1,495 research outputs found
CoSaMP: Iterative signal recovery from incomplete and inaccurate samples
Compressive sampling offers a new paradigm for acquiring signals that are
compressible with respect to an orthonormal basis. The major algorithmic
challenge in compressive sampling is to approximate a compressible signal from
noisy samples. This paper describes a new iterative recovery algorithm called
CoSaMP that delivers the same guarantees as the best optimization-based
approaches. Moreover, this algorithm offers rigorous bounds on computational
cost and storage. It is likely to be extremely efficient for practical problems
because it requires only matrix-vector multiplies with the sampling matrix. For
many cases of interest, the running time is just O(N*log^2(N)), where N is the
length of the signal.Comment: 30 pages. Revised. Presented at Information Theory and Applications,
31 January 2008, San Dieg
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
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
Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions
Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed—either explicitly or
implicitly—to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired low-rank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, robustness, and/or speed. These claims are supported by extensive numerical experiments and a detailed error analysis. The specific benefits of randomized techniques depend on the computational environment. Consider the model problem of finding the k dominant components of the singular value decomposition of an m × n matrix. (i) For a dense input matrix, randomized algorithms require O(mn log(k))
floating-point operations (flops) in contrast to O(mnk) for classical algorithms. (ii) For a sparse input matrix, the flop count matches classical Krylov subspace methods, but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures. (iii) For a matrix that is too large to fit in fast memory, the randomized techniques require only a constant number of passes over the data, as opposed to O(k) passes for classical algorithms. In fact, it is sometimes possible to perform matrix approximation with a single pass over the data
Dictionary learning with large step gradient descent for sparse representations
This is the accepted version of an article published in Lecture Notes in Computer Science Volume 7191, 2012, pp 231-238. The final publication is available at link.springer.com
http://www.springerlink.com/content/l1k4514765283618
Improved sparse approximation over quasi-incoherent dictionaries
This paper discusses a new greedy algorithm for solving the sparse approximation problem over quasi-incoherent dictionaries. These dictionaries consist of waveforms that are uncorrelated "on average," and they provide a natural generalization of incoherent dictionaries. The algorithm provides strong guarantees on the quality of the approximations it produces, unlike most other methods for sparse approximation. Moreover, very efficient implementations are possible via approximate nearest-neighbor data structure
Algorithmic linear dimension reduction in the l_1 norm for sparse vectors
This paper develops a new method for recovering m-sparse signals that is
simultaneously uniform and quick. We present a reconstruction algorithm whose
run time, O(m log^2(m) log^2(d)), is sublinear in the length d of the signal.
The reconstruction error is within a logarithmic factor (in m) of the optimal
m-term approximation error in l_1. In particular, the algorithm recovers
m-sparse signals perfectly and noisy signals are recovered with polylogarithmic
distortion. Our algorithm makes O(m log^2 (d)) measurements, which is within a
logarithmic factor of optimal. We also present a small-space implementation of
the algorithm. These sketching techniques and the corresponding reconstruction
algorithms provide an algorithmic dimension reduction in the l_1 norm. In
particular, vectors of support m in dimension d can be linearly embedded into
O(m log^2 d) dimensions with polylogarithmic distortion. We can reconstruct a
vector from its low-dimensional sketch in time O(m log^2(m) log^2(d)).
Furthermore, this reconstruction is stable and robust under small
perturbations
A foundation for analytical developments in the logarithmic region of turbulent channels
An analytical framework for studying the logarithmic region of turbulent
channels is formulated. We build on recent findings (Moarref et al., J. Fluid
Mech., 734, 2013) that the velocity fluctuations in the logarithmic region can
be decomposed into a weighted sum of geometrically self-similar resolvent
modes. The resolvent modes and the weights represent the linear amplification
mechanisms and the scaling influence of the nonlinear interactions in the
Navier-Stokes equations (NSE), respectively (McKeon & Sharma, J. Fluid Mech.,
658, 2010). Originating from the NSE, this framework provides an analytical
support for Townsend's attached-eddy model. Our main result is that
self-similarity enables order reduction in modeling the logarithmic region by
establishing a quantitative link between the self-similar structures and the
velocity spectra. Specifically, the energy intensities, the Reynolds stresses,
and the energy budget are expressed in terms of the resolvent modes with speeds
corresponding to the top of the logarithmic region. The weights of the triad
modes -the modes that directly interact via the quadratic nonlinearity in the
NSE- are coupled via the interaction coefficients that depend solely on the
resolvent modes (McKeon et al., Phys. Fluids, 25, 2013). We use the hierarchies
of self-similar modes in the logarithmic region to extend the notion of triad
modes to triad hierarchies. It is shown that the interaction coefficients for
the triad modes that belong to a triad hierarchy follow an exponential
function. The combination of these findings can be used to better understand
the dynamics and interaction of flow structures in the logarithmic region. The
compatibility of the proposed model with theoretical and experimental results
is further discussed.Comment: Submitted to J. Fluid Mec
- …