2,541 research outputs found
Simultaneous sparse approximation via greedy pursuit
A simple sparse approximation problem requests an approximation of a given input signal as a linear combination of T elementary signals drawn from a large, linearly dependent collection. An important generalization is simultaneous sparse approximation. Now one must approximate several input signals at once using different linear combinations of the same T elementary signals. This formulation appears, for example, when analyzing multiple observations of a sparse signal that have been contaminated with noise. A new approach to this problem is presented here: a greedy pursuit algorithm called simultaneous orthogonal matching pursuit. The paper proves that the algorithm calculates simultaneous approximations whose error is within a constant factor of the optimal simultaneous approximation error. This result requires that the collection of elementary signals be weakly correlated, a property that is also known as incoherence. Numerical experiments demonstrate that the algorithm often succeeds, even when the inputs do not meet the hypotheses of the proof
Simultaneously Sparse Solutions to Linear Inverse Problems with Multiple System Matrices and a Single Observation Vector
A linear inverse problem is proposed that requires the determination of
multiple unknown signal vectors. Each unknown vector passes through a different
system matrix and the results are added to yield a single observation vector.
Given the matrices and lone observation, the objective is to find a
simultaneously sparse set of unknown vectors that solves the system. We will
refer to this as the multiple-system single-output (MSSO) simultaneous sparsity
problem. This manuscript contrasts the MSSO problem with other simultaneous
sparsity problems and conducts a thorough initial exploration of algorithms
with which to solve it. Seven algorithms are formulated that approximately
solve this NP-Hard problem. Three greedy techniques are developed (matching
pursuit, orthogonal matching pursuit, and least squares matching pursuit) along
with four methods based on a convex relaxation (iteratively reweighted least
squares, two forms of iterative shrinkage, and formulation as a second-order
cone program). The algorithms are evaluated across three experiments: the first
and second involve sparsity profile recovery in noiseless and noisy scenarios,
respectively, while the third deals with magnetic resonance imaging
radio-frequency excitation pulse design.Comment: 36 pages; manuscript unchanged from July 21, 2008, except for updated
references; content appears in September 2008 PhD thesi
Computational Methods for Sparse Solution of Linear Inverse Problems
The goal of the sparse approximation problem is to approximate a target signal using a linear combination of a few elementary signals drawn from a fixed collection. This paper surveys the major practical algorithms for sparse approximation. Specific attention is paid to computational issues, to the circumstances in which individual methods tend to perform well, and to the theoretical guarantees available. Many fundamental questions in electrical engineering, statistics, and applied mathematics can be posed as sparse approximation problems, making these algorithms versatile and relevant to a plethora of applications
Applications of sparse approximation in communications
Sparse approximation problems abound in many scientific, mathematical, and engineering applications. These problems are defined by two competing notions: we approximate a signal vector as a linear combination of elementary atoms and we require that the approximation be both as accurate and as concise as possible. We introduce two natural and direct applications of these problems and algorithmic solutions in communications. We do so by constructing enhanced codebooks from base codebooks. We show that we can decode these enhanced codebooks in the presence of Gaussian noise. For MIMO wireless communication channels, we construct simultaneous sparse approximation problems and demonstrate that our algorithms can both decode the transmitted signals and estimate the channel parameters
Self-Dictionary Sparse Regression for Hyperspectral Unmixing: Greedy Pursuit and Pure Pixel Search are Related
This paper considers a recently emerged hyperspectral unmixing formulation
based on sparse regression of a self-dictionary multiple measurement vector
(SD-MMV) model, wherein the measured hyperspectral pixels are used as the
dictionary. Operating under the pure pixel assumption, this SD-MMV formalism is
special in that it allows simultaneous identification of the endmember spectral
signatures and the number of endmembers. Previous SD-MMV studies mainly focus
on convex relaxations. In this study, we explore the alternative of greedy
pursuit, which generally provides efficient and simple algorithms. In
particular, we design a greedy SD-MMV algorithm using simultaneous orthogonal
matching pursuit. Intriguingly, the proposed greedy algorithm is shown to be
closely related to some existing pure pixel search algorithms, especially, the
successive projection algorithm (SPA). Thus, a link between SD-MMV and pure
pixel search is revealed. We then perform exact recovery analyses, and prove
that the proposed greedy algorithm is robust to noise---including its
identification of the (unknown) number of endmembers---under a sufficiently low
noise level. The identification performance of the proposed greedy algorithm is
demonstrated through both synthetic and real-data experiments
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