1,723 research outputs found
Minimizing Rational Functions by Exact Jacobian SDP Relaxation Applicable to Finite Singularities
This paper considers the optimization problem of minimizing a rational
function. We reformulate this problem as polynomial optimization by the
technique of homogenization. These two problems are shown to be equivalent
under some generic conditions. The exact Jacobian SDP relaxation method
proposed by Nie is used to solve the resulting polynomial optimization. We also
prove that the assumption of nonsingularity in Nie's method can be weakened as
the finiteness of singularities. Some numerical examples are given to
illustrate the efficiency of our method.Comment: 23 page
Universality for the largest eigenvalue of sample covariance matrices with general population
This paper is aimed at deriving the universality of the largest eigenvalue of
a class of high-dimensional real or complex sample covariance matrices of the
form . Here, is
an random matrix with independent entries such that , . On
dimensionality, we assume that and as
. For a class of general deterministic positive-definite
matrices , under some additional assumptions on the
distribution of 's, we show that the limiting behavior of the largest
eigenvalue of is universal, via pursuing a Green function
comparison strategy raised in [Probab. Theory Related Fields 154 (2012)
341-407, Adv. Math. 229 (2012) 1435-1515] by Erd\H{o}s, Yau and Yin for Wigner
matrices and extended by Pillai and Yin [Ann. Appl. Probab. 24 (2014) 935-1001]
to sample covariance matrices in the null case (). Consequently, in
the standard complex case (), combing this universality
property and the results known for Gaussian matrices obtained by El Karoui in
[Ann. Probab. 35 (2007) 663-714] (nonsingular case) and Onatski in [Ann. Appl.
Probab. 18 (2008) 470-490] (singular case), we show that after an appropriate
normalization the largest eigenvalue of converges weakly to the
type 2 Tracy-Widom distribution . Moreover, in the real case, we
show that when is spiked with a fixed number of subcritical spikes,
the type 1 Tracy-Widom limit holds for the normalized largest
eigenvalue of , which extends a result of F\'{e}ral and
P\'{e}ch\'{e} in [J. Math. Phys. 50 (2009) 073302] to the scenario of
nondiagonal and more generally distributed .Comment: Published in at http://dx.doi.org/10.1214/14-AOS1281 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Universality for a global property of the eigenvectors of Wigner matrices
Let be an real (resp. complex) Wigner matrix and
be its spectral decomposition. Set
, where is a real (resp.
complex) unit vector. Under the assumption that the elements of have 4
matching moments with those of GOE (resp. GUE), we show that the process
converges weakly to the Brownian bridge for any
such that as ,
where for the real case and for the complex case. Such a
result indicates that the othorgonal (resp. unitary) matrices with columns
being the eigenvectors of Wigner matrices are asymptotically Haar distributed
on the orthorgonal (resp. unitary) group from a certain perspective.Comment: typos correcte
Tracy-Widom law for the extreme eigenvalues of sample correlation matrices
Let the sample correlation matrix be , where with
. We assume to be a collection of independent symmetric distributed
random variables with sub-exponential tails. Moreover, for any , we assume
to be identically distributed. We assume and
with some as . In this
paper, we provide the Tracy-Widom law () for both the largest and
smallest eigenvalues of . If are i.i.d. standard normal, we can
derive the for both the largest and smallest eigenvalues of the matrix
, where with , .Comment: 35 pages, a major revisio
A Note on Rate of Convergence in Probability to Semicircular Law
In the present paper, we prove that under the assumption of the finite sixth
moment for elements of a Wigner matrix, the convergence rate of its empirical
spectral distribution to the Wigner semicircular law in probability is
when the dimension tends to infinity.Comment: 13 page
- …