Replica Symmetry Breaking in Compressive Sensing

For noisy compressive sensing systems, the asymptotic distortion with respect to an arbitrary distortion function is determined when a general class of least-square based reconstruction schemes is employed. The sampling matrix is considered to belong to a large ensemble of random matrices including i.i.d. and projector matrices, and the source vector is assumed to be i.i.d. with a desired distribution. We take a statistical mechanical approach by representing the asymptotic distortion as a macroscopic parameter of a spin glass and employing the replica method for the large-system analysis. In contrast to earlier studies, we evaluate the general replica ansatz which includes the RS ansatz as well as RSB. The generality of the solution enables us to study the impact of symmetry breaking. Our numerical investigations depict that for the reconstruction scheme with the "zero-norm" penalty function, the RS fails to predict the asymptotic distortion for relatively large compression rates; however, the one-step RSB ansatz gives a valid prediction of the performance within a larger regime of compression rates.Comment: 7 pages, 3 figures, presented at ITA 201

RSB Decoupling Property of MAP Estimators

The large-system decoupling property of a MAP estimator is studied when it estimates the i.i.d. vector $\boldsymbol{x}$ from the observation $\boldsymbol{y}=\mathbf{A}\boldsymbol{x}+\boldsymbol{z}$ with $\mathbf{A}$ being chosen from a wide range of matrix ensembles, and the noise vector $\boldsymbol{z}$ being i.i.d. and Gaussian. Using the replica method, we show that the marginal joint distribution of any two corresponding input and output symbols converges to a deterministic distribution which describes the input-output distribution of a single user system followed by a MAP estimator. Under the $b$RSB assumption, the single user system is a scalar channel with additive noise where the noise term is given by the sum of an independent Gaussian random variable and $b$ correlated interference terms. As the $b$RSB assumption reduces to RS, the interference terms vanish which results in the formerly studied RS decoupling principle.Comment: 5 pages, presented in Information Theory Workshop 201

Correlated continuous-time random walks: combining scale-invariance with long-range memory for spatial and temporal dynamics

Standard continuous time random walk (CTRW) models are renewal processes in the sense that at each jump a new, independent pair of jump length and waiting time are chosen. Globally, anomalous diffusion emerges through action of the generalized central limit theorem leading to scale-free forms of the jump length or waiting time distributions. Here we present a modified version of recently proposed correlated CTRW processes, where we incorporate a power-law correlated noise on the level of both jump length and waiting time dynamics. We obtain a very general stochastic model, that encompasses key features of several paradigmatic models of anomalous diffusion: discontinuous, scale-free displacements as in Levy flights, scale-free waiting times as in subdiffusive CTRWs, and the long-range temporal correlations of fractional Brownian motion (FBM). We derive the exact solutions for the single-time probability density functions and extract the scaling behaviours. Interestingly, we find that different combinations of the model parameters lead to indistinguishable shapes of the emerging probability density functions and identical scaling laws. Our model will be useful to describe recent experimental single particle tracking data, that feature a combination of CTRW and FBM properties.Comment: 25 pages, IOP style, 5 figure

Aging renewal theory and application to random walks

The versatility of renewal theory is owed to its abstract formulation. Renewals can be interpreted as steps of a random walk, switching events in two-state models, domain crossings of a random motion, etc. We here discuss a renewal process in which successive events are separated by scale-free waiting time periods. Among other ubiquitous long time properties, this process exhibits aging: events counted initially in a time interval [0,t] statistically strongly differ from those observed at later times [t_a,t_a+t]. In complex, disordered media, processes with scale-free waiting times play a particularly prominent role. We set up a unified analytical foundation for such anomalous dynamics by discussing in detail the distribution of the aging renewal process. We analyze its half-discrete, half-continuous nature and study its aging time evolution. These results are readily used to discuss a scale-free anomalous diffusion process, the continuous time random walk. By this we not only shed light on the profound origins of its characteristic features, such as weak ergodicity breaking. Along the way, we also add an extended discussion on aging effects. In particular, we find that the aging behavior of time and ensemble averages is conceptually very distinct, but their time scaling is identical at high ages. Finally, we show how more complex motion models are readily constructed on the basis of aging renewal dynamics.Comment: 21 pages, 7 figures, RevTe

Broad-Band Soft X-ray Polarimetry

We developed an instrument design capable of measuring linear X-ray polarization over a broad-band using conventional spectroscopic optics. A set of multilayer-coated flats reflects the dispersed X-rays to the instrument detectors. The intensity variation with position angle is measured to determine three Stokes parameters: I, Q, and U -- all as a function of energy. By laterally grading the multilayer optics and matching the dispersion of the gratings, one may take advantage of high multilayer reflectivities and achieve modulation factors > 50% over the entire 0.2 to 0.8 keV band. This instrument could be used in a small orbiting mission or scaled up for the International X-ray Observatory. Laboratory work has begun that would demonstrate the capabilities of key components.Comment: 6 pages, 3 figures (2 color); to appear in proceedings of "The Coming of Age of X-ray Polarimetry

Aminosäuren – Leitlinie Parenterale Ernährung, Kapitel 4

Protein catabolism should be reduced and protein synthesis promoted with parenteral nutrion (PN). Amino acid (AA) solutions should always be infused with PN. Standard AA solutions are generally used, whereas specially adapted AA solutions may be required in certain conditions such as severe disorders of AA utilisation or in inborn errors of AA metabolism. An AA intake of 0.8 g/kg/day is generally recommended for adult patients with a normal metabolism, which may be increased to 1.2–1.5 g/kg/day, or to 2.0 or 2.5 g/kg/day in exceptional cases. Sufficient non-nitrogen energy sources should be added in order to assure adequate utilisation of AA. A nitrogen calorie ratio of 1:130 to 1:170 (g N/kcal) or 1:21 to 1:27 (g AA/kcal) is recommended under normal metabolic conditions. In critically ill patients glutamine should be administered parenterally if indicated in the form of peptides, for example 0.3–0.4 g glutamine dipeptide/kg body weight/day (=0.2–0.26 g glutamine/kg body weight/day). No recommendation can be made for glutamine supplementation in PN for patients with acute pancreatitis or after bone marrow transplantation (BMT), and in newborns. The application of arginine is currently not warranted as a supplement in PN in adults. N-acetyl AA are only of limited use as alternative AA sources. There is currently no indication for use of AA solutions with an increased content of glycine, branched-chain AAs (BCAA) and ornithine-α-ketoglutarate (OKG) in all patients receiving PN. AA solutions with an increased proportion of BCAA are recommended in the treatment of hepatic encephalopathy (III–IV).Ein Proteinkatabolismus soll bei parenteraler Ernährung (PE) vermindert und anabole Stoffwechselprozesse gefördert werden. Standard-Aminosäure (AS)-Lösungen werden empfohlen, falls nicht in Sondersituationen z. B. bei schweren AS-Verwertungsstörungen oder bei angeborenen Stoffwechselstörungen spezifisch adaptierte AS-Lösungen eingesetzt werden müssen. Für erwachsene Patienten in ausgeglichenem Stoffwechselzustand wird eine AS-Zufuhr von 0,8 g/kg/Tag empfohlen, die auf 1,2–1,5 g/kg/Tag oder in Ausnahmefällen auch auf 2,0–2,5 g/kg/Tag gesteigert werden kann. Zur Gewährleistung einer angemessenen Utilisation von AS sollten ausreichend Nicht-Stickstoff-Energieträger zugegeben werden. Das angestrebte Verhältnis zwischen Stickstoff- und Energiezufuhr (Stickstoff-Kalorien-Verhältnis) sollte unter Normalbedingungen 1:100–1:130 (g N:kcal) bzw. 1:16–1:21 (g AS:kcal) betragen. Glutamin sollte parenteral bei kritisch Kranken, sofern indiziert, in Form von Peptiden verabreicht werden, wie z.B. 0,3–0,4 g Glutamin-Dipepetid/kg KG/Tag (entsprechend 0,2–0,26 g Glutamin/kg KG/Tag). Für Patienten mit akuter Pankreatitis, nach Knochenmarkstransplantation sowie für Neugeborene kann derzeit keine Empfehlung für eine Glutaminsupplementierung mit der PE ausgesprochen werden. Der Einsatz von Arginin als Supplement in der PE beim Erwachsenen ist derzeit nicht gerechtfertigt. Den N-azetylierten AS kommen als alternative Aminosäurenquellen zur Zeit nur eine begrenzte Bedeutung zu. Für eine generelle Verwendung von AS-Lösungen mit einem erhöhten Gehalt von Glyzin und verzweigtkettigten AS (VKAS) wie auch für Ornithin-α-Ketoglutarat (OKG) besteht keine gesicherte Indikation. Die Wirksamkeit von AS-Lösungen mit erhöhtem Anteil an VKAS in der Behandlung der hepatischen Enzephalopathie (III–IV) wird empfohlen