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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Recent work has shown that convolutional networks can be substantially deeper, more accurate, and efficient to train if they contain shorter connections between layers close to the input and those close to the output. In this paper, we embrace this observation and introduce the Dense Convolutional Network (DenseNet), which connects each layer to every other layer in a feed-forward fashion. Whereas traditional convolutional networks with L layers have L connections - one between each layer and its subsequent layer - our network has L(L+1) 2 direct connections. For each layer, the feature-maps of all preceding layers are used as inputs, and its own feature-maps are used as inputs into all subsequent layers. DenseNets have several compelling advantages: they alleviate the vanishing-gradient problem, strengthen feature propagation, encourage feature reuse, and substantially reduce the number of parameters. We evaluate our proposed architecture on four highly competitive object recognition benchmark tasks (CIFAR-10, CIFAR-100, SVHN, and ImageNet). DenseNets obtain significant improvements over the state-of-the-art on most of them, whilst requiring less computation to achieve high performance. Code and pre-trained models are available at this https URL . Skip connections made the training of very deep neural networks possible and have become an indispendable component in a variety of neural architectures. A completely satisfactory explanation for their success remains elusive. Here, we present a novel explanation for the benefits of skip connections in training very deep neural networks. We argue that skip connections help break symmetries inherent in the loss landscapes of deep networks, leading to drastically simplified landscapes. In particular, skip connections between adjacent layers in a multilayer network break the permutation symmetry of nodes in a given layer, and the recently proposed DenseNet architecture, where each layer projects skip connections to every layer above it, also breaks the rescaling symmetry of connectivity matrices between different layers. This hypothesis is supported by evidence from a toy model with binary weights and from experiments with fully-connected networks suggesting (i) that skip connections do not necessarily improve training unless they help break symmetries and (ii) that alternative ways of breaking the symmetries also lead to significant performance improvements in training deep networks, hence there is nothing special about skip connections in this respect. We find, however, that skip connections confer additional benefits over and above symmetry-breaking, such as the ability to deal effectively with the vanishing gradients problem. The parameter space of neural networks has a Riemannian metric structure. The natural Riemannian gradient should be used instead of the conventional gradient, since the former denotes the true steepest descent direction of a loss function in the Riemannian space. The behavior of the stochastic gradient learning algorithm is much more effective if the natural gradient is used. The present paper studies the information-geometrical structure of perceptrons and other networks, and prove that the on-line learning method based on the natural gradient is asymptotically as efficient as the optimal batch algorithm. Adaptive modification of the learning constant is proposed and analyzed in terms of the Riemannian measure and is shown to be efficient. The natural gradient is finally applied to blind separation of mixtured independent signal sources. Deeper neural networks are more difficult to train. We present a residual learning framework to ease the training of networks that are substantially deeper than those used previously. We explicitly reformulate the layers as learning residual functions with reference to the layer inputs, instead of learning unreferenced functions. We provide comprehensive empirical evidence showing that these residual networks are easier to optimize, and can gain accuracy from considerably increased depth. On the ImageNet dataset we evaluate residual nets with a depth of up to 152 layers---8x deeper than VGG nets but still having lower complexity. An ensemble of these residual nets achieves 3.57 error on the ImageNet test set. This result won the 1st place on the ILSVRC 2015 classification task. We also present analysis on CIFAR-10 with 100 and 1000 layers. The depth of representations is of central importance for many visual recognition tasks. Solely due to our extremely deep representations, we obtain a 28 relative improvement on the COCO object detection dataset. Deep residual nets are foundations of our submissions to ILSVRC & COCO 2015 competitions, where we also won the 1st places on the tasks of ImageNet detection, ImageNet localization, COCO detection, and COCO segmentation. We describe four algorithms for neural network training, each adapted to different scalability constraints. These algorithms are mathematically principled and invariant under a number of transformations in data and network representation, from which performance is thus independent. These algorithms are obtained from the setting of differential geometry, and are based on either the natural gradient using the Fisher information matrix, or on Hessian methods, scaled down in a specific way to allow for scalability while keeping some of their key mathematical properties. Despite the widespread practical success of deep learning methods, our theoretical understanding of the dynamics of learning in deep neural networks remains quite sparse. We attempt to bridge the gap between the theory and practice of deep learning by systematically analyzing learning dynamics for the restricted case of deep linear neural networks. Despite the linearity of their input-output map, such networks have nonlinear gradient descent dynamics on weights that change with the addition of each new hidden layer. We show that deep linear networks exhibit nonlinear learning phenomena similar to those seen in simulations of nonlinear networks, including long plateaus followed by rapid transitions to lower error solutions, and faster convergence from greedy unsupervised pretraining initial conditions than from random initial conditions. We provide an analytical description of these phenomena by finding new exact solutions to the nonlinear dynamics of deep learning. Our theoretical analysis also reveals the surprising finding that as the depth of a network approaches infinity, learning speed can nevertheless remain finite: for a special class of initial conditions on the weights, very deep networks incur only a finite, depth independent, delay in learning speed relative to shallow networks. We show that, under certain conditions on the training data, unsupervised pretraining can find this special class of initial conditions, while scaled random Gaussian initializations cannot. We further exhibit a new class of random orthogonal initial conditions on weights that, like unsupervised pre-training, enjoys depth independent learning times. We further show that these initial conditions also lead to faithful propagation of gradients even in deep nonlinear networks, as long as they operate in a special regime known as the edge of chaos.