Variational Autoencoder (VAE) is one of the most popular generative models, and enormous advances have been explored in recent years. Due to the increasing complexity of the raw data and the model architecture, deep networks are needed in VAE models while few works discuss their impacts. According to our observation, VAE does not always benefit from deeper architecture: 1) Deeper encoder makes VAE learn more comprehensible latent representations, while results in blurry reconstruction samples; 2) Deeper decoder ensures more high-quality generations, while the latent representations become abstruse; 3) When encoder and decoder both go deeper, abstruse latent representation occurs with blurry reconstruction samples at same time. In this paper, we deduce a Fisher information measure for the corresponding analysis. With such measure, we demonstrate that information loss is ineluctable in feed-forward networks and causes the previous three types of degeneration, especially when the network goes deeper. We also demonstrate that skip connections benefit the preservation of information amount, thus propose a VAE enhanced by skip connections, named SCVAE. In the experiments, SCVAE is shown to mitigate the information loss and to achieve a promising performance in both encoding and decoding tasks. Moreover, SCVAE can be adaptive to other state-of-the-art variants of VAE for further amelioration.
It is noteworthy that neural networks and VAE possess similarities and differences when going deeper. @cite_9 @cite_14 point out the vanishing-gradient problem that prevents the neural network from going deeper, and introduce residual connections to help training of very deep neural networks. @cite_11 @cite_4 point out deep neural network is defective by degeneration and claim degeneration occurs when networks lack sufficient information to provide for learning dynamics, which corresponds to the third degeneration observed in Figure . Apart from this phenomenon, the other two degeneration problems are similar to information preference . Different from that, deeper encoder or decoder is supposed to be a powerful approximator of distribution, but the expressiveness does not correspond to our expectation. Fisher Information can be applied to measure the quality of parameters in neural networks as mentioned in @cite_7 @cite_15 . Inspired by these works, we investigate the observed problems in perspective of Fisher Information measure and demonstrate the existence of information loss in deep VAE. Skip connection is applied as a solution for information preservation, but not for avoiding gradient vanishing problem .
@cite_14 @cite_4 @cite_7 @cite_9 @cite_15 @cite_11
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Deep neural network is difficult to train and this predicament becomes worse as the depth increases. The essence of this problem exists in the magnitude of backpropagated errors that will result in gradient vanishing or exploding phenomenon. We show that a variant of regularizer which utilizes orthonormality among different filter banks can alleviate this problem. Moreover, we design a backward error modulation mechanism based on the quasi-isometry assumption between two consecutive parametric layers. Equipped with these two ingredients, we propose several novel optimization solutions that can be utilized for training a specific-structured (repetitively triple modules of Conv-BNReLU) extremely deep convolutional neural network (CNN) WITHOUT any shortcuts identity mappings from scratch. Experiments show that our proposed solutions can achieve distinct improvements for a 44-layer and a 110-layer plain networks on both the CIFAR-10 and ImageNet datasets. Moreover, we can successfully train plain CNNs to match the performance of the residual counterparts. Besides, we propose new principles for designing network structure from the insights evoked by orthonormality. Combined with residual structure, we achieve comparative performance on the ImageNet dataset. Deep neural network is difficult to train and this predicament becomes worse as the depth increases. The essence of this problem exists in the magnitude of backpropagated errors that will result in gradient vanishing or exploding phenomenon. We show that a variant of regularizer which utilizes orthonormality among different filter banks can alleviate this problem. Moreover, we design a backward error modulation mechanism based on the quasi-isometry assumption between two consecutive parametric layers. Equipped with these two ingredients, we propose several novel optimization solutions that can be utilized for training a specific-structured (repetitively triple modules of Conv-BNReLU) extremely deep convolutional neural network (CNN) WITHOUT any shortcuts identity mappings from scratch. Experiments show that our proposed solutions can achieve distinct improvements for a 44-layer and a 110-layer plain networks on both the CIFAR-10 and ImageNet datasets. Moreover, we can successfully train plain CNNs to match the performance of the residual counterparts. Besides, we propose new principles for designing network structure from the insights evoked by orthonormality. Combined with residual structure, we achieve comparative performance on the ImageNet dataset. We investigate the use of Deep Neural Networks for the classification of image datasets where texture features are important for generating class-conditional discriminative representations. To this end, we first derive the size of the feature space for some standard textural features extracted from the input dataset and then use the theory of Vapnik-Chervonenkis dimension to show that hand-crafted feature extraction creates low-dimensional representations which help in reducing the overall excess error rate. As a corollary to this analysis, we derive for the first time upper bounds on the VC dimension of Convolutional Neural Network as well as Dropout and Dropconnect networks and the relation between excess error rate of Dropout and Dropconnect networks. The concept of intrinsic dimension is used to validate the intuition that texture-based datasets are inherently higher dimensional as compared to handwritten digits or other object recognition datasets and hence more difficult to be shattered by neural networks. We then derive the mean distance from the centroid to the nearest and farthest sampling points in an n-dimensional manifold and show that the Relative Contrast of the sample data vanishes as dimensionality of the underlying vector space tends to infinity. Previous theoretical work on deep learning and neural network optimization tend to focus on avoiding saddle points and local minima. However, the practical observation is that, at least in the case of the most successful Deep Convolutional Neural Networks (DCNNs), practitioners can always increase the network size to fit the training data (an extreme example would be [1]). The most successful DCNNs such as VGG and ResNets are best used with a degree of "overparametrization". In this work, we characterize with a mix of theory and experiments, the landscape of the empirical risk of overparametrized DCNNs. We first prove in the regression framework the existence of a large number of degenerate global minimizers with zero empirical error (modulo inconsistent equations). The argument that relies on the use of Bezout theorem is rigorous when the RELUs are replaced by a polynomial nonlinearity (which empirically works as well). As described in our Theory III [2] paper, the same minimizers are degenerate and thus very likely to be found by SGD that will furthermore select with higher probability the most robust zero-minimizer. We further experimentally explored and visualized the landscape of empirical risk of a DCNN on CIFAR-10 during the entire training process and especially the global minima. Finally, based on our theoretical and experimental results, we propose an intuitive model of the landscape of DCNN's empirical loss surface, which might not be as complicated as people commonly believe.