18,204 research outputs found
The geometric mean of two matrices from a computational viewpoint
The geometric mean of two matrices is considered and analyzed from a
computational viewpoint. Some useful theoretical properties are derived and an
analysis of the conditioning is performed. Several numerical algorithms based
on different properties and representation of the geometric mean are discussed
and analyzed and it is shown that most of them can be classified in terms of
the rational approximations of the inverse square root functions. A review of
the relevant applications is given
Symmetric spaces and Lie triple systems in numerical analysis of differential equations
A remarkable number of different numerical algorithms can be understood and
analyzed using the concepts of symmetric spaces and Lie triple systems, which
are well known in differential geometry from the study of spaces of constant
curvature and their tangents. This theory can be used to unify a range of
different topics, such as polar-type matrix decompositions, splitting methods
for computation of the matrix exponential, composition of selfadjoint numerical
integrators and dynamical systems with symmetries and reversing symmetries. The
thread of this paper is the following: involutive automorphisms on groups
induce a factorization at a group level, and a splitting at the algebra level.
In this paper we will give an introduction to the mathematical theory behind
these constructions, and review recent results. Furthermore, we present a new
Yoshida-like technique, for self-adjoint numerical schemes, that allows to
increase the order of preservation of symmetries by two units. Since all the
time-steps are positive, the technique is particularly suited to stiff
problems, where a negative time-step can cause instabilities
Time Reversal and n-qubit Canonical Decompositions
For n an even number of qubits and v a unitary evolution, a matrix
decomposition v=k1 a k2 of the unitary group is explicitly computable and
allows for study of the dynamics of the concurrence entanglement monotone. The
side factors k1 and k2 of this Concurrence Canonical Decomposition (CCD) are
concurrence symmetries, so the dynamics reduce to consideration of the a
factor. In this work, we provide an explicit numerical algorithm computing v=k1
a k2 for n odd. Further, in the odd case we lift the monotone to a two-argument
function, allowing for a theory of concurrence dynamics in odd qubits. The
generalization may also be studied using the CCD, leading again to maximal
concurrence capacity for most unitaries. The key technique is to consider the
spin-flip as a time reversal symmetry operator in Wigner's axiomatization; the
original CCD derivation may be restated entirely in terms of this time
reversal. En route, we observe a Kramers' nondegeneracy: the existence of a
nondegenerate eigenstate of any time reversal symmetric n-qubit Hamiltonian
demands (i) n even and (ii) maximal concurrence of said eigenstate. We provide
examples of how to apply this work to study the kinematics and dynamics of
entanglement in spin chain Hamiltonians.Comment: 20 pages, 3 figures; v2 (17pp.): major revision, new abstract,
introduction, expanded bibliograph
Generalized power method for sparse principal component analysis
In this paper we develop a new approach to sparse principal component
analysis (sparse PCA). We propose two single-unit and two block optimization
formulations of the sparse PCA problem, aimed at extracting a single sparse
dominant principal component of a data matrix, or more components at once,
respectively. While the initial formulations involve nonconvex functions, and
are therefore computationally intractable, we rewrite them into the form of an
optimization program involving maximization of a convex function on a compact
set. The dimension of the search space is decreased enormously if the data
matrix has many more columns (variables) than rows. We then propose and analyze
a simple gradient method suited for the task. It appears that our algorithm has
best convergence properties in the case when either the objective function or
the feasible set are strongly convex, which is the case with our single-unit
formulations and can be enforced in the block case. Finally, we demonstrate
numerically on a set of random and gene expression test problems that our
approach outperforms existing algorithms both in quality of the obtained
solution and in computational speed.Comment: Submitte
Optimal map of the modular structure of complex networks
Modular structure is pervasive in many complex networks of interactions
observed in natural, social and technological sciences. Its study sheds light
on the relation between the structure and function of complex systems.
Generally speaking, modules are islands of highly connected nodes separated by
a relatively small number of links. Every module can have contributions of
links from any node in the network. The challenge is to disentangle these
contributions to understand how the modular structure is built. The main
problem is that the analysis of a certain partition into modules involves, in
principle, as many data as number of modules times number of nodes. To confront
this challenge, here we first define the contribution matrix, the mathematical
object containing all the information about the partition of interest, and
after, we use a Truncated Singular Value Decomposition to extract the best
representation of this matrix in a plane. The analysis of this projection allow
us to scrutinize the skeleton of the modular structure, revealing the structure
of individual modules and their interrelations.Comment: 21 pages, 10 figure
Generalized power method for sparse principal component analysis
In this paper we develop a new approach to sparse principal component analysis (sparse PCA). We propose two single-unit and two block optimization formulations of the sparse PCA problem, aimed at extracting a single sparse dominant principal component of a data matrix, or more components at once, respectively. While the initial formulations involve nonconvex functions, and are therefore computationally intractable, we rewrite them into the form of an optimization program involving maximization of a convex function on a compact set. The dimension of the search space is decreased enormously if the data matrix has many more columns (variables) than rows. We then propose and analyze a simple gradient method suited for the task. It appears that our algorithm has best convergence properties in the case when either the objective function or the feasible set are strongly convex, which is the case with our single-unit formulations and can be enforced in the block case. Finally, we demonstrate numerically on a set of random and gene expression test problems that our approach outperforms existing algorithms both in quality of the obtained solution and in computational speed.sparse PCA, power method, gradient ascent, strongly convex sets, block algorithms.
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