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

Untrained Graph Neural Networks for Denoising

“…Lemma 1. Let the matrix X be defined as in (16), set and δ to small positive numbers, and denote by V K and W K the K leading eigenvectors in the respective eigendecompositions of à and X . Under Assumption 1, there exists an orthonormal matrix Q and an integer N ,δ such that, for N > N ,δ , the bound…”

“…Finally, replacing (28) in ( 27) the proof concludes. and let X be given by (16). Denote by H a graph filter defined as a polynomial of the expected adjacency matrix Ã, and let X be the expected squared Jacobian using the graph filter H, i.e., X = 0.5 11…”

“…The upsampling operators are based on hierarchical clustering algorithms [15], [23]- [25] so that, in contrast with [26], matrix inversions are not required, avoiding the related numerical issues. Our work is substantially different from [15], [16], which deal with graph encoder-decoder architectures. On top of our theoretical analysis and extensive numerical simulations, additional differences to prior work are that: (a) our graph decoder is an untrained network, and thus, it does not need a training phase; (b) we only require a decoder-like architecture for denoising graph signals, so it is not necessary to jointly design and train two different architectures as done in [15], [16].…”

“…The importance of leveraging the graph influence when using deep non-linear architectures is reflected in the wide range of GNNs that co-exist in the literature, including graph convolutional NNs (GCNNs) [10]- [12], graph recurrent NNs [13], graph autoencoders [14]- [16], graph generative adversarial networks [17], [18], or simplicial NNs [19]- [21], to name a few. Incorporating the graph structure into deep nonlinear models involves a wide range of options when designing the architecture.…”