Estimation of high-dimensional low-rank matrices

Suppose that we observe entries or, more generally, linear combinations of entries of an unknown $m\times T$-matrix $A$ corrupted by noise. We are particularly interested in the high-dimensional setting where the number $mT$ of unknown entries can be much larger than the sample size $N$. Motivated by several applications, we consider estimation of matrix $A$ under the assumption that it has small rank. This can be viewed as dimension reduction or sparsity assumption. In order to shrink toward a low-rank representation, we investigate penalized least squares estimators with a Schatten-$p$ quasi-norm penalty term, $p\leq1$. We study these estimators under two possible assumptions---a modified version of the restricted isometry condition and a uniform bound on the ratio "empirical norm induced by the sampling operator/Frobenius norm." The main results are stated as nonasymptotic upper bounds on the prediction risk and on the Schatten-$q$ risk of the estimators, where $q\in[p,2]$. The rates that we obtain for the prediction risk are of the form $rm/N$ (for $m=T$), up to logarithmic factors, where $r$ is the rank of $A$. The particular examples of multi-task learning and matrix completion are worked out in detail. The proofs are based on tools from the theory of empirical processes. As a by-product, we derive bounds for the $k$th entropy numbers of the quasi-convex Schatten class embeddings $S_p^M\hookrightarrow S_2^M$, $p<1$, which are of independent interest.Comment: Published in at http://dx.doi.org/10.1214/10-AOS860 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

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Økologisk hvidkløver - hvordan kan det lade sig gøre?

Økologisk hvidkløver - hvordan kan det lade sig gøre

Symmetry Protected Topological phases and Generalized Cohomology

We discuss the classification of SPT phases in condensed matter systems. We review Kitaev's argument that SPT phases are classified by a generalized cohomology theory, valued in the spectrum of gapped physical systems. We propose a concrete description of that spectrum and of the corresponding cohomology theory. We compare our proposal to pre-existing constructions in the literature.Comment: 27 pages, 10 figures. v2: citation updat

Poultry house remodeling

May, 1938

Low cost poultry housing

"July, 1941

Probabilistic state preparation of a single molecular ion by projection measurement

We show how to prepare a single molecular ion in a specific internal quantum state in a situation where the molecule is trapped and sympathetically cooled by an atomic ion and where its internal degrees of freedom are initially in thermal equilibrium with the surroundings. The scheme is based on conditional creation of correlation between the internal state of the molecule and the translational state of the collective motion of the two ions, followed by a projection measurement of this collective mode by atomic ion shelving techniques. State preparation in a large number of internal states is possible.Comment: 4 pages, 2 figures, 2 table

Fluxes of microbes, organic aerosols, dust, sea-salt Na ions, non-sea-salt Ca ions, and methanesulfonate onto Greenland and Antarctic ice

Using a spectrofluorimeter with 224-nm laser excitation and six emission bands from 300 to 420 nm to measure fluorescence intensities at 0.3-mm depth intervals in ice cores, we report results of the first comparative study of concentrations of microbial cells (using the spectrum of protein-bound tryptophan (Trp) as a proxy) and of aerosols with autofluorescence spectra different from Trp (denoted &quot;non-Trp&quot;) as a function of depth in ice cores from West Antarctica (WAIS Divide and Siple Dome) and Greenland (GISP2). The ratio of fluxes of microbial cells onto West Antarctic (WAIS Divide) versus Greenland sites is 0.13&amp;plusmn;0.06; the ratio of non-Trp aerosols onto WAIS Divide versus Greenland sites is 0.16&amp;plusmn;0.08; and the ratio of non-sea-salt Ca&lt;sup&gt;2+&lt;/sup&gt; ions (a proxy for dust grains) onto WAIS Divide versus Greenland sites is 0.06&amp;plusmn;0.03. All of these are roughly comparable to the ratio of fluxes of dust onto Antarctic versus Greenland sites (0.08&amp;plusmn;0.05). By contrast to those values, which are considerably lower than unity, the ratio of fluxes of methanesulfonate (MSA) onto Antarctic versus Greenland sites is 1.9&amp;plusmn;0.4 and the ratio of sea-salt Na&lt;sup&gt;2+&lt;/sup&gt; ions onto WAIS Divide versus Greenland sites is 3.0&amp;plusmn;2. These ratios are more than an order of magnitude higher than those in the first grouping. We infer that the correlation of microbes and non-Trp aerosols with non-sea-salt Ca and dust suggests a largely terrestrial rather than marine origin. The lower fluxes of microbes, non-Trp aerosols, non-sea-salt Ca and dust onto WAIS Divide ice than onto Greenland ice may be due to the smaller areas of their source regions and less favorable wind patterns for transport onto Antarctic ice than onto Greenland ice. The correlated higher relative fluxes of MSA and marine Na onto Antarctic versus Greenland ice is consistent with the view that both originate largely on or around sea ice, with the Antarctic sea ice being far more extensive than that around Greenland

Experiments towards quantum information with trapped Calcium ions

Ground state cooling and coherent manipulation of ions in an rf-(Paul) trap is the prerequisite for quantum information experiments with trapped ions. With resolved sideband cooling on the optical S1/2 - D5/2 quadrupole transition we have cooled one and two 40Ca+ ions to the ground state of vibration with up to 99.9% probability. With a novel cooling scheme utilizing electromagnetically induced transparency on the S1/2 - P1/2 manifold we have achieved simultaneous ground state cooling of two motional sidebands 1.7 MHz apart. Starting from the motional ground state we have demonstrated coherent quantum state manipulation on the S1/2 - D5/2 quadrupole transition at 729 nm. Up to 30 Rabi oscillations within 1.4 ms have been observed in the motional ground state and in the n=1 Fock state. In the linear quadrupole rf-trap with 700 kHz trap frequency along the symmetry axis (2 MHz in radial direction) the minimum ion spacing is more than 5 micron for up to 4 ions. We are able to cool two ions to the ground state in the trap and individually address the ions with laser pulses through a special optical addressing channel.Comment: Proceedings of the ICAP 2000, Firenz

Exploiting the capacity of 1mm PMMA step-index polymer optical fibers

Three different techniques are discussed that are currently under investigation at Siemens Corporate Technology – Information and Communications in order to exploit the bandwidth capacity of 1 mm PMMA Step-Index Polymer Optical Fiber (SI-POF). By using Adaptive Multitone Modulation (AMTM) a record result of 540 Mb/s transmission over 100 m of SI-POF is achieved