276,340 research outputs found
Laplacian spectral characterization of roses
A rose graph is a graph consisting of cycles that all meet in one vertex. We
show that except for two specific examples, these rose graphs are determined by
the Laplacian spectrum, thus proving a conjecture posed by Lui and Huang [F.J.
Liu and Q.X. Huang, Laplacian spectral characterization of 3-rose graphs,
Linear Algebra Appl. 439 (2013), 2914--2920]. We also show that if two rose
graphs have a so-called universal Laplacian matrix with the same spectrum, then
they must be isomorphic. In memory of Horst Sachs (1927-2016), we show the
specific case of the latter result for the adjacency matrix by using Sachs'
theorem and a new result on the number of matchings in the disjoint union of
paths
Robust variable screening for regression using factor profiling
Sure Independence Screening is a fast procedure for variable selection in
ultra-high dimensional regression analysis. Unfortunately, its performance
greatly deteriorates with increasing dependence among the predictors. To solve
this issue, Factor Profiled Sure Independence Screening (FPSIS) models the
correlation structure of the predictor variables, assuming that it can be
represented by a few latent factors. The correlations can then be profiled out
by projecting the data onto the orthogonal complement of the subspace spanned
by these factors. However, neither of these methods can handle the presence of
outliers in the data. Therefore, we propose a robust screening method which
uses a least trimmed squares method to estimate the latent factors and the
factor profiled variables. Variable screening is then performed on factor
profiled variables by using regression MM-estimators. Different types of
outliers in this model and their roles in variable screening are studied. Both
simulation studies and a real data analysis show that the proposed robust
procedure has good performance on clean data and outperforms the two nonrobust
methods on contaminated data
Eigenvectors of random matrices: A survey
Eigenvectors of large matrices (and graphs) play an essential role in
combinatorics and theoretical computer science. The goal of this survey is to
provide an up-to-date account on properties of eigenvectors when the matrix (or
graph) is random.Comment: 64 pages, 1 figure; added Section 7 on localized eigenvector
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