78 research outputs found
Rank Awareness in Joint Sparse Recovery
This paper revisits the sparse multiple measurement vector (MMV) problem, where the aim is to recover a set of jointly sparse multichannel vectors from incomplete measurements. This problem is an extension of single channel sparse recovery, which lies at the heart of compressed sensing. Inspired by the links to array signal processing, a new family of MMV algorithms is considered that highlight the role of rank in determining the difficulty of the MMV recovery problem. The simplest such method is a discrete version of MUSIC which is guaranteed to recover the sparse vectors in the full rank MMV setting, under mild conditions. This idea is extended to a rank aware pursuit algorithm that naturally reduces to Order Recursive Matching Pursuit (ORMP) in the single measurement case while also providing guaranteed recovery in the full rank setting. In contrast, popular MMV methods such as Simultaneous Orthogonal Matching Pursuit (SOMP) and mixed norm minimization techniques are shown to be rank blind in terms of worst case analysis. Numerical simulations demonstrate that the rank aware techniques are significantly better than existing methods in dealing with multiple measurements
Joint Sparsity Recovery for Spectral Compressed Sensing
Compressed Sensing (CS) is an effective approach to reduce the required
number of samples for reconstructing a sparse signal in an a priori basis, but
may suffer severely from the issue of basis mismatch. In this paper we study
the problem of simultaneously recovering multiple spectrally-sparse signals
that are supported on the same frequencies lying arbitrarily on the unit
circle. We propose an atomic norm minimization problem, which can be regarded
as a continuous counterpart of the discrete CS formulation and be solved
efficiently via semidefinite programming. Through numerical experiments, we
show that the number of samples per signal may be further reduced by harnessing
the joint sparsity pattern of multiple signals
Joint Sparsity with Different Measurement Matrices
We consider a generalization of the multiple measurement vector (MMV)
problem, where the measurement matrices are allowed to differ across
measurements. This problem arises naturally when multiple measurements are
taken over time, e.g., and the measurement modality (matrix) is time-varying.
We derive probabilistic recovery guarantees showing that---under certain (mild)
conditions on the measurement matrices---l2/l1-norm minimization and a variant
of orthogonal matching pursuit fail with a probability that decays
exponentially in the number of measurements. This allows us to conclude that,
perhaps surprisingly, recovery performance does not suffer from the individual
measurements being taken through different measurement matrices. What is more,
recovery performance typically benefits (significantly) from diversity in the
measurement matrices; we specify conditions under which such improvements are
obtained. These results continue to hold when the measurements are subject to
(bounded) noise.Comment: Allerton 201
Model Selection for Nonnegative Matrix Factorization by Support Union Recovery
Nonnegative matrix factorization (NMF) has been widely used in machine
learning and signal processing because of its non-subtractive, part-based
property which enhances interpretability. It is often assumed that the latent
dimensionality (or the number of components) is given. Despite the large amount
of algorithms designed for NMF, there is little literature about automatic
model selection for NMF with theoretical guarantees. In this paper, we propose
an algorithm that first calculates an empirical second-order moment from the
empirical fourth-order cumulant tensor, and then estimates the latent
dimensionality by recovering the support union (the index set of non-zero rows)
of a matrix related to the empirical second-order moment. By assuming a
generative model of the data with additional mild conditions, our algorithm
provably detects the true latent dimensionality. We show on synthetic examples
that our proposed algorithm is able to find an approximately correct number of
components
Random Access in C-RAN for User Activity Detection with Limited-Capacity Fronthaul
Cloud-Radio Access Network (C-RAN) is characterized by a hierarchical
structure in which the baseband processing functionalities of remote radio
heads (RRHs) are implemented by means of cloud computing at a Central Unit
(CU). A key limitation of C-RANs is given by the capacity constraints of the
fronthaul links connecting RRHs to the CU. In this letter, the impact of this
architectural constraint is investigated for the fundamental functions of
random access and active User Equipment (UE) identification in the presence of
a potentially massive number of UEs. In particular, the standard C-RAN approach
based on quantize-and-forward and centralized detection is compared to a scheme
based on an alternative CU-RRH functional split that enables local detection.
Both techniques leverage Bayesian sparse detection. Numerical results
illustrate the relative merits of the two schemes as a function of the system
parameters.Comment: 6 pages, 3 figures, under revision in IEEE Signal Processing Letter
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