Linear regression under fixed-rank constraints: A Riemannian approach

In this paper, we tackle the problem of learning a linear regression model whose parameter is a fixed-rank matrix. We study the Riemannian manifold geometry of the set of fixed-rank matrices and develop efficient line-search algorithms. The proposed algorithms have many applications, scale to high-dimensional problems, enjoy local convergence properties and confer a geometric basis to recent contributions on learning fixed-rank matrices. Numerical experiments on benchmarks suggest that the proposed algorithms compete with the state-of-the-art, and that manifold optimization offers a versatile framework for the design of rank-constrained machine learning algorithms. Copyright 2011 by the author(s)/owner(s).

Regression on fixed-rank positive semidefinite matrices: A Riemannian approach

The paper addresses the problem of learning a regression model parameterized by a fixed-rank positive semidefinite matrix. The focus is on the nonlinear nature of the search space and on scalability to high-dimensional problems. The mathematical developments rely on the theory of gradient descent algorithms adapted to the Riemannian geometry that underlies the set of fixedrank positive semidefinite matrices. In contrast with previous contributions in the literature, no restrictions are imposed on the range space of the learned matrix. The resulting algorithms maintain a linear complexity in the problem size and enjoy important invariance properties. We apply the proposed algorithms to the problem of learning a distance function parameterized by a positive semidefinite matrix. Good performance is observed on classical benchmarks. © 2011 Gilles Meyer, Silvere Bonnabel and Rodolphe Sepulchre.

Low-rank optimization on the cone of positive semidefinite matrices

We propose an algorithm for solving optimization problems defined on a subset of the cone of symmetric positive semidefinite matrices. This algorithm relies on the factorization X = Y Y T , where the number of columns of Y fixes an upper bound on the rank of the positive semidefinite matrix X. It is thus very effective for solving problems that have a low-rank solution. The factorization X = Y Y T leads to a reformulation of the original problem as an optimization on a particular quotient manifold. The present paper discusses the geometry of that manifold and derives a second-order optimization method with guaranteed quadratic convergence. It furthermore provides some conditions on the rank of the factorization to ensure equivalence with the original problem. In contrast to existing methods, the proposed algorithm converges monotonically to the sought solution. Its numerical efficiency is evaluated on two applications: the maximal cut of a graph and the problem of sparse principal component analysis. © 2010 Society for Industrial and Applied Mathematics.

Riemannian metric and geometric mean for positive semidefinite matrices of fixed rank

This paper introduces a new metric and mean on the set of positive semidefinite matrices of fixed-rank. The proposed metric is derived from a well-chosen Riemannian quotient geometry that generalizes the reductive geometry of the positive cone and the associated natural metric. The resulting Riemannian space has strong geometrical properties: it is geodesically complete, and the metric is invariant with respect to all transformations that preserve angles (orthogonal transformations, scalings, and pseudoinversion). A meaningful approximation of the associated Riemannian distance is proposed, that can be efficiently numerically computed via a simple algorithm based on SVD. The induced mean preserves the rank, possesses the most desirable characteristics of a geometric mean, and is easy to compute. © 2009 Society for Industrial and Applied Mathematics.

Stable multivariate rational interpolation for parameter-dependent aerospace models

A multivariate, robust, rational interpolation method for propagating uncertainties in several dimensions is presented. The algorithm for selecting numerator and denominator polynomial orders is based on recent work that uses a singular value decomposition approach. In this paper we extend this algorithm to higher dimensions and demonstrate its efficacy in terms of convergence and accuracy, both as a method for response suface generation and interpolation. To obtain stable approximants for continuous functions, we use an L2 error norm indicator to rank optimal numerator and denominator solutions. For discontinous functions, a second criterion setting an upper limit on the approximant value is employed. Analytical examples demonstrate that, for the same stencil, rational methods can yield more rapid convergence compared to pseudospectral or collocation approaches for certain problems. © 2012 AIAA.

Low-rank optimization with trace norm penalty

The paper addresses the problem of low-rank trace norm minimization. We propose an algorithm that alternates between fixed-rank optimization and rank-one updates. The fixed-rank optimization is characterized by an efficient factorization that makes the trace norm differentiable in the search space and the computation of duality gap numerically tractable. The search space is nonlinear but is equipped with a Riemannian structure that leads to efficient computations. We present a second-order trust-region algorithm with a guaranteed quadratic rate of convergence. Overall, the proposed optimization scheme converges superlinearly to the global solution while maintaining complexity that is linear in the number of rows and columns of the matrix. To compute a set of solutions efficiently for a grid of regularization parameters we propose a predictor-corrector approach that outperforms the naive warm-restart approach on the fixed-rank quotient manifold. The performance of the proposed algorithm is illustrated on problems of low-rank matrix completion and multivariate linear regression. © 2013 Society for Industrial and Applied Mathematics.

Reducing organic substances from anaerobic decomposition of hydrophytes

Oxidation-reduction properties of surface sediments are tightly associated with the geochemistry of substances, and reducing organic substances (ROS) from hydrophytes residues may play an important role in these processes. In this study, composition, dynamics, and properties of ROS from anaerobic decomposition of Eichhornia crassipes (Mart.) Solms, Potamogenton crispus Linn, Vallisneria natans (Lour.) Hara, Lemna trisulca Linn and Microcystis flos-aquae (Wittr) Kirch were investigated using differential pulse voltammetry (DPV). The type of hydrophytes determined both the reducibility and composition of ROS. At the peak time of ROS production, the anaerobic decomposition of M. flos-aquae produced 6 types of ROS, among which 3 belonged to strongly reducing organic substance (SROS), whereas there were only 3-4 types of ROS from the other hydrophytes, 2 of them exhibiting strong reducibility. The order of potential of hydrophytes to produce ROS was estimated to be: M. flos-aquae > E. crassipes > L. trisulca > P. crispus approximate to V. natans, based on the summation of SROS and weakly reducing organic substances (WROS). The dynamic pattern of SROS production was greatly different from WROS. The total SROS appeared periodic fluctuation with reducibility gradually weakening with incubation time, whereas the total WROS increased with incubation time. Reducibility of ROS from hydrophytes was readily affected by acid, base and ligands, suggesting that their properties were related to these aspects. In addition to the reducibility, we believe that more attention should be paid to the other behaviors of ROS in surface sediments.

Sorptive interaction between goethite and strongly reducing organic substances from anaerobic decomposition of green manures

Strongly reducing organic substances (SROS) and iron oxides exist widely in soils and sediments and have been implicated in many soil and sediment processes. In the present work, the sorptive interaction between goethite and SROS derived from anaerobic decomposition of green manures was investigated by differential pulse voltammetry (DPV). Both green manures, Astragaltus sinicus (Astragalus) and Vicia varia (Vicia) were chosen to be anaerobically decomposed by the mixed microorganisms isolated from paddy soils for 30 d to prepare different SROS. Goethite used in experiments was synthesized in laboratory. The anaerobic incubation solutions from green manures at different incubation time were arranged to react with goethite, in which SROS concentration and Fe(II) species were analyzed. The anaerobic decomposition of Astragalus generally produced SROS more in amount but weaker in reducibility than that of Vicia in the same incubation time. The available SROS from Astragalus that could interact with goethite was 0.69 +/- 0.04, 0.84 +/- 0.04 and 1.09 +/- 0.03 cmol kg(-1) as incubated for 10, 15 and 30 d, respectively, for Vicia, it was 0.12 +/- 0.03, 0.46 +/- 0.02 and 0.70 +/- 0.02 cmol kg(-1). One of the fates of SROS as they interacted with goethite was oxidation. The amounts of oxidizable SROS from Astragalus decreased over increasing incubation time from 0.51 +/- 0.05 cmol kg(-1) at day 10 to 0.39 +/- 0.04 cmol kg(-1) at day 30, but for Vicia, it increased with the highest reaching to 0.58 +/- 0.04 cmol kg(-1) at day 30. Another fate of these substances was sorption by goethite. The SROS from Astragalus were sorbed more readily than those from Vicia, and closely depended upon the incubation time, whereas for those from Vicia, the corresponding values were remarkably less and apparently unchangeable with incubation time. The extent of goethite dissolution induced by the anaerobic solution from Vicia was greater than that from Astragalus, showing its higher reactivity. (c) 2008 Published by Elsevier Ltd.

Fixed-rank matrix factorizations and Riemannian low-rank optimization

Motivated by the problem of learning a linear regression model whose parameter is a large fixed-rank non-symmetric matrix, we consider the optimization of a smooth cost function defined on the set of fixed-rank matrices. We adopt the geometric framework of optimization on Riemannian quotient manifolds. We study the underlying geometries of several well-known fixed-rank matrix factorizations and then exploit the Riemannian quotient geometry of the search space in the design of a class of gradient descent and trust-region algorithms. The proposed algorithms generalize our previous results on fixed-rank symmetric positive semidefinite matrices, apply to a broad range of applications, scale to high-dimensional problems, and confer a geometric basis to recent contributions on the learning of fixed-rank non-symmetric matrices. We make connections with existing algorithms in the context of low-rank matrix completion and discuss the usefulness of the proposed framework. Numerical experiments suggest that the proposed algorithms compete with state-of-the-art algorithms and that manifold optimization offers an effective and versatile framework for the design of machine learning algorithms that learn a fixed-rank matrix. © 2013 Springer-Verlag Berlin Heidelberg.

Coulomb blockade double-dot Aharonov-Bohm interferometer: Harmonic decomposition of the interference pattern

For the solid-state double-dot interferometer, the phase shifted interference pattern induced by the interplay of inter-dot Coulomb correlation and multiple reflections is analyzed by harmonic decomposition. Unexpected result is uncovered, and is discussed in connection with the which-path detection and electron loss. (C) 2009 Elsevier B.V. All rights reserved.

Purification of single-walled carbon nanotubes synthesized by the catalytic decomposition of hydrocarbons

A procedure for purifying single-walled carbon nanotubes (SWNTs) synthesized by the catalytic decomposition of hydrocarbons has been developed. Based on the results from SEM observations, EDS analysis and Raman measurements, it was found that amorphous carbon, catalyst particles, vapor-grown carbon nanofibers and multi-walled carbon nanotubes were removed from the ropes of SWNTs without damaging the SWNT bundles, and a 40% yield of the SWNTs with a purity of about 95% was achieved after purification. (C) 2000 Elsevier Science Ltd. All rights reserved.

Molecular Dynamics Simulation on Characteristics of Decomposition of Methane Hydrate in the Presence of Hydrogen Peroxide Solution

In this work, the characteristics of the decomposition of methane hydrate Structure I (SI) in the presence of hydrogen peroxide solution is investigated using the molecular dynamics simulation. The mechanism of the transformation process from the solid hydrate to the liquid is analyzed with the effect of hydrogen peroxide (HP) solution. In addition, the effect of ethylene glycol (EG) with the same molar concentration with HP on the methane hydrate dissociation is also studied. The results illustrate that both HP and EG promote well the hydrate dissociation. The work provides the important reference value for the experimental investigation into the promotion effect of HP on the hydrate dissociation.