Order Recursive Matching Pursuit Research Articles

Dictionary Learning for Two-Dimensional Kendall Shapes

We propose a novel sparse dictionary learning method for planar shapes in the sense of Kendall, namely configurations of landmarks in the plane considered up to similitudes. Our shape dictionary method provides a good trade-off between algorithmic simplicity and faithfulness with respect to the nonlinear geometric structure of Kendall's shape space. Remarkably, it boils down to a classical dictionary learning formulation modified using complex weights. Existing dictionary learning methods extended to nonlinear spaces map the manifold either to a reproducing kernel Hilbert space or to a tangent space. The first approach is unnecessarily heavy in the case of Kendall's shape space and causes the geometrical understanding of shapes to be lost, while the second one induces distortions and theoretical complexity. Our approach does not suffer from these drawbacks. Instead of embedding the shape space into a linear space, we rely on the hyperplane of centered configurations, including preshapes from which shapes are defined as rotation orbits. In this linear space, the dictionary atoms are scaled and rotated using complex weights before summation. Furthermore, our formulation is more general than Kendall's original one: it applies to discretely defined configurations of landmarks as well as continuously defined interpolating curves. We implemented our algorithm by adapting the method of optimal directions combined to a Cholesky-optimized order recursive matching pursuit. An interesting feature of our shape dictionary is that it produces visually realistic atoms, while guaranteeing reconstruction accuracy. Its efficiency can mostly be attributed to a clear formulation of the framework with complex numbers. We illustrate the strong potential of our approach for the characterization of datasets of shapes up to similitudes and the analysis of patterns in deforming two-dimensional shapes.

Nearly Optimal Restricted Isometry Condition for Rank Aware Order Recursive Matching Pursuit

In this paper, we analyze the performance guarantee of the rank aware order recursive matching pursuit (RA-ORMP) algorithm in recovering a group of jointly sparse vectors. Specifically, we show that RA-ORMP accurately reconstructs any group of r linearly independent jointly K-sparse vectors, provided that the sampling matrix has unit l 2 -norm columns and satisfies the restricted isometry property (RIP) of order K + 1 with δ K+1 <; √r/ (√/(K +(r/4)) + √(r/4). We show the near-optimality of the proposed guarantee by providing a group of r jointly K-sparse vectors that cannot be recovered by RA-ORMP under δ K+1 ≥ √(r/K). We also present a condition of RA-ORMP in the more realistic scenarios where a sampling matrix might not have unit l 2 -norm columns and the measurements are contaminated with noise.

A Fast and Noise-Robust Algorithm for Joint Sparse Recovery Through Information Transfer

In multiple measurement vector (MMV) problems, L measurement vectors each of which has length M are available for recovering jointly sparse signals that have a common support set of size K. In this paper, a fast and noise-robust greedy algorithm is proposed for joint sparse recovery in MMV problems, by exploiting a posteriori probability ratios for every index of sparse input signals. The essence of the algorithm is to transfer the information through iterations, which contributes to the performance improvement of support detection. When L is sufficiently large at M = K +1, we investigate the asymptotic performance of exact support recovery using a Gaussian assumption, power-law approximations, and order statistics, where the techniques are inspired by experimental results. In this case, we also present a sufficient condition on the number of measurements, or M = K + 1 = Ω (log N), for theoretical support recovery guarantee. The theoretical analysis reveals that the proposed algorithm can achieve reliable joint sparse recovery asymptotically at the theoretical limit of M = K + 1 with high probability. By examining the performance for various M, K, and L, simulation results demonstrate that if M is not too small, the proposed algorithm can be reliable, fast, and noise-robust, compared to the conventional ones, such as simultaneous orthogonal matching pursuit (SOMP), subspace augmented MUSIC (SA-MUSIC), and rank-aware order recursive matching pursuit (RA-ORMP).

Rank Awareness in Joint Sparse Recovery

This paper revisits the sparse multiple measurement vector (MMV) problem, where the aim is to recover a set of jointly sparse multichannel vectors from incomplete measurements. This problem is an extension of single channel sparse recovery, which lies at the heart of compressed sensing. Inspired by the links to array signal processing, a new family of MMV algorithms is considered that highlight the role of rank in determining the difficulty of the MMV recovery problem. The simplest such method is a discrete version of MUSIC which is guaranteed to recover the sparse vectors in the full rank MMV setting, under mild conditions. This idea is extended to a rank aware pursuit algorithm that naturally reduces to Order Recursive Matching Pursuit (ORMP) in the single measurement case while also providing guaranteed recovery in the full rank setting. In contrast, popular MMV methods such as Simultaneous Orthogonal Matching Pursuit (SOMP) and mixed norm minimization techniques are shown to be rank blind in terms of worst case analysis. Numerical simulations demonstrate that the rank aware techniques are significantly better than existing methods in dealing with multiple measurements.

Estimation of multipath time delays based on bi-sparse constraint

In this paper, a novel method to estimate the multi-path time delays of broadband signal transmission is proposed. This new method extends the existed basis selection methods. The difference between the problems discussed here and that of basis selection is the occurrence of an unknown weight matrix in the measurement equation, which represents the frequency characteristics of the received signal. Existed basis selection methods are insufficient to solve the problem. We introduced an iterative algorithm in consideration of bi-sparse constraint on solutions of a constructed equation and developed a method based on order recursive matching pursuit (ORMP) to obtain the solution. Both solution vectors and weight matrix are updated iteratively in our algorithm. Validation of the proposed method is confirmed by simulations.

Forward sequential algorithms for best basis selection

The problem of signal representation in terms of basis vectors from a large, over-complete, spanning dictionary has been the focus of much research. Achieving a succinct, or 'sparse', representation is known as the problem of best basis representation. Methods are considered which seek to solve this problem by sequentially building up a basis set for the signal. Three distinct algorithm types have appeared in the literature which are here termed basic matching pursuit (BMP), order recursive matching pursuit (ORMP) and modified matching pursuit (MMP). The algorithms are first described and then their computation is closely examined. Modifications are made to each of the procedures which improve their computational efficiency. The complexity of each algorithm is considered in two contexts; one where the dictionary is variable (time-dependent) and the other where the dictionary is fixed (time-independent). Experimental results are presented which demonstrate that the ORMP method is the best procedure in terms of its ability to give the most compact signal representation, followed by MMP and then BMP which gives the poorest results. Finally, weighing the performance of each algorithm, its computational complexity and the type of dictionary available, recommendations are made as to which algorithm should be used for a given problem.