Recovery of Sparse Signals Using Multiple Orthogonal Least Squares

We study the problem of recovering sparse signals from compressed linear measurements. This problem, often referred to as sparse recovery or sparse reconstruction, has generated a great deal of interest in recent years. To recover the sparse signals, we propose a new method called multiple orthogonal least squares (MOLS), which extends the well-known orthogonal least squares (OLS) algorithm by allowing multiple $L$ indices to be chosen per iteration. Owing to inclusion of multiple support indices in each selection, the MOLS algorithm converges in much fewer iterations and improves the computational efficiency over the conventional OLS algorithm. Theoretical analysis shows that MOLS ($L > 1$) performs exact recovery of all $K$-sparse signals within $K$ iterations if the measurement matrix satisfies the restricted isometry property (RIP) with isometry constant $\delta_{LK} < \frac{\sqrt{L}}{\sqrt{K} + 2 \sqrt{L}}.$ The recovery performance of MOLS in the noisy scenario is also studied. It is shown that stable recovery of sparse signals can be achieved with the MOLS algorithm when the signal-to-noise ratio (SNR) scales linearly with the sparsity level of input signals

Structured Sparsity Models for Multiparty Speech Recovery from Reverberant Recordings

We tackle the multi-party speech recovery problem through modeling the acoustic of the reverberant chambers. Our approach exploits structured sparsity models to perform room modeling and speech recovery. We propose a scheme for characterizing the room acoustic from the unknown competing speech sources relying on localization of the early images of the speakers by sparse approximation of the spatial spectra of the virtual sources in a free-space model. The images are then clustered exploiting the low-rank structure of the spectro-temporal components belonging to each source. This enables us to identify the early support of the room impulse response function and its unique map to the room geometry. To further tackle the ambiguity of the reflection ratios, we propose a novel formulation of the reverberation model and estimate the absorption coefficients through a convex optimization exploiting joint sparsity model formulated upon spatio-spectral sparsity of concurrent speech representation. The acoustic parameters are then incorporated for separating individual speech signals through either structured sparse recovery or inverse filtering the acoustic channels. The experiments conducted on real data recordings demonstrate the effectiveness of the proposed approach for multi-party speech recovery and recognition.Comment: 31 page

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

Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit

This paper demonstrates theoretically and empirically that a greedy algorithm called Orthogonal Matching Pursuit (OMP) can reliably recover a signal with $m$ nonzero entries in dimension $d$ given ${rm O}(m ln d)$ random linear measurements of that signal. This is a massive improvement over previous results, which require ${rm O}(m^{2})$ measurements. The new results for OMP are comparable with recent results for another approach called Basis Pursuit (BP). In some settings, the OMP algorithm is faster and easier to implement, so it is an attractive alternative to BP for signal recovery problems

Projection-Based and Look Ahead Strategies for Atom Selection

In this paper, we improve iterative greedy search algorithms in which atoms are selected serially over iterations, i.e., one-by-one over iterations. For serial atom selection, we devise two new schemes to select an atom from a set of potential atoms in each iteration. The two new schemes lead to two new algorithms. For both the algorithms, in each iteration, the set of potential atoms is found using a standard matched filter. In case of the first scheme, we propose an orthogonal projection strategy that selects an atom from the set of potential atoms. Then, for the second scheme, we propose a look ahead strategy such that the selection of an atom in the current iteration has an effect on the future iterations. The use of look ahead strategy requires a higher computational resource. To achieve a trade-off between performance and complexity, we use the two new schemes in cascade and develop a third new algorithm. Through experimental evaluations, we compare the proposed algorithms with existing greedy search and convex relaxation algorithms.Comment: sparsity, compressive sensing; IEEE Trans on Signal Processing 201

Dynamic Iterative Pursuit

For compressive sensing of dynamic sparse signals, we develop an iterative pursuit algorithm. A dynamic sparse signal process is characterized by varying sparsity patterns over time/space. For such signals, the developed algorithm is able to incorporate sequential predictions, thereby providing better compressive sensing recovery performance, but not at the cost of high complexity. Through experimental evaluations, we observe that the new algorithm exhibits a graceful degradation at deteriorating signal conditions while capable of yielding substantial performance gains as conditions improve.Comment: 6 pages, 7 figures. Accepted for publication in IEEE Transactions on Signal Processin