Convolutional neural networks (CNNs) have achieved remarkable performance in various fields, particularly in the domain of computer vision. However, why this architecture works well remains to be a mystery. In this work we move a small step toward understanding the success of CNNs by investigating the learning dynamics of a two-layer nonlinear convolutional neural network over some specific data distributions. Rather than the typical Gaussian assumption for input data distribution, we consider a more realistic setting that each data point (e.g. image) contains a specific pattern determining its class label. Within this setting, we both theoretically and empirically show that some convolutional filters will learn the key patterns in data and the norm of these filters will dominate during the training process with stochastic gradient descent. And with any high probability, when the number of iterations is sufficiently large, the CNN model could obtain 100 accuracy over the considered data distributions. Our experiments demonstrate that for practical image classification tasks our findings still hold to some extent.
The works @cite_12 @cite_15 @cite_7 @cite_20 focused on analyzing the learning a two-layer CNN for regression problem with Gaussian inputs, under the teacher-student network settings. They showed that the dynamics could converge to true parameter or zero loss. @cite_1 considered learning a two-layer CNN for some binary classification task assuming a restricted input distribution. Multi-layer linear CNNs were studied in @cite_14 for binary classification problem with linear separable data distribution. And they proved that the dynamics could converge to a linear predictor related to the @math bridge penalty in the frequency domain.
@cite_14 @cite_7 @cite_1 @cite_15 @cite_12 @cite_20
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This paper aims to accelerate the test-time computation of convolutional neural networks (CNNs), especially very deep CNNs [1] that have substantially impacted the computer vision community. Unlike previous methods that are designed for approximating linear filters or linear responses, our method takes the nonlinear units into account. We develop an effective solution to the resulting nonlinear optimization problem without the need of stochastic gradient descent (SGD). More importantly, while previous methods mainly focus on optimizing one or two layers, our nonlinear method enables an asymmetric reconstruction that reduces the rapidly accumulated error when multiple (e.g., @math 10) layers are approximated. For the widely used very deep VGG-16 model [1] , our method achieves a whole-model speedup of 4 @math with merely a 0.3 percent increase of top-5 error in ImageNet classification. Our 4 @math accelerated VGG-16 model also shows a graceful accuracy degradation for object detection when plugged into the Fast R-CNN detector [2] . Convolutional Neural Network (CNN) image classifiers are traditionally designed to have sequential convolutional layers with a single output layer. This is based on the assumption that all target classes should be treated equally and exclusively. However, some classes can be more difficult to distinguish than others, and classes may be organized in a hierarchy of categories. At the same time, a CNN is designed to learn internal representations that abstract from the input data based on its hierarchical layered structure. So it is natural to ask if an inverse of this idea can be applied to learn a model that can predict over a classification hierarchy using multiple output layers in decreasing order of class abstraction. In this paper, we introduce a variant of the traditional CNN model named the Branch Convolutional Neural Network (B-CNN). A B-CNN model outputs multiple predictions ordered from coarse to fine along the concatenated convolutional layers corresponding to the hierarchical structure of the target classes, which can be regarded as a form of prior knowledge on the output. To learn with B-CNNs a novel training strategy, named the Branch Training strategy (BT-strategy), is introduced which balances the strictness of the prior with the freedom to adjust parameters on the output layers to minimize the loss. In this way we show that CNN based models can be forced to learn successively coarse to fine concepts in the internal layers at the output stage, and that hierarchical prior knowledge can be adopted to boost CNN models' classification performance. Our models are evaluated to show that the B-CNN extensions improve over the corresponding baseline CNN on the benchmark datasets MNIST, CIFAR-10 and CIFAR-100. Whereas CNNs have demonstrated immense progress in many vision problems, they suffer from a dependence on monumental amounts of labeled training data. On the other hand, dictionary learning does not scale to the size of problems that CNNs can handle, despite being very effective at low-level vision tasks such as denoising and inpainting. Recently, interest has grown in adapting dictionary learning methods for supervised tasks such as classification and inverse problems. We propose two new network layers that are based on dictionary learning: a sparse factorization layer and a convolutional sparse factorization layer, analogous to fully-connected and convolutional layers, respectively. Using our derivations, these layers can be dropped in to existing CNNs, trained together in an end-to-end fashion with back-propagation, and leverage semisupervision in ways classical CNNs cannot. We experimentally compare networks with these two new layers against a baseline CNN. Our results demonstrate that networks with either of the sparse factorization layers are able to outperform classical CNNs when supervised data are few. They also show performance improvements in certain tasks when compared to the CNN with no sparse factorization layers with the same exact number of parameters. Recently, plugging trainable structural layers into deep convolutional neural networks (CNNs) as image representations has made promising progress. However, there has been little work on inserting parametric probability distributions, which can effectively model feature statistics, into deep CNNs in an end-to-end manner. This paper proposes a Global Gaussian Distribution embedding Network (G2DeNet) to take a step towards addressing this problem. The core of G2DeNet is a novel trainable layer of a global Gaussian as an image representation plugged into deep CNNs for end-to-end learning. The challenge is that the proposed layer involves Gaussian distributions whose space is not a linear space, which makes its forward and backward propagations be non-intuitive and non-trivial. To tackle this issue, we employ a Gaussian embedding strategy which respects the structures of both Riemannian manifold and smooth group of Gaussians. Based on this strategy, we construct the proposed global Gaussian embedding layer and decompose it into two sub-layers: the matrix partition sub-layer decoupling the mean vector and covariance matrix entangled in the embedding matrix, and the square-rooted, symmetric positive definite matrix sub-layer. In this way, we can derive the partial derivatives associated with the proposed structural layer and thus allow backpropagation of gradients. Experimental results on large scale region classification and fine-grained recognition tasks show that G2DeNet is superior to its counterparts, capable of achieving state-of-the-art performance.