Sparse Recovery with Very Sparse Compressed Counting

Compressed sensing (sparse signal recovery) often encounters nonnegative data (e.g., images). Recently we developed the methodology of using (dense) Compressed Counting for recovering nonnegative K-sparse signals. In this paper, we adopt very sparse Compressed Counting for nonnegative signal recovery. Our design matrix is sampled from a maximally-skewed p-stable distribution (0<p<1), and we sparsify the design matrix so that on average (1-g)-fraction of the entries become zero. The idea is related to very sparse stable random projections (Li et al 2006 and Li 2007), the prior work for estimating summary statistics of the data. In our theoretical analysis, we show that, when p->0, it suffices to use M= K/(1-exp(-gK) log N measurements, so that all coordinates can be recovered in one scan of the coordinates. If g = 1 (i.e., dense design), then M = K log N. If g= 1/K or 2/K (i.e., very sparse design), then M = 1.58K log N or M = 1.16K log N. This means the design matrix can be indeed very sparse at only a minor inflation of the sample complexity. Interestingly, as p->1, the required number of measurements is essentially M = 2.7K log N, provided g= 1/K. It turns out that this result is a general worst-case bound

Observation of tunneling gap in epitaxial ultrathin films of pyrite-type copper disulfide

We report scanning tunneling microscopy investigation on epitaxial ultrathin films of pyrite-type copper disulfide. Layer by layer growth of CuS2 films with a preferential orientation of (111) on SrTiO3(001) and Bi2Sr2CaCu2O8+{\delta} substrates is achieved by molecular beam epitaxy growth. For ultrathin films on both kinds of substrates, we observed symmetric tunneling gap around Fermi level that persists up to ~ 15 K. The tunneling gap degrades with either increasing temperature or increasing thickness, suggesting new matter states at the extreme two dimensional limit.Comment: 4 Figure