66 resultados para Matrices doublement stochastiques
Resumo:
The generalization of the geometric mean of positive scalars to positive definite matrices has attracted considerable attention since the seminal work of Ando. The paper generalizes this framework of matrix means by proposing the definition of a rank-preserving mean for two or an arbitrary number of positive semi-definite matrices of fixed rank. The proposed mean is shown to be geometric in that it satisfies all the expected properties of a rank-preserving geometric mean. The work is motivated by operations on low-rank approximations of positive definite matrices in high-dimensional spaces.© 2012 Elsevier Inc. All rights reserved.
Resumo:
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.
Resumo:
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.
Resumo:
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.
Resumo:
This paper reports on the synthesis of zinc oxide (ZnO) nanostructures and examines the performance of nanocomposite thin-film transistors (TFTs) fabricated using ZnO dispersed in both n- and p-type polymer host matrices. The ZnO nanostructures considered here comprise nanowires and tetrapods and were synthesized using vapor phase deposition techniques involving the carbothermal reduction of solid-phase zinc-containing compounds. Measurement results of nanocomposite TFTs based on dispersion of ZnO nanorods in an n-type organic semiconductor ([6, 6]-phenyl-C61-butyric acid methyl ester) show electron field-effect mobilities in the range 0.3-0.6 cm2V-1 s-1. representing an approximate enhancement by as much as a factor of 40 from the pristine state. The on/off current ratio of the nanocomposite TFTs approach 106 at saturation with off-currents on the order of 10 pA. The results presented here, although preliminary, show a highly promising enhancement for realization of high-performance solution-processable n-type organic TFTs. © 2008 IEEE.
Resumo:
Physical forces generated by cells drive morphologic changes during development and can feedback to regulate cellular phenotypes. Because these phenomena typically occur within a 3-dimensional (3D) matrix in vivo, we used microelectromechanical systems (MEMS) technology to generate arrays of microtissues consisting of cells encapsulated within 3D micropatterned matrices. Microcantilevers were used to simultaneously constrain the remodeling of a collagen gel and to report forces generated during this process. By concurrently measuring forces and observing matrix remodeling at cellular length scales, we report an initial correlation and later decoupling between cellular contractile forces and changes in tissue morphology. Independently varying the mechanical stiffness of the cantilevers and collagen matrix revealed that cellular forces increased with boundary or matrix rigidity whereas levels of cytoskeletal and extracellular matrix (ECM) proteins correlated with levels of mechanical stress. By mapping these relationships between cellular and matrix mechanics, cellular forces, and protein expression onto a bio-chemo-mechanical model of microtissue contractility, we demonstrate how intratissue gradients of mechanical stress can emerge from collective cellular contractility and finally, how such gradients can be used to engineer protein composition and organization within a 3D tissue. Together, these findings highlight a complex and dynamic relationship between cellular forces, ECM remodeling, and cellular phenotype and describe a system to study and apply this relationship within engineered 3D microtissues.