180 research outputs found
Sparsity-based Recovery of Finite Alphabet Solutions to Underdetermined Linear Systems
International audienceWe consider the problem of estimating a deterministic finite alphabet vector f from underdetermined measurements y = A f, where A is a given (random) n x M matrix. Two new convex optimization methods are introduced for the recovery of finite alphabet signals via l1-norm minimization. The first method is based on regularization. In the second approach, the problem is formulated as the recovery of sparse signals after a suitable sparse transform. The regularization-based method is less complex than the transform-based one. When the alphabet size equals 2 and (n,N) grows proportionally, the conditions under which the signal will be recovered with high probability are the same for the two methods. When p > 2, the behavior of the transform-based method is established. Experimental results support this theoretical result and show that the transform method outperforms the regularization-based one
The Simplest Solution to an Underdetermined System of Linear Equations
Consider a d*n matrix A, with d<n. The problem of solving for x in y=Ax is
underdetermined, and has infinitely many solutions (if there are any). Given y,
the minimum Kolmogorov complexity solution (MKCS) of the input x is defined to
be an input z (out of many) with minimum Kolmogorov-complexity that satisfies
y=Az. One expects that if the actual input is simple enough, then MKCS will
recover the input exactly. This paper presents a preliminary study of the
existence and value of the complexity level up to which such a complexity-based
recovery is possible. It is shown that for the set of all d*n binary matrices
(with entries 0 or 1 and d<n), MKCS exactly recovers the input for an
overwhelming fraction of the matrices provided the Kolmogorov complexity of the
input is O(d). A weak converse that is loose by a log n factor is also
established for this case. Finally, we investigate the difficulty of finding a
matrix that has the property of recovering inputs with complexity of O(d) using
MKCS.Comment: Proceedings of the IEEE International Symposium on Information Theory
Seattle, Washington, July 9-14, 200
Recovery of binary sparse signals from compressed linear measurements via polynomial optimization
The recovery of signals with finite-valued components from few linear
measurements is a problem with widespread applications and interesting
mathematical characteristics. In the compressed sensing framework, tailored
methods have been recently proposed to deal with the case of finite-valued
sparse signals. In this work, we focus on binary sparse signals and we propose
a novel formulation, based on polynomial optimization. This approach is
analyzed and compared to the state-of-the-art binary compressed sensing
methods
Underdetermined Source Separation of Finite Alphabet Signals Via L1 Minimization
International audienceThis paper addresses the underdetermined source separation problem of finite alphabet signals. We present a new framework for recovering finite alphabet signals. We formulate this problem as a recovery of sparse signals from highly incomplete measurements. It is known that sparse solutions can be obtained by L_1 minimization, through convex optimization. This relaxation procedure in our problem fails in recovering sparse solutions. However, this does not impact the reconstruction of the finite alphabet signals. Simulation results are presented to show that this approach provides good recovery properties and good images separation performance
Identifiability for Blind Source Separation of Multiple Finite Alphabet Linear Mixtures
We give under weak assumptions a complete combinatorial characterization of
identifiability for linear mixtures of finite alphabet sources, with unknown
mixing weights and unknown source signals, but known alphabet. This is based on
a detailed treatment of the case of a single linear mixture. Notably, our
identifiability analysis applies also to the case of unknown number of sources.
We provide sufficient and necessary conditions for identifiability and give a
simple sufficient criterion together with an explicit construction to determine
the weights and the source signals for deterministic data by taking advantage
of the hierarchical structure within the possible mixture values. We show that
the probability of identifiability is related to the distribution of a hitting
time and converges exponentially fast to one when the underlying sources come
from a discrete Markov process. Finally, we explore our theoretical results in
a simulation study. Our work extends and clarifies the scope of scenarios for
which blind source separation becomes meaningful
Sparse Signal Processing Concepts for Efficient 5G System Design
As it becomes increasingly apparent that 4G will not be able to meet the
emerging demands of future mobile communication systems, the question what
could make up a 5G system, what are the crucial challenges and what are the key
drivers is part of intensive, ongoing discussions. Partly due to the advent of
compressive sensing, methods that can optimally exploit sparsity in signals
have received tremendous attention in recent years. In this paper we will
describe a variety of scenarios in which signal sparsity arises naturally in 5G
wireless systems. Signal sparsity and the associated rich collection of tools
and algorithms will thus be a viable source for innovation in 5G wireless
system design. We will discribe applications of this sparse signal processing
paradigm in MIMO random access, cloud radio access networks, compressive
channel-source network coding, and embedded security. We will also emphasize
important open problem that may arise in 5G system design, for which sparsity
will potentially play a key role in their solution.Comment: 18 pages, 5 figures, accepted for publication in IEEE Acces
Reconstruction par transformation parcimonieuse de solutions à alphabet fini de systèmes linéaires sous-déterminés
National audienceNous considérons le problème d'estimer un vecteur à alphabet fini à partir d'un système sous-déterminé y = Af , où A est une matrice aléatoire générique réelle donnée de dimension n × N . Une méthode originale par optimisation convexe est proposée pour reconstruire le vecteur par minimisation L1 . Cette méthode est basée sur une transformation du problème dans un domaine où la solution recherchée est parcimonieuse. Le comportement théorique de cette méthode est donné et illustrée expérimentalement
Compression-Based Compressed Sensing
Modern compression algorithms exploit complex structures that are present in
signals to describe them very efficiently. On the other hand, the field of
compressed sensing is built upon the observation that "structured" signals can
be recovered from their under-determined set of linear projections. Currently,
there is a large gap between the complexity of the structures studied in the
area of compressed sensing and those employed by the state-of-the-art
compression codes. Recent results in the literature on deterministic signals
aim at bridging this gap through devising compressed sensing decoders that
employ compression codes. This paper focuses on structured stochastic processes
and studies the application of rate-distortion codes to compressed sensing of
such signals. The performance of the formerly-proposed compressible signal
pursuit (CSP) algorithm is studied in this stochastic setting. It is proved
that in the very low distortion regime, as the blocklength grows to infinity,
the CSP algorithm reliably and robustly recovers instances of a stationary
process from random linear projections as long as their count is slightly more
than times the rate-distortion dimension (RDD) of the source. It is also
shown that under some regularity conditions, the RDD of a stationary process is
equal to its information dimension (ID). This connection establishes the
optimality of the CSP algorithm at least for memoryless stationary sources, for
which the fundamental limits are known. Finally, it is shown that the CSP
algorithm combined by a family of universal variable-length fixed-distortion
compression codes yields a family of universal compressed sensing recovery
algorithms
Sparsity Enhanced Decision Feedback Equalization
For single-carrier systems with frequency domain equalization, decision
feedback equalization (DFE) performs better than linear equalization and has
much lower computational complexity than sequence maximum likelihood detection.
The main challenge in DFE is the feedback symbol selection rule. In this paper,
we give a theoretical framework for a simple, sparsity based thresholding
algorithm. We feed back multiple symbols in each iteration, so the algorithm
converges fast and has a low computational cost. We show how the initial
solution can be obtained via convex relaxation instead of linear equalization,
and illustrate the impact that the choice of the initial solution has on the
bit error rate performance of our algorithm. The algorithm is applicable in
several existing wireless communication systems (SC-FDMA, MC-CDMA, MIMO-OFDM).
Numerical results illustrate significant performance improvement in terms of
bit error rate compared to the MMSE solution
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