Graph Signal Separation Based on Smoothness or Sparsity in the Frequency Domain

In this paper, we study the problem of demixing an observed signal, which is the summation of a set of signals that live on a multi-layer graph, by proposing several methods to decompose the observed signal into structured components. For this purpose, we build on two of the most widely-used graph signal models' assumptions, namely smoothness and sparsity in the graph spectral domain. We firstly show that a vector can be
uniquely
decomposed as the summation of a set of smooth graph signals, up to the indeterminacy of their DC values. So, if the original signals are known to be smooth, it is expected that with such a decomposition all of the original signals are retrieved. From the blind source separation (BSS) point of view, this is like the separation of a set of graph signals from a single mixture, contrary to traditional BSS in which at least two observed mixtures are typically required. Then, the approach is generalized to a wider family of graph signals, which are not necessarily smooth, but exhibit some sparse frequency characteristics in the graph spectral domain. Numerical simulations confirm the good performance of our approach in separating a mixture of graph signals.