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We consider the problem of selecting the "best" subset of exactly k columns from an m × n matrix A. In particular, we present and analyze a novel two-stage algorithm that runs in O(min{mn 2 , m 2 n}) time and returns as output an m × k matrix C consisting of exactly k columns of A. In the first stage (the randomized stage), the algorithm randomly selects Θ(k log k) columns according to a judiciously-chosen probability distribution that depends on information in the top-k right singular subspace of A. In the second stage (the deterministic stage), the algorithm applies a deterministic column-selection procedure to select and return exactly k columns from the set of columns selected in the first stage. Let C be the m × k matrix containing those k columns, let P C denote the projection matrix onto the span of those columns, and let A k denote the "best" rank-k approximation to the matrix A as computed with the singular value decomposition. Then, we prove that, with probability at least 0.8,

Aβ(1-42) is the highly pathologic isoform of amyloid-β, the peptide constituent of fibrils and neurotoxic oligomers involved in Alzheimer's disease. Recent studies on the structural features of Aβ in water have suggested that the system can be described as an ensemble of distinct conformational species in fast exchange. Here, we use replica exchange molecular dynamics simulations (REMD) to characterize the conformations accessible to Aβ42 in explicit water solvent, under the ff99SB forcefield. Monitoring the correlation between J-coupling ( ) and residual dipolar coupling (RDC) data calculated from the REMD trajectories to their experimental values, as determined by NMR indicates that the simulations are converging towards sampling an ensemble that is representative of the experimental data after 60ns/replica of simulation time. We further validate the converged MD-derived ensemble through direct comparison with and RDC experimental data. Our analysis indicates that the ff99SB-derived REMD ensemble can reproduce the experimental J-coupling values with high accuracy and further provide good agreement with the RDC data. Our results indicate that the peptide is sampling a highly diverse range of conformations: by implementing statistical learning techniques (Laplacian Eigenmaps, Spectral Clustering, and Laplacian Scores) we are able to obtain an otherwise hidden structure in the complex conformational space of the peptide. Using these methods we characterize the peptide conformations and extract their intrinsic characteristics, identify a small number of different conformations that characterize the whole ensemble, and identify a small number of protein interactions (such as contacts between the peptide termini) that are the most discriminative of the different conformations and thus can be used in designing experimental probes of transitions between such molecular states. This is a study of an important intrinsically disordered peptide © 2010 Published by Elsevier Ltd.To whom correspondence should be addressed. angel@rpi.edu Tel. 518-276-6310 Fax. 518-276-6680. Publisher's Disclaimer: This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. system that provides an atomic-level description of structural features and interactions that are relevant during the early stages of the oligomerization and fibril nucleation pathways. NIH Public Access

Abstract. We consider low-rank reconstruction of a matrix using a subset of its columns and we present asymptotically optimal algorithms for both spectral norm and Frobenius norm reconstruction. The main tools we introduce to obtain our results are: (i) the use of fast approximate SVD-like decompositions for column-based matrix reconstruction, and (ii) two deterministic algorithms for selecting rows from matrices with orthonormal columns, building upon the sparse representation theorem for decompositions of the identity that appeared in [1].

Abstract. We consider low-rank reconstruction of a matrix using a subset of its columns and we present asymptotically optimal algorithms for both spectral norm and Frobenius norm reconstruction. The main tools we introduce to obtain our results are: (i) the use of fast approximate SVD-like decompositions for column-based matrix reconstruction, and (ii) two deterministic algorithms for selecting rows from matrices with orthonormal columns, building upon the sparse representation theorem for decompositions of the identity that appeared in [1].