11,906 research outputs found
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
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
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
Sparse Sampling for Inverse Problems with Tensors
We consider the problem of designing sparse sampling strategies for
multidomain signals, which can be represented using tensors that admit a known
multilinear decomposition. We leverage the multidomain structure of tensor
signals and propose to acquire samples using a Kronecker-structured sensing
function, thereby circumventing the curse of dimensionality. For designing such
sensing functions, we develop low-complexity greedy algorithms based on
submodular optimization methods to compute near-optimal sampling sets. We
present several numerical examples, ranging from multi-antenna communications
to graph signal processing, to validate the developed theory.Comment: 13 pages, 7 figure
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