991 research outputs found
Forest matrices around the Laplacian matrix
We study the matrices Q_k of in-forests of a weighted digraph G and their
connections with the Laplacian matrix L of G. The (i,j) entry of Q_k is the
total weight of spanning converging forests (in-forests) with k arcs such that
i belongs to a tree rooted at j. The forest matrices, Q_k, can be calculated
recursively and expressed by polynomials in the Laplacian matrix; they provide
representations for the generalized inverses, the powers, and some eigenvectors
of L. The normalized in-forest matrices are row stochastic; the normalized
matrix of maximum in-forests is the eigenprojection of the Laplacian matrix,
which provides an immediate proof of the Markov chain tree theorem. A source of
these results is the fact that matrices Q_k are the matrix coefficients in the
polynomial expansion of adj(a*I+L). Thereby they are precisely Faddeev's
matrices for -L.
Keywords: Weighted digraph; Laplacian matrix; Spanning forest; Matrix-forest
theorem; Leverrier-Faddeev method; Markov chain tree theorem; Eigenprojection;
Generalized inverse; Singular M-matrixComment: 19 pages, presented at the Edinburgh (2001) Conference on Algebraic
Graph Theor
Some Preconditioning Techniques for Saddle Point Problems
Saddle point problems arise frequently in many applications in science and engineering, including constrained optimization, mixed finite element formulations of partial differential equations, circuit analysis, and so forth. Indeed the formulation of most problems with constraints gives rise to saddle point systems. This paper provides a concise overview of iterative approaches for the solution of such systems which are of particular importance in the context of large scale computation. In particular we describe some of the most useful preconditioning techniques for Krylov subspace solvers applied to saddle point problems, including block and constrained preconditioners.\ud
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The work of Michele Benzi was supported in part by the National Science Foundation grant DMS-0511336
Polynomials with Lorentzian Signature, and Computing Permanents via Hyperbolic Programming
We study the class of polynomials whose Hessians evaluated at any point of a
closed convex cone have Lorentzian signature. This class is a generalization to
the remarkable class of Lorentzian polynomials. We prove that hyperbolic
polynomials and conic stable polynomials belong to this class, and the set of
polynomials with Lorentzian signature is closed. Finally, we develop a method
for computing permanents of nonsingular matrices which belong to a class that
includes nonsingular -locally singular matrices via hyperbolic programming
Robust Low-Rank Subspace Segmentation with Semidefinite Guarantees
Recently there is a line of research work proposing to employ Spectral
Clustering (SC) to segment (group){Throughout the paper, we use segmentation,
clustering, and grouping, and their verb forms, interchangeably.}
high-dimensional structural data such as those (approximately) lying on
subspaces {We follow {liu2010robust} and use the term "subspace" to denote both
linear subspaces and affine subspaces. There is a trivial conversion between
linear subspaces and affine subspaces as mentioned therein.} or low-dimensional
manifolds. By learning the affinity matrix in the form of sparse
reconstruction, techniques proposed in this vein often considerably boost the
performance in subspace settings where traditional SC can fail. Despite the
success, there are fundamental problems that have been left unsolved: the
spectrum property of the learned affinity matrix cannot be gauged in advance,
and there is often one ugly symmetrization step that post-processes the
affinity for SC input. Hence we advocate to enforce the symmetric positive
semidefinite constraint explicitly during learning (Low-Rank Representation
with Positive SemiDefinite constraint, or LRR-PSD), and show that factually it
can be solved in an exquisite scheme efficiently instead of general-purpose SDP
solvers that usually scale up poorly. We provide rigorous mathematical
derivations to show that, in its canonical form, LRR-PSD is equivalent to the
recently proposed Low-Rank Representation (LRR) scheme {liu2010robust}, and
hence offer theoretic and practical insights to both LRR-PSD and LRR, inviting
future research. As per the computational cost, our proposal is at most
comparable to that of LRR, if not less. We validate our theoretic analysis and
optimization scheme by experiments on both synthetic and real data sets.Comment: 10 pages, 4 figures. Accepted by ICDM Workshop on Optimization Based
Methods for Emerging Data Mining Problems (OEDM), 2010. Main proof simplified
and typos corrected. Experimental data slightly adde
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Observer Design for Interconnected Systems and Implementation via Differential-Algebraic Equations
A new approach to the design of observers of nonlinear dynamical systems is presented. Generally, linear or nonlinear control systems are expressed as explicit systems of differential equations and solved either analytically or numerically. If numerically, they are implemented using standard ordinary differential equation (ODE) solvers. In this thesis, a system is decomposed and modeled as an interconnection between two observer subsystems, particularly, as canonical DAE observers. In general, control design engineers may be faced with a formidable problem of solving this system analytically or in obtaining closed-form solutions. To attest to the complexity and complications in treating a system of interconnected DAE observer systems, a scaled-down version of a publication on “Small-Gain Theorem” is included in the appendix for the reader’s perusal. (A brief introduction to “Small-Gain Theorem” can be found in Chapter 4). The premise of this thesis is to demonstrate that, where the design of an observer plays a major role involving output feedback, there may be advantages in formulating a control system as a differential-algebraic equation (DAE), especially in the case of interconnected subsystems. An implicit system of interconnected DAE observers is considered and shown implementable using an existing DAE solver, whose resolution allows one the capability of computing input and output bounds. This is based on fixed or variable timesteps within the operating interval of each subsystem to ensure input-output stability (IOS) and the observability property of the interconnected observer system. The observer design method is based on the extended linearization approach. The basic background is provided for the design process of an interconnected observer system using DAE. Note, the application of the new approach has not been considered previously for the case of an interconnected DAE observer system
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