Performance Analysis of Sparse Recovery Based on Constrained Minimal Singular Values

The stability of sparse signal reconstruction is investigated in this paper. We design efficient algorithms to verify the sufficient condition for unique $\ell_1$ sparse recovery. One of our algorithm produces comparable results with the state-of-the-art technique and performs orders of magnitude faster. We show that the $\ell_1$-constrained minimal singular value ($\ell_1$-CMSV) of the measurement matrix determines, in a very concise manner, the recovery performance of $\ell_1$-based algorithms such as the Basis Pursuit, the Dantzig selector, and the LASSO estimator. Compared with performance analysis involving the Restricted Isometry Constant, the arguments in this paper are much less complicated and provide more intuition on the stability of sparse signal recovery. We show also that, with high probability, the subgaussian ensemble generates measurement matrices with $\ell_1$-CMSVs bounded away from zero, as long as the number of measurements is relatively large. To compute the $\ell_1$-CMSV and its lower bound, we design two algorithms based on the interior point algorithm and the semi-definite relaxation

Sparse Recovery from Combined Fusion Frame Measurements

Sparse representations have emerged as a powerful tool in signal and information processing, culminated by the success of new acquisition and processing techniques such as Compressed Sensing (CS). Fusion frames are very rich new signal representation methods that use collections of subspaces instead of vectors to represent signals. This work combines these exciting fields to introduce a new sparsity model for fusion frames. Signals that are sparse under the new model can be compressively sampled and uniquely reconstructed in ways similar to sparse signals using standard CS. The combination provides a promising new set of mathematical tools and signal models useful in a variety of applications. With the new model, a sparse signal has energy in very few of the subspaces of the fusion frame, although it does not need to be sparse within each of the subspaces it occupies. This sparsity model is captured using a mixed l1/l2 norm for fusion frames. A signal sparse in a fusion frame can be sampled using very few random projections and exactly reconstructed using a convex optimization that minimizes this mixed l1/l2 norm. The provided sampling conditions generalize coherence and RIP conditions used in standard CS theory. It is demonstrated that they are sufficient to guarantee sparse recovery of any signal sparse in our model. Moreover, a probabilistic analysis is provided using a stochastic model on the sparse signal that shows that under very mild conditions the probability of recovery failure decays exponentially with increasing dimension of the subspaces

Room Helps: Acoustic Localization With Finite Elements

Acoustic source localization often relies on the free-space/far-field model. Recent work exploiting spatio-temporal sparsity promises to go beyond these scenarios, however, it requires the knowledge of the transfer functions from each possible source location to each microphone. We propose a method for indoor acoustic source localization in which the physical modeling is implicit. By approximating the wave equation with the finite element method (FEM), we naturally get a sparse recovery formulation of the source localization. We demonstrate how exploiting the bandwidth leads to improved performance and surprising results, such as localization of multiple sources with one microphone, or hearing around corners. Numerical simulation results show the feasibility of such schemes

Distributed target localization via spatial sparsity

We propose an approximation framework for distributed target localization in sensor networks. We represent the unknown target positions on a location grid as a sparse vector, whose support encodes the multiple target locations. The location vector is linearly related to multiple sensor measurements through a sensing matrix, which can be locally estimated at each sensor. We show that we can successfully determine multiple target locations by using linear dimensionality-reducing projections of sensor measurements. The overall communication bandwidth requirement per sensor is logarithmic in the number of grid points and linear in the number of targets, ameliorating the communication requirements. Simulations results demonstrate the performance of the proposed framework

A Note on Block-Sparse Signal Recovery with Coherent Tight Frames

This note discusses the recovery of signals from undersampled data in the situation that such signals are nearly block sparse in terms of an overcomplete and coherent tight frame D. By introducing the notion of block D-restricted isometry property (D-RIP), we establish several sufficient conditions for the proposed mixed l2/l1-analysis method to guarantee stable recovery of nearly block-sparse signals in terms of D. One of the main results of this note shows that if the measurement matrix satisfies the block D-RIP with constants δk<0.307, then the signals which are nearly block k-sparse in terms of D can be stably recovered via mixed l2/l1-analysis in the presence of noise

The Synchronized Short-Time-Fourier-Transform: Properties and Definitions for Multichannel Source Separation.

This paper proposes the use of a synchronized linear transform, the synchronized short-time-Fourier-transform (sSTFT), for time-frequency analysis of anechoic mixtures. We address the short comings of the commonly used time-frequency linear transform in multichannel settings, namely the classical short-time-Fourier-transform (cSTFT). We propose a series of desirable properties for the linear transform used in a multichannel source separation scenario: stationary invertibility, relative delay, relative attenuation, and finally delay invariant relative windowed-disjoint orthogonality (DIRWDO). Multisensor source separation techniques which operate in the time-frequency domain, have an inherent error unless consideration is given to the multichannel properties proposed in this paper. The sSTFT preserves these relationships for multichannel data. The crucial innovation of the sSTFT is to locally synchronize the analysis to the observations as opposed to a global clock. Improvement in separation performance can be achieved because assumed properties of the time-frequency transform are satisfied when it is appropriately synchronized. Numerical experiments show the sSTFT improves instantaneous subsample relative parameter estimation in low noise conditions and achieves good synthesis