Article

# Nearly-Linear Time Algorithms for Preconditioning and Solving Symmetric, Diagonally Dominant Linear Systems

SIAM Journal on Matrix Analysis and Applications (Impact Factor: 1.34). 07/2006; DOI: 10.1137/090771430

Source: arXiv

- [Show abstract] [Hide abstract]

**ABSTRACT:**Spectral clustering is arguably one of the most important algorithms in data mining and machine intelligence; however, its computational complexity makes challenging to use in large scale data analysis. Recently, several approximation algorithms for spectral clustering have been developed in order to alleviate the relevant costs, but theoretical results are lacking. In this paper, we present an approximation algorithm for spectral clustering with strong theoretical evidence of its performance. Our algorithm is based on approximating the eigenvectors of the Laplacian matrix using random projections, a.k.a randomized sketching. Our experimental results demonstrate that the proposed approximation algorithm compares remarkably well to the exact algorithm.11/2013; - [Show abstract] [Hide abstract]

**ABSTRACT:**We present the first parallel algorithm for solving systems of linear equations in symmetric, diagonally dominant (SDD) matrices that runs in polylogarithmic time and nearly-linear work. The heart of our algorithm is a construction of a sparse approximate inverse chain for the input matrix: a sequence of sparse matrices whose product approximates its inverse. Whereas other fast algorithms for solving systems of equations in SDD matrices exploit low-stretch spanning trees, our algorithm only requires spectral graph sparsifiers.11/2013; - [Show abstract] [Hide abstract]

**ABSTRACT:**In this thesis I present experimental demonstrations of room-temperature, single-photon sources with definite linear and circular polarizations. Definite photon polarization increases the efficiency of quantum communication systems. In contrast with cryogenic-temperature single-photon sources based on epitaxial quantum dots requiring expensive MBE and nanofabrication, my method utilizes a mature liquid crystal technology, which I made consistent with single-emitter fluorescence microscopy. The structures I have prepared are planar-aligned cholesteric liquid crystals forming 1-D photonic bandgaps for circularly-polarized light, which were used to achieve definite circularly-polarized fluorescence of single emitters doped in this environment. I also used planar-aligned nematic liquid crystals to align single molecules with linear dipole moments and achieved definite linearly-polarized fluorescence. I used single nanocrystal quantum dots, single nanodiamond color-centers, rare-earth-doped nanocrystals, and single terrylene and DiIC18(3) dye molecules as emitters. For nanocrystal quantum dots I observed circular polarization dissymmetry factors as large as ge = --1.6. In addition, I observed circularly-polarized resonances in the fluorescence of emitters within a cholesteric microcavity, with cavity quality factors of up to Q ˜ 250. I also showed that the fluorescence of DiIC18(3) dye molecules in planar-aligned nematic cells exhibits definite linear polarization, with a degree of polarization of rho = --0.58 +/- 0.03. Distributed Bragg reflectors form another type of microcavity that can be used to realize a single-photon source. I characterized the fluorescence from nanocrystal quantum dots doped in the defect layers of such microcavites, both organic and inorganic. Finally, to demonstrate the single-photon properties of single-emitter-doped cholesteric and nematic liquid crystal structures and distributed Bragg reflector microcavities, I present observations of photon antibunching from emitters doped in each of these structures. These experimental observations include photon antibunching from: nanocrystal quantum dots and nanodiamond color-centers doped in a cholesteric microcavity; terrylene and DiIC 18(3) dye molecules doped in nematic structures, and nanocrystal quantum dots doped in the distributed Bragg reflector microcavity. A value of the zero-time second-order coherence as low as g(2)(0) = 0.001 +/- 0.03 was measured. These results represent an important step forward in the realization of room temperature single-photon sources with definite polarization for secure quantum communication.01/2011;

Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. The impact factor represents a rough estimation of the journal's impact factor and does not reflect the actual current impact factor. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.