5 research outputs found
Signals on Graphs: Uncertainty Principle and Sampling
In many applications, the observations can be represented as a signal defined
over the vertices of a graph. The analysis of such signals requires the
extension of standard signal processing tools. In this work, first, we provide
a class of graph signals that are maximally concentrated on the graph domain
and on its dual. Then, building on this framework, we derive an uncertainty
principle for graph signals and illustrate the conditions for the recovery of
band-limited signals from a subset of samples. We show an interesting link
between uncertainty principle and sampling and propose alternative signal
recovery algorithms, including a generalization to frame-based reconstruction
methods. After showing that the performance of signal recovery algorithms is
significantly affected by the location of samples, we suggest and compare a few
alternative sampling strategies. Finally, we provide the conditions for perfect
recovery of a useful signal corrupted by sparse noise, showing that this
problem is also intrinsically related to vertex-frequency localization
properties.Comment: This article is the revised version submitted to the IEEE
Transactions on Signal Processing on May, 2016; first revision was submitted
on January, 2016; original manuscript was submitted on July, 2015. The work
includes 16 pages, 8 figure
Distributed Adaptive Learning of Graph Signals
The aim of this paper is to propose distributed strategies for adaptive
learning of signals defined over graphs. Assuming the graph signal to be
bandlimited, the method enables distributed reconstruction, with guaranteed
performance in terms of mean-square error, and tracking from a limited number
of sampled observations taken from a subset of vertices. A detailed mean square
analysis is carried out and illustrates the role played by the sampling
strategy on the performance of the proposed method. Finally, some useful
strategies for distributed selection of the sampling set are provided. Several
numerical results validate our theoretical findings, and illustrate the
performance of the proposed method for distributed adaptive learning of signals
defined over graphs.Comment: To appear in IEEE Transactions on Signal Processing, 201
On the degrees of freedom of signals on graphs
Continuous-time signals are well known for not being perfectly localized in both time and frequency domains. Conversely, a signal defined over the vertices of a graph can be perfectly localized in both vertex and frequency domains. We derive the conditions ensuring the validity of this property and then, building on this theory, we provide the conditions for perfect reconstruction of a graph signal from its samples. Next, we provide a finite step algorithm for the reconstruction of a band-limited signal from its samples and then we show the effect of sampling a non perfectly band-limited signal and show how to select the bandwidth that minimizes the mean square reconstruction error