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 $k,N$, an $m$-by-$N$ measurement matrix, $\Phi$, and a decoding algorithm, $\mathcal{D}$. Given a vector, $x$, the system approximates $x$ by $\widehat x =\mathcal{D}(\Phi x)$, which must satisfy $\| \widehat x - x\|_2\le C \|x - x_k\|_2$, where $x_k$ denotes the optimal $k$-term approximation to $x$. For each vector $x$, the system must succeed with probability at least 3/4. Among the goals in designing such systems are minimizing the number $m$ of measurements and the runtime of the decoding algorithm, $\mathcal{D}$. In this paper, we give a system with $m=O(k \log(N/k))$ measurements--matching a lower bound, up to a constant factor--and decoding time $O(k\log^c N)$, matching a lower bound up to $\log(N)$ factors. We also consider the encode time (i.e., the time to multiply $\Phi$ by $x$), the time to update measurements (i.e., the time to multiply $\Phi$ by a 1-sparse $x$), and the robustness and stability of the algorithm (adding noise before and after the measurements). Our encode and update times are optimal up to $\log(N)$ 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, $f_{RL}$, as a function of host galaxy stellar mass, $M_*$, 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 $f^{LE}_{RL} \propto M^{2.5}_*$ 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 $f^{HE}_{RL} \propto M^{1.5}_*$ 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