19 research outputs found
Sampling Large Data on Graphs
We consider the problem of sampling from data defined on the nodes of a
weighted graph, where the edge weights capture the data correlation structure.
As shown recently, using spectral graph theory one can define a cut-off
frequency for the bandlimited graph signals that can be reconstructed from a
given set of samples (i.e., graph nodes). In this work, we show how this
cut-off frequency can be computed exactly. Using this characterization, we
provide efficient algorithms for finding the subset of nodes of a given size
with the largest cut-off frequency and for finding the smallest subset of nodes
with a given cut-off frequency. In addition, we study the performance of random
uniform sampling when compared to the centralized optimal sampling provided by
the proposed algorithms.Comment: To be presented at GlobalSIP 201
Sampling and Reconstruction of Graph Signals via Weak Submodularity and Semidefinite Relaxation
We study the problem of sampling a bandlimited graph signal in the presence
of noise, where the objective is to select a node subset of prescribed
cardinality that minimizes the signal reconstruction mean squared error (MSE).
To that end, we formulate the task at hand as the minimization of MSE subject
to binary constraints, and approximate the resulting NP-hard problem via
semidefinite programming (SDP) relaxation. Moreover, we provide an alternative
formulation based on maximizing a monotone weak submodular function and propose
a randomized-greedy algorithm to find a sub-optimal subset. We then derive a
worst-case performance guarantee on the MSE returned by the randomized greedy
algorithm for general non-stationary graph signals. The efficacy of the
proposed methods is illustrated through numerical simulations on synthetic and
real-world graphs. Notably, the randomized greedy algorithm yields an
order-of-magnitude speedup over state-of-the-art greedy sampling schemes, while
incurring only a marginal MSE performance loss
Graph Signal Processing: Overview, Challenges and Applications
Research in Graph Signal Processing (GSP) aims to develop tools for
processing data defined on irregular graph domains. In this paper we first
provide an overview of core ideas in GSP and their connection to conventional
digital signal processing. We then summarize recent developments in developing
basic GSP tools, including methods for sampling, filtering or graph learning.
Next, we review progress in several application areas using GSP, including
processing and analysis of sensor network data, biological data, and
applications to image processing and machine learning. We finish by providing a
brief historical perspective to highlight how concepts recently developed in
GSP build on top of prior research in other areas.Comment: To appear, Proceedings of the IEE
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