Many underlying relationships among data in several areas of science and engineering, e.g., computer vision, molecular chemistry, molecular biology, pattern recognition, and data mining, can be represented in terms of graphs. In this paper, we propose a new neural network model, called graph neural network (GNN) model, that extends existing neural network methods for processing the data represented in graph domains. This GNN model, which can directly process most of the practically useful types of graphs, e.g., acyclic, cyclic, directed, and undirected, implements a function tau(G,n) isin IR m that maps a graph G and one of its nodes n into an m-dimensional Euclidean space. A supervised learning algorithm is derived to estimate the parameters of the proposed GNN model. The computational cost of the proposed algorithm is also considered. Some experimental results are shown to validate the proposed learning algorithm, and to demonstrate its generalization capabilities. Disciplines Physical Sciences and Mathematics

We present a hybrid neural-network for human face recognition which compares favourably with other methods. The system combines local image sampling, a self-organizing map (SOM) neural network, and a convolutional neural network. The SOM provides a quantization of the image samples into a topological space where inputs that are nearby in the original space are also nearby in the output space, thereby providing dimensionality reduction and invariance to minor changes in the image sample, and the convolutional neural network provides partial invariance to translation, rotation, scale, and deformation. The convolutional network extracts successively larger features in a hierarchical set of layers. We present results using the Karhunen-Loeve transform in place of the SOM, and a multilayer perceptron (MLP) in place of the convolutional network for comparison. We use a database of 400 images of 40 individuals which contains quite a high degree of variability in expression, pose, and facial details. We analyze the computational complexity and discuss how new classes could be added to the trained recognizer.

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Recent developments in the area of neural networks produced models capable of dealing with structured data. Here, we propose the first fully unsupervised model, namely an extension of traditional self-organizing maps (SOMs), for the processing of labeled directed acyclic graphs (DAGs). The extension is obtained by using the unfolding procedure adopted in recurrent and recursive neural networks, with the replicated neurons in the unfolded network comprising of a full SOM. This approach enables the discovery of similarities among objects including vectors consisting of numerical data. The capabilities of the model are analyzed in detail by utilizing a relatively large data set taken from an artificial benchmark problem involving visual patterns encoded as labeled DAGs. The experimental results demonstrate clearly that the proposed model is capable of exploiting both information conveyed in the labels attached to each node of the input DAGs and information encoded in the DAG topology. Abstract-Recent developments in the area of neural networks produced models capable of dealing with structured data. Here, we propose the first fully unsupervised model, namely an extension of traditional self-organizing maps (SOMs), for the processing of labeled directed acyclic graphs (DAGs). The extension is obtained by using the unfolding procedure adopted in recurrent and recursive neural networks, with the replicated neurons in the unfolded network comprising of a full SOM. This approach enables the discovery of similarities among objects including vectors consisting of numerical data. The capabilities of the model are analyzed in detail by utilizing a relatively large data set taken from an artificial benchmark problem involving visual patterns encoded as labeled DAGs. The experimental results demonstrate clearly that the proposed model is capable of exploiting both information conveyed in the labels attached to each node of the input DAGs and information encoded in the DAG topology. Disciplines Physical Sciences and MathematicsIndex Terms-Clustering, data mining which involves novel types of data/knowledge, data reduction techniques, discovering similarities, innovative algorithms, processing labeled graphs, recurrent neural networks, recursive neural networks, self organizing maps (SOMs), vector quantization (VQ).

In this paper, we will consider the approximation properties of a recently introduced neural network model called graph neural network (GNN), which can be used to process-structured data inputs, e.g., acyclic graphs, cyclic graphs, and directed or undirected graphs. This class of neural networks implements a function tau(G,n) is an element of IR(m) that maps a graph G and one of its nodes n onto an m-dimensional Euclidean space. We characterize the functions that can be approximated by GNNs, in probability, up to any prescribed degree of precision. This set contains the maps that satisfy a property called preservation of the unfolding equivalence, and includes most of the practically useful functions on graphs; the only known exception is when the input graph contains particular patterns of symmetries when unfolding equivalence may not be preserved. The result can be considered an extension of the universal approximation property established for the classic feedforward neural networks (FNNs). Some experimental examples are used to show the computational capabilities of the proposed model.