On the Power of Adaptivity in Matrix Completion and Approximation

We consider the related tasks of matrix completion and matrix approximation from missing data and propose adaptive sampling procedures for both problems. We show that adaptive sampling allows one to eliminate standard incoherence assumptions on the matrix row space that are necessary for passive sampling procedures. For exact recovery of a low-rank matrix, our algorithm judiciously selects a few columns to observe in full and, with few additional measurements, projects the remaining columns onto their span. This algorithm exactly recovers an $n \times n$ rank $r$ matrix using $O(nr\mu_0 \log^2(r))$ observations, where $\mu_0$ is a coherence parameter on the column space of the matrix. In addition to completely eliminating any row space assumptions that have pervaded the literature, this algorithm enjoys a better sample complexity than any existing matrix completion algorithm. To certify that this improvement is due to adaptive sampling, we establish that row space coherence is necessary for passive sampling algorithms to achieve non-trivial sample complexity bounds. For constructing a low-rank approximation to a high-rank input matrix, we propose a simple algorithm that thresholds the singular values of a zero-filled version of the input matrix. The algorithm computes an approximation that is nearly as good as the best rank-$r$ approximation using $O(nr\mu \log^2(n))$ samples, where $\mu$ is a slightly different coherence parameter on the matrix columns. Again we eliminate assumptions on the row space

Detecting Activations over Graphs using Spanning Tree Wavelet Bases

We consider the detection of activations over graphs under Gaussian noise, where signals are piece-wise constant over the graph. Despite the wide applicability of such a detection algorithm, there has been little success in the development of computationally feasible methods with proveable theoretical guarantees for general graph topologies. We cast this as a hypothesis testing problem, and first provide a universal necessary condition for asymptotic distinguishability of the null and alternative hypotheses. We then introduce the spanning tree wavelet basis over graphs, a localized basis that reflects the topology of the graph, and prove that for any spanning tree, this approach can distinguish null from alternative in a low signal-to-noise regime. Lastly, we improve on this result and show that using the uniform spanning tree in the basis construction yields a randomized test with stronger theoretical guarantees that in many cases matches our necessary conditions. Specifically, we obtain near-optimal performance in edge transitive graphs, $k$-nearest neighbor graphs, and $\epsilon$-graphs

Effects of DC electric field on particle transportation and deposition in evaporating droplets

Microfluidics(uF) is the science of manipulating and controlling the fluids usually in the range of microliters (10-6L) to picolitres (10-12L). Physical parameters, such as surface tension and contact angle do not play a significant effect on macro scale but play a crucial role at the microscopic level. Microfluidics is viewed as an essential tool for life science research and flexible electronics. A deeper understanding of physical parameters of microfluidics would result in more efficient and lower cost devices (‘Lab-on-a-Chip’ devices) and foldable electronics. In short, the concepts of microfluidics can be used to reduce the cost, size, and ease of usage in a wide variety of futuristic products. In our study we are going to explore the effects of electric fields on a particle transport in evaporating droplet and deposition patterns left behind evaporating droplets. We will be studying droplets evaporating on ‘Electrowetting on Dielectric’ (EWOD) devices where the droplet is separated from the active electrode by dielectric layers. These types of devices are relevant in a variety of applications such as medical diagnostics and optics. We believe that understanding this phenomenon will impact printing and the development of flexible electronics. This work will further the understanding of transport and deposition of particles in evaporating droplets under applied electric field by understanding the effects of particle concentrations and different dielectric layer. Previous works have demonstrated that particle transport in evaporating droplets and their resultant deposition patterns can be altered under the presence of the electric fields. Applied electric fields have potential to provide real-time control of the particle transport in evaporating droplets by allowing an instantaneous control of the contact line dynamics, electrophoretic manipulation of particles inside the droplet, changes in interface shape, dielectrophoretic manipulation and particle motion inside the droplet due to forces induced by the electric field and evaporation. Our work is going to provide a deeper insight into the effects of DC electric fields on droplets with varying particle concentrations on different dielectric layer. We will better understand the effects of changing the variables (i.e., hydrophobicity ,polarity and particle concentration) under applied DC electric field

Trion and exciton dynamics in two dimensional semiconductors

Two-dimensional semiconducting systems have become increasingly important for a variety of applications including photo-detectors, high-power transistors and optoelectronics. With the discovery of the indirect-to-direct bandgap transition in atomically thin transition metal dichalcogenide (TMDs’) materials, a plethora of further applications and advances await. Optical properties in these materials are especially interesting to measure, due to presence of spin-valley coupling giving rise to valleytronic applications, and enhanced light emission (and absorption) with applications in optoelectronics. Optical studies in semiconductors near the bandgap primarily relate to the fundamental optical excitation of semiconductors, an exciton (a Coulomb-bound electron-hole pair). If the Coulomb interaction is strong enough, excitons may capture an extra electron or hole, forming charged excitons known as trions. Trions have shown to carry longer-lived spin information, and can drift under an electric field. The interaction between excitons and trions, is thus a technologically important issue in optoelectronics. The purpose of this dissertation is to measure the interactions between excitons and trions in a variety of two-dimensional systems, primarily in the new class of semiconducting two-dimensional materials, TMDs’. The interactions are measured for their character (coherent or incoherent) and dynamics. Utilizing a two-color pump-probe setup we uncover coherent coupling, between excitons and trions in monolayer molybdenum diselenide, an order of magnitude larger than traditional semiconductors (like gallium arsenide). Incoherent relaxation pathways towards trions, are measured via resonant excitation of excitons. A mobility edge within the exciton resonance is uncovered, with applications in quantifying transport properties of materials under study. Further, valley sensitive measurements are carried out on monolayer tungsten diselenide, revealing the long-lived trion spin polarization and ultrafast exciton valley relaxation. The possible spectroscopy feature of biexcitons is discussed in monolayer tungsten diselenide. Finally, measurements are extended to high mobility gallium arsenide quantum well systems, and electron-density dependent spin scattering mechanisms are uncovered. We further discuss the possibility to suppress spin relaxation, via gate voltage, in these gallium arsenide quantum wells.Physic