We study deep neural networks with polynomial activations, particularly their expressive power. For a fixed architecture and activation degree, a polynomial neural network defines an algebraic map from weights to polynomials. The image of this map is the functional space associated to the network, and it is an irreducible algebraic variety upon taking closure. This paper proposes the dimension of this variety as a precise measure of the expressive power of polynomial neural networks. We obtain several theoretical results regarding this dimension as a function of architecture, including an exact formula for high activation degrees, as well as upper and lower bounds on layer widths in order for deep polynomials networks to fill the ambient functional space. We also present computational evidence that it is profitable in terms of expressiveness for layer widths to increase monotonically and then decrease monotonically. Finally, we link our study to favorable optimization properties when training weights, and we draw intriguing connections with tensor and polynomial decompositions.
Most of the aforementioned studies make strong hypotheses on the network architecture. In particular, @cite_27 @cite_10 focus on , or @cite_5 . These are networks composed of units that compute either the product or a weighted sum of their inputs. @cite_9 , the authors introduce a model of . This is a particular class of arithmetic circuits that includes networks with layers of 1D convolutions and product pooling. This model does not allow for non-linear activations (beside the product pooling), although the follow-up paper @cite_25 extends some results to ReLU activations with sum pooling. Interestingly, these networks are related to Hierarchical Tucker (HT) decomposition of tensors.
@cite_9 @cite_27 @cite_5 @cite_10 @cite_25
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Convolutional rectifier networks, i.e. convolutional neural networks with rectified linear activation and max or average pooling, are the cornerstone of modern deep learning. However, despite their wide use and success, our theoretical understanding of the expressive properties that drive these networks is partial at best. On the other hand, we have a much firmer grasp of these issues in the world of arithmetic circuits. Specifically, it is known that convolutional arithmetic circuits possess the property of "complete depth efficiency", meaning that besides a negligible set, all functions realizable by a deep network of polynomial size, require exponential size in order to be realized (or approximated) by a shallow network. In this paper we describe a construction based on generalized tensor decompositions, that transforms convolutional arithmetic circuits into convolutional rectifier networks. We then use mathematical tools available from the world of arithmetic circuits to prove new results. First, we show that convolutional rectifier networks are universal with max pooling but not with average pooling. Second, and more importantly, we show that depth efficiency is weaker with convolutional rectifier networks than it is with convolutional arithmetic circuits. This leads us to believe that developing effective methods for training convolutional arithmetic circuits, thereby fulfilling their expressive potential, may give rise to a deep learning architecture that is provably superior to convolutional rectifier networks but has so far been overlooked by practitioners. Two architectures that generalize convolutional neural networks (CNNs) for the processing of signals supported on graphs are introduced. We start with the selection graph neural network (GNN), which replaces linear time invariant filters with linear shift invariant graph filters to generate convolutional features and reinterprets pooling as a possibly nonlinear subsampling stage where nearby nodes pool their information in a set of preselected sample nodes. A key component of the architecture is to remember the position of sampled nodes to permit computation of convolutional features at deeper layers. The second architecture, dubbed aggregation GNN, diffuses the signal through the graph and stores the sequence of diffused components observed by a designated node. This procedure effectively aggregates all components into a stream of information having temporal structure to which the convolution and pooling stages of regular CNNs can be applied. A multinode version of aggregation GNNs is further introduced for operation in large-scale graphs. An important property of selection and aggregation GNNs is that they reduce to conventional CNNs when particularized to time signals reinterpreted as graph signals in a circulant graph. Comparative numerical analyses are performed in a source localization application over synthetic and real-world networks. Performance is also evaluated for an authorship attribution problem and text category classification. Multinode aggregation GNNs are consistently the best-performing GNN architecture. With the rapid development of Deep Convolutional Neural Networks (DCNNs), numerous works focus on designing better network architectures (i.e., AlexNet, VGG, Inception, ResNet and DenseNet etc.). Nevertheless, all these networks have the same characteristic: each convolutional layer is followed by an activation layer, a Rectified Linear Unit (ReLU) layer is the most used among them. In this work, we argue that the paired module with 1:1 convolution and ReLU ratio is not the best choice since it may result in poor generalization ability. Thus, we try to investigate the more suitable convolution and ReLU ratio for exploring the better network architectures. Specifically, inspired by Leaky ReLU, we focus on adopting the proportional module with N:M (N @math M) convolution and ReLU ratio to design the better networks. From the perspective of ensemble learning, Leaky ReLU can be considered as an ensemble of networks with different convolution and ReLU ratio. We find that the proportional module with N:M (N @math M) convolution and ReLU ratio can help networks acquire the better performance, through the analysis of a simple Leaky ReLU model. By utilizing the proportional module with N:M (N @math M) convolution and ReLU ratio, many popular networks can form more rich representations in models, since the N:M (N @math M) proportional module can utilize information more effectively. Furthermore, we apply this module in diverse DCNN models to explore whether is the N:M (N @math M) convolution and ReLU ratio indeed more effective. From our experimental results, we can find that such a simple yet effective method achieves better performance in different benchmarks with various network architectures and the experimental results verify that the superiority of the proportional module. It is widely believed that the success of deep convolutional networks is based on progressively discarding uninformative variability about the input with respect to the problem at hand. This is supported empirically by the difficulty of recovering images from their hidden representations, in most commonly used network architectures. In this paper we show that this loss of information is not a necessary condition to learn representations that generalize well on complicated problems, such as ImageNet. Via a cascade of homeomorphic layers, we build the i-RevNet, a network that can be fully inverted up to the final projection onto the classes, i.e. no information is discarded. Building an invertible architecture is difficult, for example, because the local inversion is ill-conditioned, we overcome this by providing an explicit inverse. An analysis of i-RevNet’s learned representations suggests an explanation of the good accuracy by a progressive contraction and linear separation with depth. To shed light on the nature of the model learned by the i-RevNet we reconstruct linear interpolations between natural images representations.