Untrained Graph Neural Networks for Denoising

Abstract

A fundamental problem in signal processing is to denoise a signal. While there are many well-performing methods for denoising signals defined on regular supports, such as images defined on two-dimensional grids of pixels, many important classes of signals are defined over irregular domains such as graphs. This paper introduces two untrained graph neural network architectures for graph signal denoising, provides theoretical guarantees for their denoising capabilities in a simple setup, and numerically validates the theoretical results in more general scenarios. The two architectures differ on how they incorporate the information encoded in the graph, with one relying on graph convolutions and the other employing graph upsampling operators based on hierarchical clustering. Each architecture implements a different prior over the targeted signals.To numerically illustrate the validity of the theoretical results and to compare the performance of the proposed architectures with other denoising alternatives, we present several experimental results with real and synthetic datasets.

Theoretical Perspectives on Deep Learning Methods in Inverse Problems

“…Finally, untrained neural networks have also been used to reconstruct graph signals [110], as well as continuouslyindexed objects through fitting probabilistic models [150], [114]. We expect that there is significant potential for further theoretical (and practical) developments in these directions.…”

Theoretical Perspectives on Deep Learning Methods in Inverse Problems

“…Finally, untrained neural networks have also been used to reconstruct graph signals [102], as well as continuouslyindexed objects through fitting probabilistic models [139], [106]. We expect that there is significant potential for further theoretical (and practical) developments in these directions.…”

Untrained Neural Network Priors for Inverse Imaging Problems: A Survey

“…For instance, the Deep Matching Prior [67] uses an UNNP to learn priors for semantically similar pairs of input images. Under-parameterized UNNPinspired untrained graph neural networks were introduced in [39]. The illustration of different UNNP architectures proposed in the literature can be found in Figure 4.…”

“…4: Different UNNP architectures proposed in the literature. Relevant papers: (a) [6] ; (b) [12] ; (c) [7], [25], [49] ; (d) [37], [68], [69] ; (e) [28], [70] ; (f) [67] ; (g) [39]. Fig.…”

“…[27] [28] [29], [30], [31], [32] [33], [ [36], [37] [38], [39], [40], [41], [42], [43] [44] [45] [46] [47], [48], [49], [50] [51] [52], [53], [54] [ 55] image (in vectorized form) x 0 ∈ R n is observed via m noisy measurements y ∈ R m . More specifically, we can write…”