Simultaneous sparse approximation via greedy pursuit

A simple sparse approximation problem requests an approximation of a given input signal as a linear combination of T elementary signals drawn from a large, linearly dependent collection. An important generalization is simultaneous sparse approximation. Now one must approximate several input signals at once using different linear combinations of the same T elementary signals. This formulation appears, for example, when analyzing multiple observations of a sparse signal that have been contaminated with noise. A new approach to this problem is presented here: a greedy pursuit algorithm called simultaneous orthogonal matching pursuit. The paper proves that the algorithm calculates simultaneous approximations whose error is within a constant factor of the optimal simultaneous approximation error. This result requires that the collection of elementary signals be weakly correlated, a property that is also known as incoherence. Numerical experiments demonstrate that the algorithm often succeeds, even when the inputs do not meet the hypotheses of the proof

Improved sparse approximation over quasi-incoherent dictionaries

This paper discusses a new greedy algorithm for solving the sparse approximation problem over quasi-incoherent dictionaries. These dictionaries consist of waveforms that are uncorrelated "on average," and they provide a natural generalization of incoherent dictionaries. The algorithm provides strong guarantees on the quality of the approximations it produces, unlike most other methods for sparse approximation. Moreover, very efficient implementations are possible via approximate nearest-neighbor data structure

Algorithmic linear dimension reduction in the l_1 norm for sparse vectors

This paper develops a new method for recovering m-sparse signals that is simultaneously uniform and quick. We present a reconstruction algorithm whose run time, O(m log^2(m) log^2(d)), is sublinear in the length d of the signal. The reconstruction error is within a logarithmic factor (in m) of the optimal m-term approximation error in l_1. In particular, the algorithm recovers m-sparse signals perfectly and noisy signals are recovered with polylogarithmic distortion. Our algorithm makes O(m log^2 (d)) measurements, which is within a logarithmic factor of optimal. We also present a small-space implementation of the algorithm. These sketching techniques and the corresponding reconstruction algorithms provide an algorithmic dimension reduction in the l_1 norm. In particular, vectors of support m in dimension d can be linearly embedded into O(m log^2 d) dimensions with polylogarithmic distortion. We can reconstruct a vector from its low-dimensional sketch in time O(m log^2(m) log^2(d)). Furthermore, this reconstruction is stable and robust under small perturbations

List decoding of noisy Reed-Muller-like codes

First- and second-order Reed-Muller (RM(1) and RM(2), respectively) codes are two fundamental error-correcting codes which arise in communication as well as in probabilistically-checkable proofs and learning. In this paper, we take the first steps toward extending the quick randomized decoding tools of RM(1) into the realm of quadratic binary and, equivalently, Z_4 codes. Our main algorithmic result is an extension of the RM(1) techniques from Goldreich-Levin and Kushilevitz-Mansour algorithms to the Hankel code, a code between RM(1) and RM(2). That is, given signal s of length N, we find a list that is a superset of all Hankel codewords phi with dot product to s at least (1/sqrt(k)) times the norm of s, in time polynomial in k and log(N). We also give a new and simple formulation of a known Kerdock code as a subcode of the Hankel code. As a corollary, we can list-decode Kerdock, too. Also, we get a quick algorithm for finding a sparse Kerdock approximation. That is, for k small compared with 1/sqrt{N} and for epsilon > 0, we find, in time polynomial in (k log(N)/epsilon), a k-Kerdock-term approximation s~ to s with Euclidean error at most the factor (1+epsilon+O(k^2/sqrt{N})) times that of the best such approximation

Shellflow. I. The Convergence of the Velocity Field at 6000 km/s

We present the first results from the Shellflow program, an all-sky Tully-Fisher (TF) peculiar velocity survey of 276 Sb-Sc galaxies with redshifts between 4500 and 7000 km/s. Shellflow was designed to minimize systematic errors between observing runs and between telescopes, thereby removing the possibility of a spurious bulk flow caused by data inhomogeneity. A fit to the data yields a bulk flow amplitude V_bulk = 70{+100}{-70} km/s (1 sigma error) with respect to the Cosmic Microwave Background, i.e., consistent with being at rest. At the 95% confidence level, the flow amplitude is < 300 km/s. Our results are insensitive to which Galactic extinction maps we use, and to the parameterization of the TF relation. The larger bulk motion found in analyses of the Mark III peculiar velocity catalog are thus likely to be due to non-uniformities between the subsamples making up Mark III. The absence of bulk flow is consistent with the study of Giovanelli and collaborators and flow field predictions from the observed distribution of IRAS galaxies.Comment: Accepted version for publication in ApJ. Includes an epitaph for Jeffrey Alan Willick (Oct 8, 1959 - Jun 18, 2000

An HI survey of the Bootes Void. II. The Analysis

We discuss the results of a VLA HI survey of the Bootes void and compare the distribution and HI properties of the void galaxies to those of galaxies found in a survey of regions of mean cosmic density. The Bootes survey covers 1100 Mpc$^{3}$, or $\sim$ 1\% of the volume of the void and consists of 24 cubes of typically 2 Mpc * 2 Mpc * 1280 km/s, centered on optically known galaxies. Sixteen targets were detected in HI; 18 previously uncataloged objects were discovered directly in HI. The control sample consists of 12 cubes centered on IRAS selected galaxies with FIR luminosities similar to those of the Bootes targets and located in regions of 1 to 2 times the cosmic mean density. In addition to the 12 targets 29 companions were detected in HI. We find that the number of galaxies within 1 Mpc of the targets is the same to within a factor of two for void and control samples, and thus that the small scale clustering of galaxies is the same in regions that differ by a factor of $\sim$ 6 in density on larger scales. A dynamical analysis of the galaxies in the void suggests that on scales of a few Mpc the galaxies are gravitationally bound, forming interacting galaxy pairs, loose pairs and loose groups. One group is compact enough to qualify as a Hickson compact group. The galaxies found in the void are mostly late-type, gas rich systems. A careful scrutiny of their HI and optical properties shows them to be very similar to field galaxies of the same morphological type. This, combined with our finding that the small scale clustering of the galaxies in the void is the same as in the field, suggests that it is the near environment that mostly affects the evolution of galaxies.Comment: Latex file of abstract. The postscript version of the complete paper (0.2 Mb in gzipped format) including all the figures can be retrieved from http://www.astro.rug.nl:80/~secr/ To appear in the February 1996 issue of the Astronomical Journa