Detecting and quantifying stellar magnetic fields -- Sparse Stokes profile approximation using orthogonal matching pursuit

In the recent years, we have seen a rapidly growing number of stellar magnetic field detections for various types of stars. Many of these magnetic fields are estimated from spectropolarimetric observations (Stokes V) by using the so-called center-of-gravity (COG) method. Unfortunately, the accuracy of this method rapidly deteriorates with increasing noise and thus calls for a more robust procedure that combines signal detection and field estimation. We introduce an estimation method that provides not only the effective or mean longitudinal magnetic field from an observed Stokes V profile but also uses the net absolute polarization of the profile to obtain an estimate of the apparent (i.e., velocity resolved) absolute longitudinal magnetic field. By combining the COG method with an orthogonal-matching-pursuit (OMP) approach, we were able to decompose observed Stokes profiles with an overcomplete dictionary of wavelet-basis functions to reliably reconstruct the observed Stokes profiles in the presence of noise. The elementary wave functions of the sparse reconstruction process were utilized to estimate the effective longitudinal magnetic field and the apparent absolute longitudinal magnetic field. A multiresolution analysis complements the OMP algorithm to provide a robust detection and estimation method. An extensive Monte-Carlo simulation confirms the reliability and accuracy of the magnetic OMP approach.Comment: A&A, in press, 15 pages, 14 figure

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

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 -- L_1-minimization methods and iterative methods (Matching Pursuits). We find a simple regularized version of the Orthogonal Matching Pursuit (ROMP) which has advantages of both approaches: the speed and transparency of OMP and the strong uniform guarantees of the L_1-minimization. Our algorithm ROMP reconstructs a sparse signal in a number of iterations linear in the sparsity (in practice even logarithmic), and the reconstruction is exact provided the linear measurements satisfy the Uniform Uncertainty Principle.Comment: This is the final version of the paper, including referee suggestion

Polar Polytopes and Recovery of Sparse Representations

Suppose we have a signal y which we wish to represent using a linear combination of a number of basis atoms a_i, y=sum_i x_i a_i = Ax. The problem of finding the minimum L0 norm representation for y is a hard problem. The Basis Pursuit (BP) approach proposes to find the minimum L1 norm representation instead, which corresponds to a linear program (LP) that can be solved using modern LP techniques, and several recent authors have given conditions for the BP (minimum L1 norm) and sparse (minimum L0 solutions) representations to be identical. In this paper, we explore this sparse representation problem} using the geometry of convex polytopes, as recently introduced into the field by Donoho. By considering the dual LP we find that the so-called polar polytope P of the centrally-symmetric polytope P whose vertices are the atom pairs +-a_i is particularly helpful in providing us with geometrical insight into optimality conditions given by Fuchs and Tropp for non-unit-norm atom sets. In exploring this geometry we are able to tighten some of these earlier results, showing for example that the Fuchs condition is both necessary and sufficient for L1-unique-optimality, and that there are situations where Orthogonal Matching Pursuit (OMP) can eventually find all L1-unique-optimal solutions with m nonzeros even if ERC fails for m, if allowed to run for more than m steps

Measurements design and phenomena discrimination

The construction of measurements suitable for discriminating signal components produced by phenomena of different types is considered. The required measurements should be capable of cancelling out those signal components which are to be ignored when focusing on a phenomenon of interest. Under the hypothesis that the subspaces hosting the signal components produced by each phenomenon are complementary, their discrimination is accomplished by measurements giving rise to the appropriate oblique projector operator. The subspace onto which the operator should project is selected by nonlinear techniques in line with adaptive pursuit strategies