Overparametrized Deep Encoder-Decoder Schemes for Inputs and Outputs Defined over Graphs

There is a growing interest in the joint application of graph signal processing and neural networks (NNs) for learning problems involving complex, non-linear and/or non-Euclidean datasets. This paper proposes an overparametrized graph-aware NN architecture able to represent a non-linear mapping between two graph signals, each defined on a different graph. The considered architecture is based on two NNs and a common latent space. Specifically, we consider an overparametrized graph-aware NN encoder which maps the input graph signal to a latent space, followed by an overparametrized graph-aware NN decoder that transforms the latent representation to the output graph signal. The parameters of the two NNs are jointly tuned by applying the back-propagation algorithm with an early stopping procedure to prevent overfitting. The overall architecture can be interpreted as an overparametrized graph-aware encoder/decoder NN operating over two different graphs. A key element in the encoder (decoding) scheme is the consideration of a nested collection of parametric graph-aware (down-) up-sampling operators, whose design will be studied in detail. We show by numerical simulations that the proposed scheme outperforms the corresponding benchmark NN architectures, previously introduced in the literature.