolutional layers to a fraction of the filters, prune the redundant ones, then re-train it to restore the accuracy. In this way, a smaller version of the original network is learned and the floating-point operations (FLOPs) are reduced. By applying this method on several common CNNs, we show that a large portion of the filters can be discarded without obvious accuracy drop, leading to significant reduction of computational burdens. Concretely, we reduce the inference cost of LeNet-5 on MNIST, VGG-16 and ResNet-56 on CIFAR-10 by 95.1 , 79.7 and 60.9 , respectively. Copyright © 2018, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. Deep neural networks currently demonstrate state-of-the-art performance in several domains. At the same time, models of this class are very demanding in terms of computational resources. In particular, a large amount of memory is required by commonly used fully-connected layers, making it hard to use the models on low-end devices and stopping the further increase of the model size. In this paper we convert the dense weight matrices of the fully-connected layers to the Tensor Train format such that the number of parameters is reduced by a huge factor and at the same time the expressive power of the layer is preserved. In particular, for the Very Deep VGG networks we report the compression factor of the dense weight matrix of a fully-connected layer up to 200000 times leading to the compression factor of the whole network up to 7 times. In this paper, we propose and study a technique to reduce the number of parameters and computation time in convolutional neural networks. We use Kronecker product to exploit the local structures within convolution and fully-connected layers, by replacing the large weight matrices by combinations of multiple Kronecker products of smaller matrices. Just as the Kronecker product is a generalization of the outer product from vectors to matrices, our method is a generalization of the low rank approximation method for convolution neural networks. We also introduce combinations of different shapes of Kronecker product to increase modeling capacity. Experiments on SVHN, scene text recognition and ImageNet dataset demonstrate that we can achieve @math speedup or @math parameter reduction with less than 1 drop in accuracy, showing the effectiveness and efficiency of our method. Moreover, the computation efficiency of Kronecker layer makes using larger feature map possible, which in turn enables us to outperform the previous state-of-the-art on both SVHN(digit recognition) and CASIA-HWDB (handwritten Chinese character recognition) datasets. The fully-connected layers of deep convolutional neural networks typically contain over 90 of the network parameters. Reducing the number of parameters while preserving predictive performance is critically important for training big models in distributed systems and for deployment in embedded devices. In this paper, we introduce a novel Adaptive Fastfood transform to reparameterize the matrix-vector multiplication of fully connected layers. Reparameterizing a fully connected layer with d inputs and n outputs with the Adaptive Fastfood transform reduces the storage and computational costs costs from O(nd) to O(n) and O(n log d) respectively. Using the Adaptive Fastfood transform in convolutional networks results in what we call a deep fried convnet. These convnets are end-to-end trainable, and enable us to attain substantial reductions in the number of parameters without affecting prediction accuracy on the MNIST and ImageNet datasets. Large CNNs have delivered impressive performance in various computer vision applications. But the storage and computation requirements make it problematic for deploying these models on mobile devices. Recently, tensor decompositions have been used for speeding up CNNs. In this paper, we further develop the tensor decomposition technique. We propose a new algorithm for computing the low-rank tensor decomposition for removing the redundancy in the convolution kernels. The algorithm finds the exact global optimizer of the decomposition and is more effective than iterative methods. Based on the decomposition, we further propose a new method for training low-rank constrained CNNs from scratch. Interestingly, while achieving a significant speedup, sometimes the low-rank constrained CNNs delivers significantly better performance than their non-constrained counterparts. On the CIFAR-10 dataset, the proposed low-rank NIN model achieves @math accuracy (without data augmentation), which also improves upon state-of-the-art result. We evaluated the proposed method on CIFAR-10 and ILSVRC12 datasets for a variety of modern CNNs, including AlexNet, NIN, VGG and GoogleNet with success. For example, the forward time of VGG-16 is reduced by half while the performance is still comparable. Empirical success suggests that low-rank tensor decompositions can be a very useful tool for speeding up large CNNs. Abstract: Large CNNs have delivered impressive performance in various computer vision applications. But the storage and computation requirements make it problematic for deploying these models on mobile devices. Recently, tensor decompositions have been used for speeding up CNNs. In this paper, we further develop the tensor decomposition technique. We propose a new algorithm for computing the low-rank tensor decomposition for removing the redundancy in the convolution kernels. The algorithm finds the exact global optimizer of the decomposition and is more effective than iterative methods. Based on the decomposition, we further propose a new method for training low-rank constrained CNNs from scratch. Interestingly, while achieving a significant speedup, sometimes the low-rank constrained CNNs delivers significantly better performance than their non-constrained counterparts. On the CIFAR-10 dataset, the proposed low-rank NIN model achieves @math accuracy (without data augmentation), which also improves upon state-of-the-art result. We evaluated the proposed method on CIFAR-10 and ILSVRC12 datasets for a variety of modern CNNs, including AlexNet, NIN, VGG and GoogleNet with success. For example, the forward time of VGG-16 is reduced by half while the performance is still comparable. Empirical success suggests that low-rank tensor decompositions can be a very useful tool for speeding up large CNNs. 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. Link analysis typically focuses on a single type of connection, e.g., two journal papers are linked because they are written by the same author. However, often we want to analyze data that has multiple linkages between objects, e.g., two papers may have the same keywords and one may cite the other. The goal of this paper is to show that multilinear algebra provides a tool for multilink analysis. We analyze five years of publication data from journals published by the Society for Industrial and Applied Mathematics. We explore how papers can be grouped in the context of multiple link types using a tensor to represent all the links between them. A PARAFAC decomposition on the resulting tensor yields information similar to the SVD decomposition of a standard adjacency matrix. We show how the PARAFAC decomposition can be used to understand the structure of the document space and define paper-paper similarities based on multiple linkages. Examples are presented where the decomposed tensor data is used to find papers similar to a body of work (e.g., related by topic or similar to a particular author's papers), find related authors using linkages other than explicit co-authorship or citations, distinguish between papers written bymore » different authors with the same name, and predict the journal in which a paper was published.« less We present COBLA—Constrained Optimization Based Low-rank Approximation—a systematic method of finding an optimal low-rank approximation of a trained convolutional neural network, subject to constraints in the number of multiply-accumulate (MAC) operations and the memory footprint. COBLA optimally allocates the constrained computation resources into each layer of the approximated network. The singular value decomposition of the network weight is computed, then a binary masking variable is introduced to denote whether a particular singular value and the corresponding singular vectors are used in low-rank approximation. With this formulation, the number of the MAC operations and the memory footprint are represented as linear constraints in terms of the binary masking variables. The resulted 0–1 integer programming problem is approximately solved by sequential quadratic programming. COBLA does not introduce any hyperparameter. We empirically demonstrate that COBLA outperforms prior art using the SqueezeNet and VGG-16 architecture on the ImageNet dataset. Convolutional neural networks (CNNs) have achieved remarkable success in various computer vision tasks, which are extremely powerful to deal with massive training data by using tens of millions of parameters. However, CNNs often cost significant memory and computation consumption, which prohibits their usage in resource-limited environments such as mobile or embedded devices. To address the above issues, the existing approaches typically focus on either accelerating the convolutional layers or compressing the fully-connected layers separatedly, without pursuing a joint optimum. In this paper, we overcome such a limitation by introducing a holistic CNN compression framework, termed LRDKT, which works throughout both convolutional and fully-connected layers. First, a low-rank decomposition (LRD) scheme is proposed to remove redundancies across both convolutional kernels and fullyconnected matrices, which has a novel closed-form solver to significantly improve the efficiency of the existing iterative optimization solvers. Second, a novel knowledge transfer (KT) based training scheme is introduced. To recover the accumulated accuracy loss and overcome the vanishing gradient, KT explicitly aligns outputs and intermediate responses from a teacher (original) network to its student (compressed) network. We have comprehensively analyzed and evaluated the compression and speedup ratios of the proposed model on MNIST and ILSVRC 2012 benchmarks. In both benchmarks, the proposed scheme has demonstrated superior performance gains over the state-of-the-art methods. We also demonstrate the proposed compression scheme for the task of transfer learning, including domain adaptation and object detection, which show exciting performance gains over the state-of-the-arts. Our source code and compressed models are available at https: github.com ShaohuiLin LRDKT. The singular value decomposition (SVD) has been extensively used in engineering and statistical applications. This method was originally discovered by Eckart and Young in [Psychometrika, 1 (1936), pp. 211--218], where they considered the problem of low-rank approximation to a matrix. A natural generalization of the SVD is the problem of low-rank approximation to high order tensors, which we call the multidimensional SVD. In this paper, we investigate certain properties of this decomposition as well as numerical algorithms. We consider elliptic PDE eigenvalue problems on a tensorized domain, discretized such that the resulting matrix eigenvalue problem Ax λx exhibits Kronecker product structure. In particular, we are concerned with the case of high dimensions, where standard approaches to the solution of matrix eigenvalue problems fail due to the exponentially growing degrees of freedom. Recent work shows that this curse of dimensionality can in many cases be addressed by approximating the desired solution vector x in a low-rank tensor format. In this paper, we use the hierarchical Tucker decomposition to develop a low-rank variant of LOBPCG, a classical preconditioned eigenvalue solver. We also show how the ALS and MALS (DMRG) methods known from computational quantum physics can be adapted to the hierarchical Tucker decomposition. Finally, a combination of ALS and MALS with LOBPCG and with our low-rank variant is proposed. A number of numerical experiments indicate that such combinations represent the methods of choice. The large number of parameters in deep neural networks (DNN) for automatic speech recognition (ASR) makes speaker adaptation very challenging. It also limits the use of speaker personalization due to the huge storage cost in large-scale deployments. In this paper we address DNN adaptation and personalization issues by presenting two methods based on the singular value decomposition (SVD). The first method uses an SVD to replace the weight matrix of a speaker independent DNN by the product of two low rank matrices. Adaptation is then performed by updating a square matrix inserted between the two low-rank matrices. In the second method, we adapt the full weight matrix but only store the delta matrix - the difference between the original and adapted weight matrices. We decrease the footprint of the adapted model by storing a reduced rank version of the delta matrix via an SVD. The proposed methods were evaluated on short message dictation task. Experimental results show that we can obtain similar accuracy improvements as the previously proposed Kullback-Leibler divergence (KLD) regularized method with far fewer parameters, which only requires 0.89 of the original model storage.