High-dimensional Sparse Inverse Covariance Estimation using Greedy Methods

In this paper we consider the task of estimating the non-zero pattern of the sparse inverse covariance matrix of a zero-mean Gaussian random vector from a set of iid samples. Note that this is also equivalent to recovering the underlying graph structure of a sparse Gaussian Markov Random Field (GMRF). We present two novel greedy approaches to solving this problem. The first estimates the non-zero covariates of the overall inverse covariance matrix using a series of global forward and backward greedy steps. The second estimates the neighborhood of each node in the graph separately, again using greedy forward and backward steps, and combines the intermediate neighborhoods to form an overall estimate. The principal contribution of this paper is a rigorous analysis of the sparsistency, or consistency in recovering the sparsity pattern of the inverse covariance matrix. Surprisingly, we show that both the local and global greedy methods learn the full structure of the model with high probability given just $O(d\log(p))$ samples, which is a \emph{significant} improvement over state of the art $\ell_1$-regularized Gaussian MLE (Graphical Lasso) that requires $O(d^2\log(p))$ samples. Moreover, the restricted eigenvalue and smoothness conditions imposed by our greedy methods are much weaker than the strong irrepresentable conditions required by the $\ell_1$-regularization based methods. We corroborate our results with extensive simulations and examples, comparing our local and global greedy methods to the $\ell_1$-regularized Gaussian MLE as well as the Neighborhood Greedy method to that of nodewise $\ell_1$-regularized linear regression (Neighborhood Lasso).Comment: Accepted to AI STAT 2012 for Oral Presentatio

Violent Hard X-ray Variability of Mrk 421 Observed by NuSTAR in 2013 April

The well studied blazar Markarian 421 (Mrk 421, $z$=0.031) was the subject of an intensive multi-wavelength campaign when it flared in 2013 April. The recorded X-ray and very high energy (VHE, E$>$100 GeV) $\gamma$-ray fluxes are the highest ever measured from this object. At the peak of the activity, it was monitored by the hard X-ray focusing telescope {\it Nuclear Spectroscopic Telescope Array} ({\it NuSTAR}) and {\it Swift} X-Ray Telescope (XRT). In this work, we present a detailed variability analysis of {\it NuSTAR} and {\it Swift}-XRT observations of Mrk 421 during this flaring episode. We obtained the shortest flux doubling time of 14.01$\pm$5.03 minutes, which is the shortest hard X-ray (3$-$79 keV) variability ever recorded from Mrk 421 and is on the order of the light crossing time of the black hole's event horizon. A pattern of extremely fast variability events superposed on slowly varying flares is found in most of the {\it NuSTAR} observations. We suggest that these peculiar variability patterns may be explained by magnetic energy dissipation and reconnection in a fast moving compact emission region within the jet. Based on the fast variability, we derive a lower limit on the magnetic field strength of $B \ge 0.73 \delta_1^{-2/3} \, \nu_{19}^{1/3}$~G, where $\delta_1$ is the Doppler factor in units of 10, and $\nu_{19}$ is the characteristic X-ray synchrotron frequency in units of $10^{19}$~Hz.Comment: 23 pages, 5 figures, 2 tables, to appear in the Astrophysical Journa

Black Hole Mass Limits for Optically Dark X-ray Bright Sources in Elliptical Galaxies

Estimation of the black hole mass in bright X-ray sources of nearby galaxies is crucial to the understanding of these systems and their formation. However, the present allowed black hole mass range spans five order of magnitude (10Msun < M < 10^5 Msun) with the upper limit obtained from dynamical friction arguments. We show that the absence of a detectable optical counterpart for some of these sources, can provide a much more stringent upper limit. The argument is based only on the assumption that the outer regions of their accretion disks is a standard one. Moreover, such optically dark X-ray sources cannot be foreground stars or background active galactic nuclei, and hence must be accreting systems residing within their host galaxies. As a demonstration we search for candidates among the point-like X-ray sources detected with Chandra in thirteen nearby elliptical galaxies. We use a novel technique to search for faint optical counterparts in the HST images whereby we subtract the bright galaxy light based on isophotal modeling of the surface brightness. We show that for six sources with no detectable optical emission at the 3$-$sigma level, their black hole masses M_{BH} < 5000Msun. In particular, an ultra-luminous X-ray source (ULX) in NGC 4486 has M_{BH} < 1244 Msun. We discuss the potential of this method to provide stringent constraints on the black hole masses, and the implications on the physical nature of these sources.Comment: 11 Pages, 1 figure, Accepted for publication in Ap

Superconductivity-driven magnetization modulation in YBa2Cu3O7-{\delta} /SrTiO3/La0.67Sr0.33MnO3 heterostructures

Using spin polarized neutron reflectivity experiments, we demonstrate an unusual proximity behaviour when the superconductor (SC) and the ferromagnet (FM) are coupled through an insulator (I) in YBa2Cu3O7-{\delta} (SC)/SrTiO3 (I)/La0.67Sr0.33MnO3 (FM) heterostructures. We have observed an unexpected magnetic modulation at the interface region of the FM below the superconducting transition temperature. The magnetization of the FM layer at the I/FM interface was drastically reduced as compared to the magnetization in the rest of the FM layer. This result indicates that the Cooper pairs tunnel across the insulator and interact with the local magnetization at the interface region (extending ~ 30 {\AA}) of the FM causing modification of the magnetization at the interface. This unexpected magnetic behavior cannot be explained on the basis of the existing theoretical models. However, the length scale associated here clearly suggests the long range proximity effect as a result of tunneling of Cooper pairs.Comment: 6 figure

Comparison of History Effects in Magnetization in Weakly pinned Crystals of high-TcT_c and low-Tc_c Superconductors

A comparison of the history effects in weakly pinned single crystals of a high $T_c$ YBa$_2$Cu$_3$O$_{7 - \delta}$ (for H $\parallel$ c) and a low $T_c$ Ca$_3$Rh$_4$Sn$_{13}$, which show anomalous variations in critical current density $J_c(H)$ are presented via tracings of the minor magnetization hysteresis loops using a vibrating sample magnetometer. The sample histories focussed are, (i) the field cooled (FC), (ii) the zero field cooled (ZFC) and (iii) an isothermal reversal of field from the normal state. An understanding of the results in terms of the modulation in the plastic deformation of the elastic vortex solid and supercooling across order-disorder transition is sought.Comment: Presented in IWCC-200