In this paper, we focus on learning the underlying product graph structure from multidomain training data. We assume that the product graph is formed from a Cartesian graph product of two smaller factor graphs. We then pose the product graph learning problem as the factor graph Laplacian matrix estimation problem. To estimate the factor graph Laplacian matrices, we assume that the data is smooth with respect to the underlying product graph. When the training data is noise free or complete, learning factor graphs can be formulated as a convex optimization problem, which has an explicit solution based on the water-filling algorithm. The developed framework is illustrated using numerical experiments on synthetic data as well as real data related to air quality monitoring in India.
This paper proposes a Fast Graph Convolutional Neural Network (FGRNN) architecture to predict sequences with an underlying graph structure. The proposed architecture addresses the limitations of the standard recurrent neural network (RNN), namely, vanishing and exploding gradients, causing numerical instabilities during training. State-of-the-art architectures that combine gated RNN architectures, such as Long Short-Term Memory (LSTM) and Gated Recurrent Unit (GRU) with graph convolutions are known to improve the numerical stability during the training phase, but at the expense of the model size involving a large number of training parameters. FGRNN addresses this problem by adding a weighted residual connection with only two extra training parameters as compared to the standard RNN. Numerical experiments on the real 3D point cloud dataset corroborates the proposed architecture.
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