Convex Relaxation for Low-Dimensional Representation: Phase Transitions and Limitations

There is a growing interest in taking advantage of possible patterns and structures in data so as to extract the desired information and overcome the curse of dimensionality. In a wide range of applications, including computer vision, machine learning, medical imaging, and social networks, the signal that gives rise to the observations can be modeled to be approximately sparse and exploiting this fact can be very beneficial. This has led to an immense interest in the problem of efficiently reconstructing a sparse signal from limited linear observations. More recently, low-rank approximation techniques have become prominent tools to approach problems arising in machine learning, system identification and quantum tomography.

In sparse and low-rank estimation problems, the challenge is the inherent intractability of the objective function, and one needs efficient methods to capture the low-dimensionality of these models. Convex optimization is often a promising tool to attack such problems. An intractable problem with a combinatorial objective can often be "relaxed" to obtain a tractable but almost as powerful convex optimization problem. This dissertation studies convex optimization techniques that can take advantage of low-dimensional representations of the underlying high-dimensional data. We provide provable guarantees that ensure that the proposed algorithms will succeed under reasonable conditions, and answer questions of the following flavor:

More specifically, i) Focusing on linear inverse problems, we generalize the classical error bounds known for the least-squares technique to the lasso formulation, which incorporates the signal model. ii) We show that intuitive convex approaches do not perform as well as expected when it comes to signals that have multiple low-dimensional structures simultaneously. iii) Finally, we propose convex relaxations for the graph clustering problem and give sharp performance guarantees for a family of graphs arising from the so-called stochastic block model. We pay particular attention to the following aspects. For i) and ii), we aim to provide a general geometric framework, in which the results on sparse and low-rank estimation can be obtained as special cases. For i) and iii), we investigate the precise performance characterization, which yields the right constants in our bounds and the true dependence between the problem parameters.

Temperature dependence of dark lines in coherent population trapping

We measure the signal amplitude and linewidth of a dark line in coherent population trapping in the Rb vapour cell filled with mixed buffer gas N-2 and Ar as a function of cell temperature. We find that the dark line signal amplitude increases with temperature up to a maximum at 49 degrees C and then drops at higher temperatures due to quenching effects of N-2. The linewidth of the dark line remains basically constant, at 1080 Hz. We also measure the linewidth of the dark line as a function of laser intensity. The linewidth increases linearly with laser intensity. An intrinsic linewidth (FWHM=896 Hz at 3.4 GHz) of the Rb cell is obtained.

Double injection: high frequency noise and temperature dependence

Noise measurements from 140°K to 350°K ambient temperature and between 10kHz and 22MHz performed on a double injection silicon diode as a function of operating point indicate that the high frequency noise depends linearly on the ambient temperature T and on the differential conductance g measured at the same frequency. The noise is represented quantitatively by〈i^2〉 = α•4kTgΔf. A new interpretation demands Nyquist noise with α ≡ 1 in these devices at high frequencies. This is in accord with an equivalent circuit derived for the double injection process. The effects of diode geometry on the static I-V characteristic as well as on the ac properties are illustrated. Investigation of the temperature dependence of double injection yields measurements of the temperature variation of the common high-level lifetime τ(τ ∝ T^2), the hole conductivity mobility µ_p (µ_p ∝ T^(-2.18)) and the electron conductivity mobility µ_n(µ_n ∝ T^(-1.75)).

The temperature dependence of the development of the embryo in Mesocyclops leuckarti (Claus) and Cyclops scutifer

This work is mainly intended as an addition to the studies of the populations dynamics of Cyclops scutifer, which is part of the ”Latn ja jaure project” (a study of the principles involved in the ecosystem of a small -initially fish free- mountain lake, before and after the introduction of fish). The field work consisted of sampling in Lake Erken in Roslagen in June, July and September, as well as in Latn ja jaure in the Abisko mountains in August and September of 1965. Additional sampling was done in Latn ja jaure for the study of the horizontal, vertical and temporal distribution of Cyclops scutifer, as well as the in situ development of the different stages. These samples have been analysed in such a way as to fit into the frame work of future studies on the population dynamics of Cyclops scutifer, The main aim of the present investigation is the determination of the dependence upon tempera- tare of the development of the embryo in the subarctic Cyclops scutifer as compared with the conditions found in the warm water species Mesocyclops leuckarti.

Formulas, tables, and graphs for trapezoidal reinforced concrete beams

Not available.

Analysis of dispersion compensation for position-dependence in externally modulated CATV lightwave systems by using chirped fiber grating

The dispersion compensation characteristics of the chirped fiber grating (CFG) for different dispersion compensation positions are analyzed in externally modulated cable television (CATV) lightwave system and the analytic expression of the composite second order (CSO) distortion is derived. The analyses give a reasonable explanation for the position-dependent effect of CFG dispersion compensator, which was found in practical systems. Moreover, the theoretical result is also verified by an experiment. It is believed that the theory will be helpful in designing optical CATV fiber links with nodes at proper positions both for intensity amplification and dispersion compensation.

Current dependence of the energy gap in superconductors : a direct measurement

From the tunneling characteristics of a tin-tin oxide-lead junction, a direct measurement has been made of the energy-gap variation for a superconductor carrying a current in a compensated geometry. Throughout the region investigated – several temperatures near T_c and down to a reduced temperature t = 0.8 –the observed current dependence agrees quite well with predictions based on the Ginzburg-Landau-Gor’kov theory. Near T_c the predicted temperature dependence is also well verified, though deviations are observed at lower temperatures; even for the latter, the data are internally consistent with the temperature dependence of the experimental critical current. At the lowest temperature investigated, t = 0.8, a small “Josephson” tunneling current allowed further a direct measurement of the electron drift velocity at low current densities. From this, a preliminary experimental value of the critical velocity, believed to be the first reported, can be inferred in the basis of Ginzburg-Landau theory. For tin at t = 0.8, we find v_c = 87 m/sec. This value does not appear fully consistent with those predicted by recent theories for superconductors with short electronic mean-free-paths.

Graphs and Groups

This project is a combination of graphs and group theory in which the aim is to describe the automorphism group of some specific families of graphs. Finally, an example of the application of automorphism groups in reaction graphs is shown. The project is written in english.

Wavelength dependence of light-induced domain nucleation in MgO-doped congruent LiNbO3 crystal

Within the wavelength range from 351 to 799 nm, the different reductions of nucleation field induced by the focused continuous laser irradiation are achieved in the 5 mol % MgO-doped congruent LiNbO3 crystals. The reduction proportion increases exponentially with decreasing irradiation wavelength and decreases exponentially with increasing irradiation wavelength. At one given wavelength, the reduction proportion increases exponentially with increasing irradiation intensity. An assumption is proposed that the reduction of nucleation field is directly related to the defect structure of crystal lattice generated by the complex coaction of incident irradiation field and external electric field. (c) 2007 American Institute of Physics.

Specklegram in a multiple-mode fiber and its dependence on longitudinal modes of the laser source

A specklegram in a multimode fiber (MMF) has successfully been used as a sensor for detecting external disturbance. Our experiments showed that the sensitivity in the sensor with a multiple longitudinal-mode laser as its source was much higher than that with a single longitudinal-mode laser. In addition, the near-field pattern observations indicated that the coupling between different transverse modes in the MMF is quite weak. Based on the experimental results, a theoretical model for the speckle formation is proposed, taking a bend-caused phase factor into consideration. It is shown in the theoretical analysis that the interferences between different longitudinal modes make a larger contribution to the specklegram signals. (C) 2007 Optical Society of America.

Temperature dependence of thermal properties of Ag8In14Sb55Te23 phase-change memory materials

The dependence of thermal properties of Ag8In14Sb55Te23 phase-change memory materials in crystalline and amorphous states on temperature was measured and analyzed. The results show that in the crystalline state, the thermal properties monotonically decrease with the temperature and present obvious crystalline semiconductor characteristics. The heat capacity, thermal diffusivity, and thermal conductivity decrease from 0.35 J/g K, 1.85 mm(2)/s, and 4.0 W/m K at 300 K to 0.025 J/g K, 1.475 mm(2)/s, and 0.25 W/m K at 600 K, respectively. In the amorphous state, while the dependence of thermal properties on temperature does not present significant changes, the materials retain the glass-like thermal characteristics. Within the temperature range from 320 K to 440 K, the heat capacity fluctuates between 0.27 J/g K and 0.075 J/g K, the thermal diffusivity basically maintains at 0.525 mm(2)/s, and the thermal conductivity decreases from 1.02 W/m K at 320 K to 0.2 W/m K at 440 K. Whether in the crystalline or amorphous state, Ag8In14Sb55Te23 are more thermally active than Ge2Sb2Te5, that is, the Ag8In14Sb55Te23 composites bear stronger thermal conduction and diffusion than the Ge2Sb2Te5 phase-change memory materials.

Selected topics in financial engineering: first-exit times and dependence structures of Marshall-Olkin Kind

146 p.