637 research outputs found
Concentration of Measure Inequalities for Toeplitz Matrices with Applications
We derive Concentration of Measure (CoM) inequalities for randomized Toeplitz
matrices. These inequalities show that the norm of a high-dimensional signal
mapped by a Toeplitz matrix to a low-dimensional space concentrates around its
mean with a tail probability bound that decays exponentially in the dimension
of the range space divided by a quantity which is a function of the signal. For
the class of sparse signals, the introduced quantity is bounded by the sparsity
level of the signal. However, we observe that this bound is highly pessimistic
for most sparse signals and we show that if a random distribution is imposed on
the non-zero entries of the signal, the typical value of the quantity is
bounded by a term that scales logarithmically in the ambient dimension. As an
application of the CoM inequalities, we consider Compressive Binary Detection
(CBD).Comment: Initial Submission to the IEEE Transactions on Signal Processing on
December 1, 2011. Revised and Resubmitted on July 12, 201
Estimation of Toeplitz Covariance Matrices in Large Dimensional Regime with Application to Source Detection
In this article, we derive concentration inequalities for the spectral norm
of two classical sample estimators of large dimensional Toeplitz covariance
matrices, demonstrating in particular their asymptotic almost sure consistence.
The consistency is then extended to the case where the aggregated matrix of
time samples is corrupted by a rank one (or more generally, low rank) matrix.
As an application of the latter, the problem of source detection in the context
of large dimensional sensor networks within a temporally correlated noise
environment is studied. As opposed to standard procedures, this application is
performed online, i.e. without the need to possess a learning set of pure noise
samples.Comment: 20 pages, 3 figures, submitted to IEEE Transactions on Signal
Processin
A general approach to small deviation via concentration of measures
We provide a general approach to obtain upper bounds for small deviations in different norms, namely the supremum
and - H\"older norms. The large class of processes under
consideration takes the form , where and
are two possibly dependent stochastic processes. Our approach provides an upper
bound for small deviations whenever upper bounds for the \textit{concentration
of measures} of - norm of random vectors built from increments of the
process and \textit{large deviation} estimates for the process are
available. Using our method, among others, we obtain the optimal rates of small
deviations in supremum and - H\"older norms for fractional Brownian
motion with Hurst parameter . As an application, we discuss
the usefulness of our upper bounds for small deviations in pathwise stochastic
integral representation of random variables motivated by the hedging problem in
mathematical finance
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