34,552 research outputs found
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, or 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 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
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