9,012 research outputs found
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
Approximate Sparse Recovery: Optimizing Time and Measurements
An approximate sparse recovery system consists of parameters , an
-by- measurement matrix, , and a decoding algorithm, .
Given a vector, , the system approximates by , which must satisfy , where denotes the optimal -term approximation to . For
each vector , the system must succeed with probability at least 3/4. Among
the goals in designing such systems are minimizing the number of
measurements and the runtime of the decoding algorithm, .
In this paper, we give a system with
measurements--matching a lower bound, up to a constant factor--and decoding
time , matching a lower bound up to factors.
We also consider the encode time (i.e., the time to multiply by ),
the time to update measurements (i.e., the time to multiply by a
1-sparse ), and the robustness and stability of the algorithm (adding noise
before and after the measurements). Our encode and update times are optimal up
to factors
Four-dimensional light shaping: manipulating ultrafast spatio-temporal foci in space and time
Spectral dispersion of ultrashort pulses allows simultaneous focusing of
light in both space and time creating so-called spatio-temporal foci. Such
space-time coupling may be combined with existing holographic techniques to
give a further dimension of control when generating focal light fields. It is
shown that a phase-only hologram placed in the pupil plane of an objective and
illuminated by a spatially chirped ultrashort pulse can be used to generate
three dimensional arrays of spatio-temporally focused spots. Exploiting the
pulse front tilt generated at focus when applying simultaneous spatial and
temporal focusing (SSTF), it is possible to overlap neighbouring foci in time
to create a smooth intensity distribution. The resulting light field displays a
high level of axial confinement, with experimental demonstrations given through
two-photon microscopy and non-linear laser fabrication of glass
Networks of strong ties
Social networks transmitting covert or sensitive information cannot use all
ties for this purpose. Rather, they can only use a subset of ties that are
strong enough to be ``trusted''. In this paper we consider transitivity as
evidence of strong ties, requiring that each tie can only be used if the
individuals on either end also share at least one other contact in common. We
examine the effect of removing all non-transitive ties in two real social
network data sets. We observe that although some individuals become
disconnected, a giant connected component remains, with an average shortest
path only slightly longer than that of the original network. We also evaluate
the cost of forming transitive ties by deriving the conditions for the
emergence and the size of the giant component in a random graph composed
entirely of closed triads and the equivalent Erdos-Renyi random graph.Comment: 10 pages, 7 figure
The triggering probability of radio-loud AGN: A comparison of high and low excitation radio galaxies in hosts of different colors
Low luminosity radio-loud active galactic nuclei (AGN) are generally found in
massive red elliptical galaxies, where they are thought to be powered through
gas accretion from their surrounding hot halos in a radiatively inefficient
manner. These AGN are often referred to as "low-excitation" radio galaxies
(LERGs). When radio-loud AGN are found in galaxies with a young stellar
population and active star formation, they are usually high-power
radiatively-efficient radio AGN ("high-excitation", HERG). Using a sample of
low-redshift radio galaxies identified within the Sloan Digital Sky Survey
(SDSS), we determine the fraction of galaxies that host a radio-loud AGN,
, as a function of host galaxy stellar mass, , star formation
rate, color (defined by the 4000 \angstrom break strength), radio luminosity
and excitation state (HERG/LERG).
We find the following: 1. LERGs are predominantly found in red galaxies. 2.
The radio-loud AGN fraction of LERGs hosted by galaxies of any color follows a
power law. 3. The fraction of red galaxies
hosting a LERG decreases strongly for increasing radio luminosity. For massive
blue galaxies this is not the case. 4. The fraction of green galaxies hosting a
LERG is lower than that of either red or blue galaxies, at all radio
luminosities. 5. The radio-loud AGN fraction of HERGs hosted by galaxies of any
color follows a power law. 6. HERGs have a
strong preference to be hosted by green or blue galaxies. 7. The fraction of
galaxies hosting a HERG shows only a weak dependence on radio luminosity cut.
8. For both HERGs and LERGs, the hosting probability of blue galaxies shows a
strong dependence on star formation rate. This is not observed in galaxies of a
different color.[abridged]Comment: 7 pages, 6 figure
Skewness as a probe of non-Gaussian initial conditions
We compute the skewness of the matter distribution arising from non-linear
evolution and from non-Gaussian initial perturbations. We apply our result to a
very generic class of models with non-Gaussian initial conditions and we
estimate analytically the ratio between the skewness due to non-linear
clustering and the part due to the intrinsic non-Gaussianity of the models. We
finally extend our estimates to higher moments.Comment: 5 pages, 2 ps-figs., accepted for publication in PRD, rapid com
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