Deanna Needell

Uniform Uncertainty Principle and Signal Recovery via Regularized Orthogonal Matching Pursuit

This paper seeks to bridge the two major algorithmic approaches to sparse signal recovery from an incomplete set of linear measurements—L1-minimization methods and iterative methods (Matching Pursuits). We find a simple regularized version of Orthogonal Matching Pursuit (ROMP) which has advantages of both approaches: the speed and transparency of OMP and the strong uniform guarantees of L1-minimization. Our algorithm, ROMP, reconstructs a sparse signal in a number of iterations linear in the sparsity, and the reconstruction is exact provided the linear measurements satisfy the uniform uncertainty principle.

CoSaMP: iterative signal recovery from incomplete and inaccurate samples

Compressive sampling (CoSa) is a new paradigm for developing data sampling technologies. It is based on the principle that many types of vector-space data are compressible, which is a term of art in mathematical signal processing. The key ideas are that randomized dimension reduction preserves the information in a compressible signal and that it is possible to develop hardware devices that implement this dimension reduction efficiently. The main computational challenge in CoSa is to reconstruct a compressible signal from the reduced representation acquired by the sampling device. This extended abstract describes a recent algorithm, called, CoSaMP , that accomplishes the data recovery task. It was the first known method to offer near-optimal guarantees on resource usage.

Signal Recovery From Incomplete and Inaccurate Measurements Via Regularized Orthogonal Matching Pursuit

We demonstrate a simple greedy algorithm that can reliably recover a vector <i>v</i> ¿ ¿^d from incomplete and inaccurate measurements <i>x</i> = ¿<i>v</i> + <i>e</i>. Here, ¿ is a <i>N</i> x <i>d</i> measurement matrix with <i>N</i><<d, and <i>e</i> is an error vector. Our algorithm, Regularized Orthogonal Matching Pursuit (ROMP), seeks to provide the benefits of the two major approaches to sparse recovery. It combines the speed and ease of implementation of the greedy methods with the strong guarantees of the convex programming methods. For any measurement matrix ¿ that satisfies a quantitative restricted isometry principle, ROMP recovers a signal <i>v</i> with <i>O</i>(<i>n</i>) nonzeros from its inaccurate measurements <i>x</i> in at most <i>n</i> iterations, where each iteration amounts to solving a least squares problem. The noise level of the recovery is proportional to ¿{log<i>n</i>} ||<i>e</i>||₂. In particular, if the error term <i>e</i> vanishes the reconstruction is exact.

Stable Image Reconstruction Using Total Variation Minimization

This paper presents near-optimal guarantees for stable and robust image recovery from undersampled noisy measurements using total variation minimization. In particular, we show that from

Stochastic Gradient Descent, Weighted Sampling, and the Randomized Kaczmarz algorithm

O(s\log(N))

Randomized Kaczmarz solver for noisy linear systems

nonadaptive linear measurements, an image can be reconstructed to within the best

Signal Space CoSaMP for Sparse Recovery With Redundant Dictionaries

s

Greedy signal recovery review

-term approximation of its gradient up to a logarithmic factor, and this factor can be removed by taking slightly more measurements. Along the way, we prove a strengthened Sobolev inequality for functions lying in the null space of a suitably incoherent matrix.

Paved with Good Intentions: Analysis of a Randomized Block Kaczmarz Method

We obtain an improved finite-sample guarantee on the linear convergence of stochastic gradient descent for smooth and strongly convex objectives, improving from a quadratic dependence on the conditioning