42 research outputs found
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
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 -constrained minimal singular value (-CMSV) of the
measurement matrix determines, in a very concise manner, the recovery
performance of -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 -CMSVs bounded away from zero, as
long as the number of measurements is relatively large. To compute the
-CMSV and its lower bound, we design two algorithms based on the
interior point algorithm and the semi-definite relaxation
Computable Performance Analysis of Recovering Signals with Low-dimensional Structures
The last decade witnessed the burgeoning development in the reconstruction of signals by exploiting their low-dimensional structures, particularly, the sparsity, the block-sparsity, the low-rankness, and the low-dimensional manifold structures of general nonlinear data sets. The reconstruction performance of these signals relies heavily on the structure of the sensing matrix/operator. In many applications, there is a flexibility to select the optimal sensing matrix among a class of them. A prerequisite for optimal sensing matrix design is the computability of the performance for different recovery algorithms. I present a computational framework for analyzing the recovery performance of signals with low-dimensional structures. I define a family of goodness measures for arbitrary sensing matrices as the optimal values of a set of optimization problems. As one of the primary contributions of this work, I associate the goodness measures with the fixed points of functions defined by a series of linear programs, second-order cone programs, or semidefinite programs, depending on the specific problem. This relation with the fixed-point theory, together with a bisection search implementation, yields efficient algorithms to compute the goodness measures with global convergence guarantees. As a by-product, we implement efficient algorithms to verify sufficient conditions for exact signal recovery in the noise-free case. The implementations perform orders-of-magnitude faster than the state-of-the-art techniques. The utility of these goodness measures lies in their relation with the reconstruction performance. I derive bounds on the recovery errors of convex relaxation algorithms in terms of these goodness measures. Using tools from empirical processes and generic chaining, I analytically demonstrate that as long as the number of measurements are relatively large, these goodness measures are bounded away from zeros for a large class of random sensing matrices, a result parallel to the probabilistic analysis of the restricted isometry property. Numerical experiments show that, compared with the restricted isometry based performance bounds, our error bounds apply to a wider range of problems and are tighter, when the sparsity levels of the signals are relatively low. I expect that computable performance bounds would open doors for wide applications in compressive sensing, sensor arrays, radar, MRI, image processing, computer vision, collaborative filtering, control, and many other areas where low-dimensional signal structures arise naturally
Atomic norm denoising with applications to line spectral estimation
Motivated by recent work on atomic norms in inverse problems, we propose a
new approach to line spectral estimation that provides theoretical guarantees
for the mean-squared-error (MSE) performance in the presence of noise and
without knowledge of the model order. We propose an abstract theory of
denoising with atomic norms and specialize this theory to provide a convex
optimization problem for estimating the frequencies and phases of a mixture of
complex exponentials. We show that the associated convex optimization problem
can be solved in polynomial time via semidefinite programming (SDP). We also
show that the SDP can be approximated by an l1-regularized least-squares
problem that achieves nearly the same error rate as the SDP but can scale to
much larger problems. We compare both SDP and l1-based approaches with
classical line spectral analysis methods and demonstrate that the SDP
outperforms the l1 optimization which outperforms MUSIC, Cadzow's, and Matrix
Pencil approaches in terms of MSE over a wide range of signal-to-noise ratios.Comment: 27 pages, 10 figures. A preliminary version of this work appeared in
the Proceedings of the 49th Annual Allerton Conference in September 2011.
Numerous numerical experiments added to this version in accordance with
suggestions by anonymous reviewer