Randomized Kaczmarz solver for noisy linear systems

The Kaczmarz method is an iterative algorithm for solving systems of linear equations Ax=b. Theoretical convergence rates for this algorithm were largely unknown until recently when work was done on a randomized version of the algorithm. It was proved that for overdetermined systems, the randomized Kaczmarz method converges with expected exponential rate, independent of the number of equations in the system. Here we analyze the case where the system Ax=b is corrupted by noise, so we consider the system where Ax is approximately b + r where r is an arbitrary error vector. We prove that in this noisy version, the randomized method reaches an error threshold dependent on the matrix A with the same rate as in the error-free case. We provide examples showing our results are sharp in the general context

Sparse Randomized Kaczmarz for Support Recovery of Jointly Sparse Corrupted Multiple Measurement Vectors

While single measurement vector (SMV) models have been widely studied in signal processing, there is a surging interest in addressing the multiple measurement vectors (MMV) problem. In the MMV setting, more than one measurement vector is available and the multiple signals to be recovered share some commonalities such as a common support. Applications in which MMV is a naturally occurring phenomenon include online streaming, medical imaging, and video recovery. This work presents a stochastic iterative algorithm for the support recovery of jointly sparse corrupted MMV. We present a variant of the Sparse Randomized Kaczmarz algorithm for corrupted MMV and compare our proposed method with an existing Kaczmarz type algorithm for MMV problems. We also showcase the usefulness of our approach in the online (streaming) setting and provide empirical evidence that suggests the robustness of the proposed method to the distribution of the corruption and the number of corruptions occurring.Comment: 13 pages, 6 figure

An optimally concentrated Gabor transform for localized time-frequency components

Gabor analysis is one of the most common instances of time-frequency signal analysis. Choosing a suitable window for the Gabor transform of a signal is often a challenge for practical applications, in particular in audio signal processing. Many time-frequency (TF) patterns of different shapes may be present in a signal and they can not all be sparsely represented in the same spectrogram. We propose several algorithms, which provide optimal windows for a user-selected TF pattern with respect to different concentration criteria. We base our optimization algorithm on $l^p$-norms as measure of TF spreading. For a given number of sampling points in the TF plane we also propose optimal lattices to be used with the obtained windows. We illustrate the potentiality of the method on selected numerical examples

Symmetric Informationally Complete Quantum Measurements

We consider the existence in arbitrary finite dimensions d of a POVM comprised of d^2 rank-one operators all of whose operator inner products are equal. Such a set is called a ``symmetric, informationally complete'' POVM (SIC-POVM) and is equivalent to a set of d^2 equiangular lines in C^d. SIC-POVMs are relevant for quantum state tomography, quantum cryptography, and foundational issues in quantum mechanics. We construct SIC-POVMs in dimensions two, three, and four. We further conjecture that a particular kind of group-covariant SIC-POVM exists in arbitrary dimensions, providing numerical results up to dimension 45 to bolster this claim.Comment: 8 page

Restricted Isometries for Partial Random Circulant Matrices

In the theory of compressed sensing, restricted isometry analysis has become a standard tool for studying how efficiently a measurement matrix acquires information about sparse and compressible signals. Many recovery algorithms are known to succeed when the restricted isometry constants of the sampling matrix are small. Many potential applications of compressed sensing involve a data-acquisition process that proceeds by convolution with a random pulse followed by (nonrandom) subsampling. At present, the theoretical analysis of this measurement technique is lacking. This paper demonstrates that the $s$th order restricted isometry constant is small when the number $m$ of samples satisfies $m \gtrsim (s \log n)^{3/2}$, where $n$ is the length of the pulse. This bound improves on previous estimates, which exhibit quadratic scaling

Colour reconnection in e+e- -> W+W- at sqrt(s) = 189 - 209 GeV

The effects of the final state interaction phenomenon known as colour reconnection are investigated at centre-of-mass energies in the range sqrt(s) ~ 189-209 GeV using the OPAL detector at LEP. Colour reconnection is expected to affect observables based on charged particles in hadronic decays of W+W-. Measurements of inclusive charged particle multiplicities, and of their angular distribution with respect to the four jet axes of the events, are used to test models of colour reconnection. The data are found to exclude extreme scenarios of the Sjostrand-Khoze Type I (SK-I) model and are compatible with other models, both with and without colour reconnection effects. In the context of the SK-I model, the best agreement with data is obtained for a reconnection probability of 37%. Assuming no colour reconnection, the charged particle multiplicity in hadronically decaying W bosons is measured to be (nqqch) = 19.38+-0.05(stat.)+-0.08 (syst.).Comment: 30 pages, 9 figures, Submitted to Euro. Phys. J.

Scaling violations of quark and gluon jet fragmentation functions in e+e- annihilations at sqrt(s) = 91.2 and 183-209 GeV

Flavour inclusive, udsc and b fragmentation functions in unbiased jets, and flavour inclusive, udsc, b and gluon fragmentation functions in biased jets are measured in e+e- annihilations from data collected at centre-of-mass energies of 91.2, and 183-209 GeV with the OPAL detector at LEP. The unbiased jets are defined by hemispheres of inclusive hadronic events, while the biased jet measurements are based on three-jet events selected with jet algorithms. Several methods are employed to extract the fragmentation functions over a wide range of scales. Possible biases are studied in the results are obtained. The fragmentation functions are compared to results from lower energy e+e- experiments and with earlier LEP measurements and are found to be consistent. Scaling violations are observed and are found to be stronger for the fragmentation functions of gluon jets than for those of quarks. The measured fragmentation functions are compared to three recent theoretical next-to-leading order calculations and to the predictions of three Monte Carlo event generators. While the Monte Carlo models are in good agreement with the data, the theoretical predictions fail to describe the full set of results, in particular the b and gluon jet measurements.Comment: 46 pages, 17 figures, Submitted to Eur. Phys J.

Observation of Hadronic W Decays in t-tbar Events with the Collider Detector at Fermilab

We observe hadronic W decays in t-tbar -> W (-> l nu) + >= 4 jet events using a 109 pb-1 data sample of p-pbar collisions at sqrt{s} = 1.8 TeV collected with the Collider Detector at Fermilab (CDF). A peak in the dijet invariant mass distribution is obtained that is consistent with W decay and inconsistent with the background prediction by 3.3 standard deviations. From this peak we measure the W mass to be 77.2 +- 4.6 (stat+syst) GeV/c^2. This result demonstrates the presence of two W bosons in t-tbar candidates in the W (-> l nu) + >= 4 jet channel.Comment: 20 pages, 4 figures, submitted to PR

Tests of model of color reconnection and a search for glueballs using gluon jets with a rapidity gap

Gluon jets with a mean energy of 22 GeV and purity of 95% are selected from hadronic Z0 decay events produced in e+e- annihilations. A subsample of these jets is identified which exhibits a large gap in the rapidity distribution of particles within the jet. After imposing the requirement of a rapidity gap, the gluon jet purity is 86%. These jets are observed to demonstrate a high degree of sensitivity to the presence of color reconnection, i.e. higher order QCD processes affecting the underlying color structure. We use our data to test three QCD models which include a simulation of color reconnection: one in the Ariadne Monte Carlo, one in the Herwig Monte Carlo, and the other by Rathsman in the Pythia Monte Carlo. We find the Rathsman and Ariadne color reconnection models can describe our gluon jet measurements only if very large values are used for the cutoff parameters which serve to terminate the parton showers, and that the description of inclusive Z0 data is significantly degraded in this case. We conclude that color reconnection as implemented by these two models is disfavored. The signal from the Herwig color reconnection model is less clear and we do not obtain a definite conclusion concerning this model. In a separate study, we follow recent theoretical suggestions and search for glueball-like objects in the leading part of the gluon jets. No clear evidence is observed for these objects.Comment: 42 pages, 18 figure