30,761 research outputs found
Sampling and Reconstruction of Signals on Product Graphs
In this paper, we consider the problem of subsampling and reconstruction of
signals that reside on the vertices of a product graph, such as sensor network
time series, genomic signals, or product ratings in a social network.
Specifically, we leverage the product structure of the underlying domain and
sample nodes from the graph factors. The proposed scheme is particularly useful
for processing signals on large-scale product graphs. The sampling sets are
designed using a low-complexity greedy algorithm and can be proven to be
near-optimal. To illustrate the developed theory, numerical experiments based
on real datasets are provided for sampling 3D dynamic point clouds and for
active learning in recommender systems.Comment: 5 pages, 3 figure
Graph Vertex Sampling with Arbitrary Graph Signal Hilbert Spaces
Graph vertex sampling set selection aims at selecting a set of ver-tices of a
graph such that the space of graph signals that can be reconstructed exactly
from those samples alone is maximal. In this context, we propose to extend
sampling set selection based on spectral proxies to arbitrary Hilbert spaces of
graph signals. Enabling arbitrary inner product of graph signals allows then to
better account for vertex importance on the graph for a sampling adapted to the
application. We first state how the change of inner product impacts sampling
set selection and reconstruction, and then apply it in the context of geometric
graphs to highlight how choosing an alternative inner product matrix can help
sampling set selection and reconstruction.Comment: Accepted at ICASSP 202
Sampling and Reconstruction of Sparse Signals on Circulant Graphs - An Introduction to Graph-FRI
With the objective of employing graphs toward a more generalized theory of
signal processing, we present a novel sampling framework for (wavelet-)sparse
signals defined on circulant graphs which extends basic properties of Finite
Rate of Innovation (FRI) theory to the graph domain, and can be applied to
arbitrary graphs via suitable approximation schemes. At its core, the
introduced Graph-FRI-framework states that any K-sparse signal on the vertices
of a circulant graph can be perfectly reconstructed from its
dimensionality-reduced representation in the graph spectral domain, the Graph
Fourier Transform (GFT), of minimum size 2K. By leveraging the recently
developed theory of e-splines and e-spline wavelets on graphs, one can
decompose this graph spectral transformation into the multiresolution low-pass
filtering operation with a graph e-spline filter, and subsequent transformation
to the spectral graph domain; this allows to infer a distinct sampling pattern,
and, ultimately, the structure of an associated coarsened graph, which
preserves essential properties of the original, including circularity and,
where applicable, the graph generating set.Comment: To appear in Appl. Comput. Harmon. Anal. (2017
Sampling of graph signals via randomized local aggregations
Sampling of signals defined over the nodes of a graph is one of the crucial
problems in graph signal processing. While in classical signal processing
sampling is a well defined operation, when we consider a graph signal many new
challenges arise and defining an efficient sampling strategy is not
straightforward. Recently, several works have addressed this problem. The most
common techniques select a subset of nodes to reconstruct the entire signal.
However, such methods often require the knowledge of the signal support and the
computation of the sparsity basis before sampling. Instead, in this paper we
propose a new approach to this issue. We introduce a novel technique that
combines localized sampling with compressed sensing. We first choose a subset
of nodes and then, for each node of the subset, we compute random linear
combinations of signal coefficients localized at the node itself and its
neighborhood. The proposed method provides theoretical guarantees in terms of
reconstruction and stability to noise for any graph and any orthonormal basis,
even when the support is not known.Comment: IEEE Transactions on Signal and Information Processing over Networks,
201
Kernel-based Inference of Functions over Graphs
The study of networks has witnessed an explosive growth over the past decades
with several ground-breaking methods introduced. A particularly interesting --
and prevalent in several fields of study -- problem is that of inferring a
function defined over the nodes of a network. This work presents a versatile
kernel-based framework for tackling this inference problem that naturally
subsumes and generalizes the reconstruction approaches put forth recently by
the signal processing on graphs community. Both the static and the dynamic
settings are considered along with effective modeling approaches for addressing
real-world problems. The herein analytical discussion is complemented by a set
of numerical examples, which showcase the effectiveness of the presented
techniques, as well as their merits related to state-of-the-art methods.Comment: To be published as a chapter in `Adaptive Learning Methods for
Nonlinear System Modeling', Elsevier Publishing, Eds. D. Comminiello and J.C.
Principe (2018). This chapter surveys recent work on kernel-based inference
of functions over graphs including arXiv:1612.03615 and arXiv:1605.07174 and
arXiv:1711.0930
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