28 research outputs found
Penalized Orthogonal Iteration for Sparse Estimation of Generalized Eigenvalue Problem
We propose a new algorithm for sparse estimation of eigenvectors in
generalized eigenvalue problems (GEP). The GEP arises in a number of modern
data-analytic situations and statistical methods, including principal component
analysis (PCA), multiclass linear discriminant analysis (LDA), canonical
correlation analysis (CCA), sufficient dimension reduction (SDR) and invariant
co-ordinate selection. We propose to modify the standard generalized orthogonal
iteration with a sparsity-inducing penalty for the eigenvectors. To achieve
this goal, we generalize the equation-solving step of orthogonal iteration to a
penalized convex optimization problem. The resulting algorithm, called
penalized orthogonal iteration, provides accurate estimation of the true
eigenspace, when it is sparse. Also proposed is a computationally more
efficient alternative, which works well for PCA and LDA problems. Numerical
studies reveal that the proposed algorithms are competitive, and that our
tuning procedure works well. We demonstrate applications of the proposed
algorithm to obtain sparse estimates for PCA, multiclass LDA, CCA and SDR.
Supplementary materials are available online
Community detection in networks via nonlinear modularity eigenvectors
Revealing a community structure in a network or dataset is a central problem
arising in many scientific areas. The modularity function is an established
measure quantifying the quality of a community, being identified as a set of
nodes having high modularity. In our terminology, a set of nodes with positive
modularity is called a \textit{module} and a set that maximizes is thus
called \textit{leading module}. Finding a leading module in a network is an
important task, however the dimension of real-world problems makes the
maximization of unfeasible. This poses the need of approximation techniques
which are typically based on a linear relaxation of , induced by the
spectrum of the modularity matrix . In this work we propose a nonlinear
relaxation which is instead based on the spectrum of a nonlinear modularity
operator . We show that extremal eigenvalues of
provide an exact relaxation of the modularity measure , however at the price
of being more challenging to be computed than those of . Thus we extend the
work made on nonlinear Laplacians, by proposing a computational scheme, named
\textit{generalized RatioDCA}, to address such extremal eigenvalues. We show
monotonic ascent and convergence of the method. We finally apply the new method
to several synthetic and real-world data sets, showing both effectiveness of
the model and performance of the method
Perturbation splitting for more accurate eigenvalues
Let be a symmetric tridiagonal matrix with entries and
eigenvalues of different magnitudes. For some , small entrywise
relative perturbations induce small errors in the eigenvalues,
independently of the size of the entries of the matrix; this is
certainly true when the perturbed matrix can be written as
with small . Even if it is
not possible to express in this way the perturbations in every
entry of , much can be gained by doing so for as many as
possible entries of larger magnitude. We propose a technique which
consists of splitting multiplicative and additive perturbations
to produce new error bounds which, for some matrices, are much
sharper than the usual ones. Such bounds may be useful in the
development of improved software for the tridiagonal eigenvalue
problem, and we describe their role in the context of a mixed
precision bisection-like procedure. Using the very same idea of
splitting perturbations (multiplicative and additive), we show
that when defines well its eigenvalues, the numerical values
of the pivots in the usual decomposition may
be used to compute approximations with high relative precision.Fundação para a Ciência e Tecnologia (FCT) - POCI 201