Fast algorithms for the Sylvester equation AX−XBT=C

AbstractFor given matrices A∈Fm×m, B∈Fn×n, and C∈Fm×n over an arbitrary field F, the matrix equation AX−XBT=C has a unique solution X∈Fm×n if and only if A and B have disjoint spectra. We describe an algorithm that computes the solution X for m,n⩽N with O(Nβ·logN) arithmetic operations in F, where β>2 is such that M×M matrices can be multiplied with O(Mβ) arithmetic operations, e.g., β=2.376. It seems that before no better bound than O(m3·n3) arithmetic operations was known. The state of the art in numerical analysis is O(n3+m3) flops, but these algorithms (due to Bartels/Stewart and Golub/Nash/van Loan) involve Schur decompositions, i.e., they compute the eigenvalues of at least one of A and B, and can hence not be transferred for general F

A Preconditioned Iteration Method for Solving Sylvester Equations

A preconditioned gradient-based iterative method is derived by judicious selection of two auxil- iary matrices. The strategy is based on the Newton’s iteration method and can be regarded as a generalization of the splitting iterative method for system of linear equations. We analyze the convergence of the method and illustrate that the approach is able to considerably accelerate the convergence of the gradient-based iterative method

Non-Negative Local Sparse Coding for Subspace Clustering

Subspace sparse coding (SSC) algorithms have proven to be beneficial to clustering problems. They provide an alternative data representation in which the underlying structure of the clusters can be better captured. However, most of the research in this area is mainly focused on enhancing the sparse coding part of the problem. In contrast, we introduce a novel objective term in our proposed SSC framework which focuses on the separability of data points in the coding space. We also provide mathematical insights into how this local-separability term improves the clustering result of the SSC framework. Our proposed non-linear local SSC algorithm (NLSSC) also benefits from the efficient choice of its sparsity terms and constraints. The NLSSC algorithm is also formulated in the kernel-based framework (NLKSSC) which can represent the nonlinear structure of data. In addition, we address the possibility of having redundancies in sparse coding results and its negative effect on graph-based clustering problems. We introduce the link-restore post-processing step to improve the representation graph of non-negative SSC algorithms such as ours. Empirical evaluations on well-known clustering benchmarks show that our proposed NLSSC framework results in better clusterings compared to the state-of-the-art baselines and demonstrate the effectiveness of the link-restore post-processing in improving the clustering accuracy via correcting the broken links of the representation graph.Comment: 15 pages, IDA 2018 conferenc

Die SLICOT-Toolboxen für MatlabThe SLICOT Toolboxes for Matlab

SLICOT ist eine umfangreiche Softwarebibliothek zur numerischen Behandlung von Fragestellungen aus der System- und Regelungstheorie, die mit dem Ziel entwickelt wurde, hohe Leistungsfähigkeit mit Robustheit, Verlässlichkeit, sowie Benutzerfreundlichkeit zu vereinen. Dies wird mittels einer Kombination von Fortran-Kernroutinen und Matlab- bzw. Scilab-Schnittstellen erreicht. In dieser Übersicht soll der Funktionsumfang der folgenden SLICOT-Toolboxen beschrieben und erläutert werden: (1) Grundaufgaben der System- und Regelungstheorie, (2) Systemidentifizierung, (3) Modell- und Reglerreduktion. Der Einsatz der Toolboxen in der Praxis wird durch verschiedene Beispiele veranschaulich