57,083 research outputs found
Approximation of Rectangular Beta-Laguerre Ensembles and Large Deviations
We investigate the random eigenvalues coming from the beta-Laguerre ensemble
with parameter p, which is a generalization of the real, complex and quaternion
Wishart matrices of parameter (n,p). In the case that the sample size n is much
smaller than the dimension of the population distribution p, a common situation
in modern data, we approximate the beta-Laguerre ensemble by a beta-Hermite
ensemble which is a generalization of the real, complex and quaternion Wigner
matrices. As corollaries, when n is much smaller than p, we show that the
largest and smallest eigenvalues of the complex Wishart matrix are
asymptotically independent; we obtain the limiting distribution of the
condition numbers as a sum of two i.i.d. random variables with a Tracy-Widom
distribution, which is much different from the exact square case that n=p by
Edelman (1988); we propose a test procedure for a spherical hypothesis test. By
the same approximation tool, we obtain the asymptotic distribution of the
smallest eigenvalue of the beta-Laguerre ensemble. In the second part of the
paper, under the assumption that n is much smaller than p in a certain scale,
we prove the large deviation principles for three basic statistics: the largest
eigenvalue, the smallest eigenvalue and the empirical distribution of
eigenvalues, where the last large deviation is derived by using a non-standard
method
From synaptic interactions to collective dynamics in random neuronal networks models: critical role of eigenvectors and transient behavior
The study of neuronal interactions is currently at the center of several
neuroscience big collaborative projects (including the Human Connectome, the
Blue Brain, the Brainome, etc.) which attempt to obtain a detailed map of the
entire brain matrix. Under certain constraints, mathematical theory can advance
predictions of the expected neural dynamics based solely on the statistical
properties of such synaptic interaction matrix. This work explores the
application of free random variables (FRV) to the study of large synaptic
interaction matrices. Besides recovering in a straightforward way known results
on eigenspectra of neural networks, we extend them to heavy-tailed
distributions of interactions. More importantly, we derive analytically the
behavior of eigenvector overlaps, which determine stability of the spectra. We
observe that upon imposing the neuronal excitation/inhibition balance, although
the eigenvalues remain unchanged, their stability dramatically decreases due to
strong non-orthogonality of associated eigenvectors. It leads us to the
conclusion that the understanding of the temporal evolution of asymmetric
neural networks requires considering the entangled dynamics of both
eigenvectors and eigenvalues, which might bear consequences for learning and
memory processes in these models. Considering the success of FRV analysis in a
wide variety of branches disciplines, we hope that the results presented here
foster additional application of these ideas in the area of brain sciences.Comment: 24 pages + 4 pages of refs, 8 figure
On the limit of extreme eigenvalues of large dimensional random quaternion matrices
Since E.P.Wigner (1958) established his famous semicircle law, lots of
attention has been paid by physicists, probabilists and statisticians to study
the asymptotic properties of the largest eigenvalues for random matrices. Bai
and Yin (1988) obtained the necessary and sufficient conditions for the strong
convergence of the extreme eigenvalues of a Wigner matrix. In this paper, we
consider the case of quaternion self-dual Hermitian matrices. We prove the
necessary and sufficient conditions for the strong convergence of extreme
eigenvalues of quaternion self-dual Hermitian matrices corresponding to the
Wigner case.Comment: 16 pages, 5 figure
Approximated structured pseudospectra
Pseudospectra and structured pseudospectra are important tools for the analysis of matrices. Their computation, however, can be very demanding for all but small-matrices. A new approach to compute approximations of pseudospectra and structured pseudospectra, based on determining the spectra of many suitably chosen rank-one or projected rank-one perturbations of the given matrix is proposed. The choice of rank-one or projected rank-one perturbations is inspired by Wilkinson's analysis of eigenvalue sensitivity. Numerical examples illustrate that the proposed approach gives much better insight into the pseudospectra and structured pseudospectra than random or structured random rank-one perturbations with lower computational burden. The latter approach is presently commonly used for the determination of structured pseudospectra
Induced measures in the space of mixed quantum states
We analyze several product measures in the space of mixed quantum states. In
particular we study measures induced by the operation of partial tracing. The
natural, rotationally invariant measure on the set of all pure states of a N x
K composite system, induces a unique measure in the space of N x N mixed states
(or in the space of K x K mixed states, if the reduction takes place with
respect to the first subsystem). For K=N the induced measure is equal to the
Hilbert-Schmidt measure, which is shown to coincide with the measure induced by
singular values of non-Hermitian random Gaussian matrices pertaining to the
Ginibre ensemble. We compute several averages with respect to this measure and
show that the mean entanglement of pure states behaves as lnN-1/2.Comment: 12 latex pages, 2 figures in epsf, submited to J. Phys. A. ver.3,
some improvements and a few references adde
- …