77 research outputs found
Numerical range for random matrices
We analyze the numerical range of high-dimensional random matrices, obtaining
limit results and corresponding quantitative estimates in the non-limit case.
For a large class of random matrices their numerical range is shown to converge
to a disc. In particular, numerical range of complex Ginibre matrix almost
surely converges to the disk of radius . Since the spectrum of
non-hermitian random matrices from the Ginibre ensemble lives asymptotically in
a neighborhood of the unit disk, it follows that the outer belt of width
containing no eigenvalues can be seen as a quantification the
non-normality of the complex Ginibre random matrix. We also show that the
numerical range of upper triangular Gaussian matrices converges to the same
disk of radius , while all eigenvalues are equal to zero and we prove
that the operator norm of such matrices converges to .Comment: 23 pages, 4 figure
Tail estimates for norms of sums of log-concave random vectors
We establish new tail estimates for order statistics and for the Euclidean
norms of projections of an isotropic log-concave random vector. More generally,
we prove tail estimates for the norms of projections of sums of independent
log-concave random vectors, and uniform versions of these in the form of tail
estimates for operator norms of matrices and their sub-matrices in the setting
of a log-concave ensemble. This is used to study a quantity that
controls uniformly the operator norm of the sub-matrices with rows and
columns of a matrix with independent isotropic log-concave random rows. We
apply our tail estimates of to the study of Restricted Isometry
Property that plays a major role in the Compressive Sensing theory
Chevet type inequality and norms of submatrices
We prove a Chevet type inequality which gives an upper bound for the norm of
an isotropic log-concave unconditional random matrix in terms of expectation of
the supremum of "symmetric exponential" processes compared to the Gaussian ones
in the Chevet inequality. This is used to give sharp upper estimate for a
quantity that controls uniformly the Euclidean operator norm of
the sub-matrices with rows and columns of an isotropic log-concave
unconditional random matrix. We apply these estimates to give a sharp bound for
the Restricted Isometry Constant of a random matrix with independent
log-concave unconditional rows. We show also that our Chevet type inequality
does not extend to general isotropic log-concave random matrices
Quantitative estimates of the convergence of the empirical covariance matrix in Log-concave Ensembles
Let be an isotropic convex body in . Given \eps>0, how many
independent points uniformly distributed on are needed for the
empirical covariance matrix to approximate the identity up to \eps with
overwhelming probability? Our paper answers this question posed by Kannan,
Lovasz and Simonovits. More precisely, let be a centered random
vector with a log-concave distribution and with the identity as covariance
matrix. An example of such a vector is a random point in an isotropic
convex body. We show that for any \eps>0, there exists C(\eps)>0, such that
if N\sim C(\eps) n and are i.i.d. copies of , then
\Big\|\frac{1}{N}\sum_{i=1}^N X_i\otimes X_i - \Id\Big\| \le \epsilon, with
probability larger than .Comment: Exposition changed, several explanatory remarks added, some proofs
simplifie
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