69,499 research outputs found
Deterministic Constructions of Binary Measurement Matrices from Finite Geometry
Deterministic constructions of measurement matrices in compressed sensing
(CS) are considered in this paper. The constructions are inspired by the recent
discovery of Dimakis, Smarandache and Vontobel which says that parity-check
matrices of good low-density parity-check (LDPC) codes can be used as
{provably} good measurement matrices for compressed sensing under
-minimization. The performance of the proposed binary measurement
matrices is mainly theoretically analyzed with the help of the analyzing
methods and results from (finite geometry) LDPC codes. Particularly, several
lower bounds of the spark (i.e., the smallest number of columns that are
linearly dependent, which totally characterizes the recovery performance of
-minimization) of general binary matrices and finite geometry matrices
are obtained and they improve the previously known results in most cases.
Simulation results show that the proposed matrices perform comparably to,
sometimes even better than, the corresponding Gaussian random matrices.
Moreover, the proposed matrices are sparse, binary, and most of them have
cyclic or quasi-cyclic structure, which will make the hardware realization
convenient and easy.Comment: 12 pages, 11 figure
The branching process with logistic growth
In order to model random density-dependence in population dynamics, we
construct the random analogue of the well-known logistic process in the
branching process' framework. This density-dependence corresponds to
intraspecific competition pressure, which is ubiquitous in ecology, and
translates mathematically into a quadratic death rate. The logistic branching
process, or LB-process, can thus be seen as (the mass of) a fragmentation
process (corresponding to the branching mechanism) combined with constant
coagulation rate (the death rate is proportional to the number of possible
coalescing pairs). In the continuous state-space setting, the LB-process is a
time-changed (in Lamperti's fashion) Ornstein-Uhlenbeck type process. We obtain
similar results for both constructions: when natural deaths do not occur, the
LB-process converges to a specified distribution; otherwise, it goes extinct
a.s. In the latter case, we provide the expectation and the Laplace transform
of the absorption time, as a functional of the solution of a Riccati
differential equation. We also show that the quadratic regulatory term allows
the LB-process to start at infinity, despite the fact that births occur
infinitely often as the initial state goes to \infty. This result can be viewed
as an extension of the pure-death process starting from infinity associated to
Kingman's coalescent, when some independent fragmentation is added.Comment: Published at http://dx.doi.org/10.1214/105051605000000098 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Random-time processes governed by differential equations of fractional distributed order
We analyze here different types of fractional differential equations, under
the assumption that their fractional order is random\ with
probability density We start by considering the fractional extension
of the recursive equation governing the homogeneous Poisson process
\ We prove that, for a particular (discrete) choice of , it
leads to a process with random time, defined as The distribution of the
random time argument can be
expressed, for any fixed , in terms of convolutions of stable-laws. The new
process is itself a renewal and
can be shown to be a Cox process. Moreover we prove that the survival
probability of , as well as its
probability generating function, are solution to the so-called fractional
relaxation equation of distributed order (see \cite{Vib}%).
In view of the previous results it is natural to consider diffusion-type
fractional equations of distributed order. We present here an approach to their
solutions in terms of composition of the Brownian motion with the
random time . We thus provide an
alternative to the constructions presented in Mainardi and Pagnini
\cite{mapagn} and in Chechkin et al. \cite{che1}, at least in the double-order
case.Comment: 26 page
Small Pseudo-Random Families of Matrices: Derandomizing Approximate Quantum Encryption
A quantum encryption scheme (also called private quantum channel, or state
randomization protocol) is a one-time pad for quantum messages. If two parties
share a classical random string, one of them can transmit a quantum state to
the other so that an eavesdropper gets little or no information about the state
being transmitted. Perfect encryption schemes leak no information at all about
the message. Approximate encryption schemes leak a non-zero (though small)
amount of information but require a shorter shared random key. Approximate
schemes with short keys have been shown to have a number of applications in
quantum cryptography and information theory.
This paper provides the first deterministic, polynomial-time constructions of
quantum approximate encryption schemes with short keys. Previous constructions
(quant-ph/0307104) are probabilistic--that is, they show that if the operators
used for encryption are chosen at random, then with high probability the
resulting protocol will be a secure encryption scheme. Moreover, the resulting
protocol descriptions are exponentially long. Our protocols use keys of the
same length as (or better length than) the probabilistic constructions; to
encrypt qubits approximately, one needs bits of shared key.
An additional contribution of this paper is a connection between classical
combinatorial derandomization and constructions of pseudo-random matrix
families in a continuous space.Comment: 11 pages, no figures. In Proceedings of RANDOM 2004, Cambridge, MA,
August 200
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