4,687 research outputs found
Random processes via the combinatorial dimension: introductory notes
This is an informal discussion on one of the basic problems in the theory of
empirical processes, addressed in our preprint "Combinatorics of random
processes and sections of convex bodies", which is available at ArXiV and from
our web pages.Comment: 4 page
Sampling from large matrices: an approach through geometric functional analysis
We study random submatrices of a large matrix A. We show how to approximately
compute A from its random submatrix of the smallest possible size O(r log r)
with a small error in the spectral norm, where r = ||A||_F^2 / ||A||_2^2 is the
numerical rank of A. The numerical rank is always bounded by, and is a stable
relaxation of, the rank of A. This yields an asymptotically optimal guarantee
in an algorithm for computing low-rank approximations of A. We also prove
asymptotically optimal estimates on the spectral norm and the cut-norm of
random submatrices of A. The result for the cut-norm yields a slight
improvement on the best known sample complexity for an approximation algorithm
for MAX-2CSP problems. We use methods of Probability in Banach spaces, in
particular the law of large numbers for operator-valued random variables.Comment: Our initial claim about Max-2-CSP problems is corrected. We put an
exponential failure probability for the algorithm for low-rank
approximations. Proofs are a little more explaine
No-gaps delocalization for general random matrices
We prove that with high probability, every eigenvector of a random matrix is
delocalized in the sense that any subset of its coordinates carries a
non-negligible portion of its norm. Our results pertain to a wide
class of random matrices, including matrices with independent entries,
symmetric and skew-symmetric matrices, as well as some other naturally arising
ensembles. The matrices can be real and complex; in the latter case we assume
that the real and imaginary parts of the entries are independent.Comment: 45 page
Geometric approach to error correcting codes and reconstruction of signals
We develop an approach through geometric functional analysis to error
correcting codes and to reconstruction of signals from few linear measurements.
An error correcting code encodes an n-letter word x into an m-letter word y in
such a way that x can be decoded correctly when any r letters of y are
corrupted. We prove that most linear orthogonal transformations Q from R^n into
R^m form efficient and robust robust error correcting codes over reals. The
decoder (which corrects the corrupted components of y) is the metric projection
onto the range of Q in the L_1 norm. An equivalent problem arises in signal
processing: how to reconstruct a signal that belongs to a small class from few
linear measurements? We prove that for most sets of Gaussian measurements, all
signals of small support can be exactly reconstructed by the L_1 norm
minimization. This is a substantial improvement of recent results of Donoho and
of Candes and Tao. An equivalent problem in combinatorial geometry is the
existence of a polytope with fixed number of facets and maximal number of
lower-dimensional facets. We prove that most sections of the cube form such
polytopes.Comment: 17 pages, 3 figure
The Littlewood-Offord Problem and invertibility of random matrices
We prove two basic conjectures on the distribution of the smallest singular
value of random n times n matrices with independent entries. Under minimal
moment assumptions, we show that the smallest singular value is of order
n^{-1/2}, which is optimal for Gaussian matrices. Moreover, we give a optimal
estimate on the tail probability. This comes as a consequence of a new and
essentially sharp estimate in the Littlewood-Offord problem: for i.i.d. random
variables X_k and real numbers a_k, determine the probability P that the sum of
a_k X_k lies near some number v. For arbitrary coefficients a_k of the same
order of magnitude, we show that they essentially lie in an arithmetic
progression of length 1/p.Comment: Introduction restructured, some typos and minor errors correcte
Invertibility of random matrices: unitary and orthogonal perturbations
We show that a perturbation of any fixed square matrix D by a random unitary
matrix is well invertible with high probability. A similar result holds for
perturbations by random orthogonal matrices; the only notable exception is when
D is close to orthogonal. As an application, these results completely eliminate
a hard-to-check condition from the Single Ring Theorem by Guionnet, Krishnapur
and Zeitouni.Comment: 46 pages. A more general result on orthogonal perturbations of
complex matrices added. It rectified an inaccuracy in application to Single
Ring Theorem for orthogonal matrice
On random intersections of two convex bodies. Appendix to: "Isoperimetry of waists and local versus global asymptotic convex geometries" by R.Vershynin
In the paper "Isoperimetry of waists and local versus global asymptotic
convex geometries", it was proved that the existence of nicely bounded sections
of two symmetric convex bodies K and L implies that the intersection of
randomly rotated K and L is nicely bounded. In this appendix, we achieve a
polynomial bound on the diameter of that intersection (in the ratio of the
dimensions of the sections)
Small ball probabilities for linear images of high dimensional distributions
We study concentration properties of random vectors of the form , where
has independent coordinates and is a given matrix. We
show that the distribution of is well spread in space whenever the
distributions of are well spread on the line. Specifically, assume that
the probability that falls in any given interval of length is at most
. Then the probability that falls in any given ball of radius is at most , where denotes the stable rank
of and is an absolute constant.Comment: 18 pages. A statement of Rogozin's theorem is added. Small
corrections are mad
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