438 research outputs found
Discrete Signal Processing on Graphs: Frequency Analysis
Signals and datasets that arise in physical and engineering applications, as
well as social, genetics, biomolecular, and many other domains, are becoming
increasingly larger and more complex. In contrast to traditional time and image
signals, data in these domains are supported by arbitrary graphs. Signal
processing on graphs extends concepts and techniques from traditional signal
processing to data indexed by generic graphs. This paper studies the concepts
of low and high frequencies on graphs, and low-, high-, and band-pass graph
filters. In traditional signal processing, there concepts are easily defined
because of a natural frequency ordering that has a physical interpretation. For
signals residing on graphs, in general, there is no obvious frequency ordering.
We propose a definition of total variation for graph signals that naturally
leads to a frequency ordering on graphs and defines low-, high-, and band-pass
graph signals and filters. We study the design of graph filters with specified
frequency response, and illustrate our approach with applications to sensor
malfunction detection and data classification
Signal Recovery on Graphs: Random versus Experimentally Designed Sampling
We study signal recovery on graphs based on two sampling strategies: random
sampling and experimentally designed sampling. We propose a new class of smooth
graph signals, called approximately bandlimited, which generalizes the
bandlimited class and is similar to the globally smooth class. We then propose
two recovery strategies based on random sampling and experimentally designed
sampling. The proposed recovery strategy based on experimentally designed
sampling is similar to the leverage scores used in the matrix approximation. We
show that while both strategies are unbiased estimators for the low-frequency
components, the convergence rate of experimentally designed sampling is much
faster than that of random sampling when a graph is irregular. We validate the
proposed recovery strategies on three specific graphs: a ring graph, an
Erd\H{o}s-R\'enyi graph, and a star graph. The simulation results support the
theoretical analysis.Comment: Correct some typo
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
A Connectedness Constraint for Learning Sparse Graphs
Graphs are naturally sparse objects that are used to study many problems
involving networks, for example, distributed learning and graph signal
processing. In some cases, the graph is not given, but must be learned from the
problem and available data. Often it is desirable to learn sparse graphs.
However, making a graph highly sparse can split the graph into several
disconnected components, leading to several separate networks. The main
difficulty is that connectedness is often treated as a combinatorial property,
making it hard to enforce in e.g. convex optimization problems. In this
article, we show how connectedness of undirected graphs can be formulated as an
analytical property and can be enforced as a convex constraint. We especially
show how the constraint relates to the distributed consensus problem and graph
Laplacian learning. Using simulated and real data, we perform experiments to
learn sparse and connected graphs from data.Comment: 5 pages, presented at the European Signal Processing Conference
(EUSIPCO) 201
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