Complexity of pattern classes and Lipschitz property

Rademacher and Gaussian complexities are successfully used in learning theory for measuring the capacity of the class of functions to be learned. One of the most important properties for these complexities is their Lipschitz property: a composition of a class of functions with a fixed Lipschitz function may increase its complexity by at most twice the Lipschitz constant. The proof of this property is non-trivial (in contrast to the other properties) and it is believed that the proof in the Gaussian case is conceptually more difficult then the one for the Rademacher case. In this paper we give a detailed prove of the Lipschitz property for the Rademacher case and generalize the same idea to an arbitrary complexity (including the Gaussian). We also discuss a related topic about the Rademacher complexity of a class consisting of all the Lipschitz functions with a given Lipschitz constant. We show that the complexity is surprisingly low in the one-dimensional case. The question for higher dimensions remains open

Hard x-ray polarimeter for gamma-ray bursts and solar flares

We report on the development of a dedicated polarimeter design that is capable of studying the linear polarization of hard X-rays (50-300 keV) from gamma-ray bursts and solar flares. This compact design, based on the use of a large area position-sensitive PMT (PSPMT), is referred to as GRAPE (Gamma-RAy Polarimeter Experiment). The PSPMT is used to determine the Compton interaction location within an array of small plastic scintillator elements. Some of the photons that scatter within the plastic scintillator array are subsequently absorbed by a small centrally-located array of CsI(Tl) crystals that is read out by an independent multi-anode PMT. One feature of GRAPE that is especially attractive for studies of gamma-ray bursts is the significant off-axis response (at angles \u3e 60 degrees). The modular nature of this design lends itself toward its accomodation on a balloon or spacecraft platform. For an array of GRAPE modules, sensitivity levels below a few percent can be achieved for both gamma-ray bursts and solar flares. Here we report on the latest results from the testing of a laboratory science model

The Development of GRAPE, a Gamma Ray Polarimeter Experiment

The measurement of hard X‐ray polarization in γ‐ray bursts (GRBs) would add yet another piece of information in our effort to resolve the true nature of these enigmatic objects. Here we report on the development of a dedicated polarimeter design with a relatively large FoV that is capable of studying hard X‐ray polarization (50–300 keV) from GRBs. This compact design, based on the use of a large area position‐sensitive PMT (PSPMT), is referred to as GRAPE (Gamma‐RAy Polarimeter Experiment). The feature of GRAPE that is especially attractive for studies of GRBs is the significant off‐axis polarization response (at angles greater than 60°). For an array of GRAPE modules, current sensitivity estimates give minimum detectable polarization (MDP) levels of a few percent for the brightest GRBs

Dedicated polarimeter design for hard x-ray and soft gamma-ray astronomy

We have developed a modular design for a hard X-ray and soft gamma-ray polrimeter that we call GRAPE (Gamma RAy Polarimeter Experiment). Optimized for the energy range of 50-300 keV, the GRAPE design is a Compton polarimeter based on the use of an array of plastic scintillator scattering elements in conjunction with a centrally positioned high-Z calorimeter detector. Here we shall review the results from a laboratory model of the baseline GRAPE design. The baseline design uses a 5-inch diameter position sensitive PMT (PSPMT) for readout of the plastic scintillator array and a small array of CsI detectors for measurement of the scattered photon. An improved design, based on the use of large area multi-anode PMTs (MAPMTs), is also discussed along with plans for laboratory testing of a prototype. An array of GRAPE modules could be used as the basis for a dedicated science mission, either on a long duration balloon or on an orbital mission. With a large effective FoV, a non-imaging GRAPE mission would be ideal for studying polarization in transient sources (gamma ray bursts and solar flares). It may also prove useful for studying periodically varying sources, such as pulsars. An imaging system would improve the sensitivity of the polarization measurements for transient and periodic sources and may also permit the measurement of polarization in steady-state sources

Eigenvalue variance bounds for Wigner and covariance random matrices

This work is concerned with finite range bounds on the variance of individual eigenvalues of Wigner random matrices, in the bulk and at the edge of the spectrum, as well as for some intermediate eigenvalues. Relying on the GUE example, which needs to be investigated first, the main bounds are extended to families of Hermitian Wigner matrices by means of the Tao and Vu Four Moment Theorem and recent localization results by Erd\"os, Yau and Yin. The case of real Wigner matrices is obtained from interlacing formulas. As an application, bounds on the expected 2-Wasserstein distance between the empirical spectral measure and the semicircle law are derived. Similar results are available for random covariance matrices

Optical/near-infrared selection of red QSOs: Evidence for steep extinction curves towards galactic centers?

We present the results of a search for red QSOs using a selection based on optical imaging from SDSS and near-infrared imaging from UKIDSS. For a sample of 58 candidates 46 (79%) are confirmed to be QSOs. The QSOs are predominantly dust-reddened except a handul at redshifts z>3.5. The dust is most likely located in the QSO host galaxies. 4 (7%) of the candidates turned out to be late-type stars, and another 4 (7%) are compact galaxies. The remaining 4 objects we could not identify. In terms of their optical spectra the QSOs are similar to the QSOs selected in the FIRST-2MASS red Quasar survey except they are on average fainter, more distant and only two are detected in the FIRST survey. We estimate the amount of extinction using the SDSS QSO template reddened by SMC-like dust. It is possible to get a good match to the observed (restframe ultraviolet) spectra, but for nearly all the reddened QSOs it is not possible to match the near-IR photometry from UKIDSS. The likely reasons are that the SDSS QSO template is too red at optical wavelengths due to contaminating host galaxy light and that the assumed SMC extinction curve is too shallow. Our survey has demonstrated that selection of QSOs based on near-IR photometry is an efficent way to select QSOs, including reddened QSOs, with only small contamination from late-type stars and compact galaxies. This will be useful with ongoing and future wide-field near-IR surveys such as the VISTA and EUCLID surveys. [Abridged]Comment: 74 pages, 6 figures. Accepted for for publication in ApJ

Sparsity and Incoherence in Compressive Sampling

We consider the problem of reconstructing a sparse signal $x^0\in\R^n$ from a limited number of linear measurements. Given $m$ randomly selected samples of $U x^0$, where $U$ is an orthonormal matrix, we show that $\ell_1$ minimization recovers $x^0$ exactly when the number of measurements exceeds $$m\geq \mathrm{Const}\cdot\mu^2(U)\cdot S\cdot\log n,$$ where $S$ is the number of nonzero components in $x^0$, and $\mu$ is the largest entry in $U$ properly normalized: $\mu(U) = \sqrt{n} \cdot \max_{k,j} |U_{k,j}|$. The smaller $\mu$, the fewer samples needed. The result holds for ``most'' sparse signals $x^0$ supported on a fixed (but arbitrary) set $T$. Given $T$, if the sign of $x^0$ for each nonzero entry on $T$ and the observed values of $Ux^0$ are drawn at random, the signal is recovered with overwhelming probability. Moreover, there is a sense in which this is nearly optimal since any method succeeding with the same probability would require just about this many samples

The Meinunger "Nicht Rote" Objects

Four high-latitude slow variable stars have been noted by Meinunger (1972) as "nicht rote" ("not red") objects and thus curious. We have previously reported (Margon & Deutsch 1997) that one of these objects, CC Boo, is in fact a QSO. Here we present observations demonstrating that the remaining three are also highly variable active galactic nuclei. The most interesting object of the four is perhaps S 10765 (= NGP9 F324-0276706), which proves to be a resolved galaxy at z=0.063. Despite the rapid and large reported variability amplitude (~1.6 mag), the spectrum is that of a perfectly normal galaxy, with no emission lines or evident nonthermal continuum. We also present new spectroscopic and photometric observations for AR CVn, suggested by Meinunger to be an RR Lyrae star despite its very faint magnitude (=19.4). The object is indeed one of the most distant RR Lyrae stars known, at a galactocentric distance of ~40 kpc.Comment: Accepted for publication in Publications of the Astronomical Society of the Pacific, Volume 111, January 1999; 14 pages including 4 figures and 1 tabl