23,625 research outputs found
A Quasi-Random Approach to Matrix Spectral Analysis
Inspired by the quantum computing algorithms for Linear Algebra problems
[HHL,TaShma] we study how the simulation on a classical computer of this type
of "Phase Estimation algorithms" performs when we apply it to solve the
Eigen-Problem of Hermitian matrices. The result is a completely new, efficient
and stable, parallel algorithm to compute an approximate spectral decomposition
of any Hermitian matrix. The algorithm can be implemented by Boolean circuits
in parallel time with a total cost of Boolean
operations. This Boolean complexity matches the best known rigorous parallel time algorithms, but unlike those algorithms our algorithm is
(logarithmically) stable, so further improvements may lead to practical
implementations.
All previous efficient and rigorous approaches to solve the Eigen-Problem use
randomization to avoid bad condition as we do too. Our algorithm makes further
use of randomization in a completely new way, taking random powers of a unitary
matrix to randomize the phases of its eigenvalues. Proving that a tiny Gaussian
perturbation and a random polynomial power are sufficient to ensure almost
pairwise independence of the phases is the main technical
contribution of this work. This randomization enables us, given a Hermitian
matrix with well separated eigenvalues, to sample a random eigenvalue and
produce an approximate eigenvector in parallel time and
Boolean complexity. We conjecture that further improvements of
our method can provide a stable solution to the full approximate spectral
decomposition problem with complexity similar to the complexity (up to a
logarithmic factor) of sampling a single eigenvector.Comment: Replacing previous version: parallel algorithm runs in total
complexity and not . However, the depth of the
implementing circuit is : hence comparable to fastest
eigen-decomposition algorithms know
Strong approximation results for the empirical process of stationary sequences
We prove a strong approximation result for the empirical process associated
to a stationary sequence of real-valued random variables, under dependence
conditions involving only indicators of half lines. This strong approximation
result also holds for the empirical process associated to iterates of expanding
maps with a neutral fixed point at zero, as soon as the correlations decrease
more rapidly than for some positive . This shows that
our conditions are in some sense optimal.Comment: Published in at http://dx.doi.org/10.1214/12-AOP798 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Exact BER Performance of Asynchronous MC-DS-CDMA over Fading Channels
In this contribution an accurate average Bit Error Rate (BER) formula is derived for MC-DS-CDMA in the context of asynchronous transmissions and random spreading sequences. We consider a flat Nakagami-m fading channel for each subcarrier. Our analysis is based on the Characteristic Function (CF) and does not rely on any assumption concerning the statistical behavior of the interference. We develop a new closed-form expression for the conditional CF of the inter-carrier interference and provide a procedure for calculating the exact BER expressed in the form of a single numerical integration. The accuracy of the Standard Gaussian Approximation (SGA) technique is also evaluated. Link-level results confirm the accuracy of the SGA for most practical conditions
Growth and Structure of Stochastic Sequences
We introduce a class of stochastic integer sequences. In these sequences,
every element is a sum of two previous elements, at least one of which is
chosen randomly. The interplay between randomness and memory underlying these
sequences leads to a wide variety of behaviors ranging from stretched
exponential to log-normal to algebraic growth. Interestingly, the set of all
possible sequence values has an intricate structure.Comment: 4 pages, 4 figure
Rates of convergence in the strong invariance principle under projective criteria
We give rates of convergence in the strong invariance principle for
stationary sequences satisfying some projective criteria. The conditions are
expressed in terms of conditional expectations of partial sums of the initial
sequence. Our results apply to a large variety of examples, including mixing
processes of different kinds. We present some applications to symmetric random
walks on the circle, to functions of dependent sequences, and to a reversible
Markov chain
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