5,790 research outputs found
Adaptive Graph Signal Processing: Algorithms and Optimal Sampling Strategies
The goal of this paper is to propose novel strategies for adaptive learning
of signals defined over graphs, which are observed over a (randomly
time-varying) subset of vertices. We recast two classical adaptive algorithms
in the graph signal processing framework, namely, the least mean squares (LMS)
and the recursive least squares (RLS) adaptive estimation strategies. For both
methods, a detailed mean-square analysis illustrates the effect of random
sampling on the adaptive reconstruction capability and the steady-state
performance. Then, several probabilistic sampling strategies are proposed to
design the sampling probability at each node in the graph, with the aim of
optimizing the tradeoff between steady-state performance, graph sampling rate,
and convergence rate of the adaptive algorithms. Finally, a distributed RLS
strategy is derived and is shown to be convergent to its centralized
counterpart. Numerical simulations carried out over both synthetic and real
data illustrate the good performance of the proposed sampling and
reconstruction strategies for (possibly distributed) adaptive learning of
signals defined over graphs.Comment: Submitted to IEEE Transactions on Signal Processing, September 201
Local Measurement and Reconstruction for Noisy Graph Signals
The emerging field of signal processing on graph plays a more and more
important role in processing signals and information related to networks.
Existing works have shown that under certain conditions a smooth graph signal
can be uniquely reconstructed from its decimation, i.e., data associated with a
subset of vertices. However, in some potential applications (e.g., sensor
networks with clustering structure), the obtained data may be a combination of
signals associated with several vertices, rather than the decimation. In this
paper, we propose a new concept of local measurement, which is a generalization
of decimation. Using the local measurements, a local-set-based method named
iterative local measurement reconstruction (ILMR) is proposed to reconstruct
bandlimited graph signals. It is proved that ILMR can reconstruct the original
signal perfectly under certain conditions. The performance of ILMR against
noise is theoretically analyzed. The optimal choice of local weights and a
greedy algorithm of local set partition are given in the sense of minimizing
the expected reconstruction error. Compared with decimation, the proposed local
measurement sampling and reconstruction scheme is more robust in noise existing
scenarios.Comment: 24 pages, 6 figures, 2 tables, journal manuscrip
Local-set-based Graph Signal Reconstruction
Signal processing on graph is attracting more and more attentions. For a
graph signal in the low-frequency subspace, the missing data associated with
unsampled vertices can be reconstructed through the sampled data by exploiting
the smoothness of the graph signal. In this paper, the concept of local set is
introduced and two local-set-based iterative methods are proposed to
reconstruct bandlimited graph signal from sampled data. In each iteration, one
of the proposed methods reweights the sampled residuals for different vertices,
while the other propagates the sampled residuals in their respective local
sets. These algorithms are built on frame theory and the concept of local sets,
based on which several frames and contraction operators are proposed. We then
prove that the reconstruction methods converge to the original signal under
certain conditions and demonstrate the new methods lead to a significantly
faster convergence compared with the baseline method. Furthermore, the
correspondence between graph signal sampling and time-domain irregular sampling
is analyzed comprehensively, which may be helpful to future works on graph
signals. Computer simulations are conducted. The experimental results
demonstrate the effectiveness of the reconstruction methods in various sampling
geometries, imprecise priori knowledge of cutoff frequency, and noisy
scenarios.Comment: 28 pages, 9 figures, 6 tables, journal manuscrip
A Distributed Tracking Algorithm for Reconstruction of Graph Signals
The rapid development of signal processing on graphs provides a new
perspective for processing large-scale data associated with irregular domains.
In many practical applications, it is necessary to handle massive data sets
through complex networks, in which most nodes have limited computing power.
Designing efficient distributed algorithms is critical for this task. This
paper focuses on the distributed reconstruction of a time-varying bandlimited
graph signal based on observations sampled at a subset of selected nodes. A
distributed least square reconstruction (DLSR) algorithm is proposed to recover
the unknown signal iteratively, by allowing neighboring nodes to communicate
with one another and make fast updates. DLSR uses a decay scheme to annihilate
the out-of-band energy occurring in the reconstruction process, which is
inevitably caused by the transmission delay in distributed systems. Proof of
convergence and error bounds for DLSR are provided in this paper, suggesting
that the algorithm is able to track time-varying graph signals and perfectly
reconstruct time-invariant signals. The DLSR algorithm is numerically
experimented with synthetic data and real-world sensor network data, which
verifies its ability in tracking slowly time-varying graph signals.Comment: 30 pages, 9 figures, 2 tables, journal pape
Graph Spectral Image Processing
Recent advent of graph signal processing (GSP) has spurred intensive studies
of signals that live naturally on irregular data kernels described by graphs
(e.g., social networks, wireless sensor networks). Though a digital image
contains pixels that reside on a regularly sampled 2D grid, if one can design
an appropriate underlying graph connecting pixels with weights that reflect the
image structure, then one can interpret the image (or image patch) as a signal
on a graph, and apply GSP tools for processing and analysis of the signal in
graph spectral domain. In this article, we overview recent graph spectral
techniques in GSP specifically for image / video processing. The topics covered
include image compression, image restoration, image filtering and image
segmentation
Forecasting Time Series with VARMA Recursions on Graphs
Graph-based techniques emerged as a choice to deal with the dimensionality
issues in modeling multivariate time series. However, there is yet no complete
understanding of how the underlying structure could be exploited to ease this
task. This work provides contributions in this direction by considering the
forecasting of a process evolving over a graph. We make use of the
(approximate) time-vertex stationarity assumption, i.e., timevarying graph
signals whose first and second order statistical moments are invariant over
time and correlated to a known graph topology. The latter is combined with VAR
and VARMA models to tackle the dimensionality issues present in predicting the
temporal evolution of multivariate time series. We find out that by projecting
the data to the graph spectral domain: (i) the multivariate model estimation
reduces to that of fitting a number of uncorrelated univariate ARMA models and
(ii) an optimal low-rank data representation can be exploited so as to further
reduce the estimation costs. In the case that the multivariate process can be
observed at a subset of nodes, the proposed models extend naturally to Kalman
filtering on graphs allowing for optimal tracking. Numerical experiments with
both synthetic and real data validate the proposed approach and highlight its
benefits over state-of-the-art alternatives.Comment: submitted to the IEEE Transactions on Signal Processin
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