48 research outputs found
The power of random measurements: measuring Tr(\rho^n) on single copies of \rho
While it is known that Tr(\rho^n) can be measured directly (i.e., without
first reconstructing the density matrix) by performing joint measurements on n
copies of the same state rho, it is shown here that random measurements on
single copies suffice, too. Averaging over the random measurements directly
yields estimates of Tr(\rho^n), even when it is not known what measurements
were actually performed (so that one cannot reconstruct \rho)
Homological perturbation theory for nonperturbative integrals
We use the homological perturbation lemma to produce explicit formulas
computing the class in the twisted de Rham complex represented by an arbitrary
polynomial. This is a non-asymptotic version of the method of Feynman diagrams.
In particular, we explain that phenomena usually thought of as particular to
asymptotic integrals in fact also occur exactly: integrals of the type
appearing in quantum field theory can be reduced in a totally algebraic fashion
to integrals over an Euler--Lagrange locus, provided this locus is understood
in the scheme-theoretic sense, so that imaginary critical points and
multiplicities of degenerate critical points contribute.Comment: 22 pages. Minor revisions from previous versio
Non-Parametric Approximations for Anisotropy Estimation in Two-dimensional Differentiable Gaussian Random Fields
Spatially referenced data often have autocovariance functions with elliptical
isolevel contours, a property known as geometric anisotropy. The anisotropy
parameters include the tilt of the ellipse (orientation angle) with respect to
a reference axis and the aspect ratio of the principal correlation lengths.
Since these parameters are unknown a priori, sample estimates are needed to
define suitable spatial models for the interpolation of incomplete data. The
distribution of the anisotropy statistics is determined by a non-Gaussian
sampling joint probability density. By means of analytical calculations, we
derive an explicit expression for the joint probability density function of the
anisotropy statistics for Gaussian, stationary and differentiable random
fields. Based on this expression, we obtain an approximate joint density which
we use to formulate a statistical test for isotropy. The approximate joint
density is independent of the autocovariance function and provides conservative
probability and confidence regions for the anisotropy parameters. We validate
the theoretical analysis by means of simulations using synthetic data, and we
illustrate the detection of anisotropy changes with a case study involving
background radiation exposure data. The approximate joint density provides (i)
a stand-alone approximate estimate of the anisotropy statistics distribution
(ii) informed initial values for maximum likelihood estimation, and (iii) a
useful prior for Bayesian anisotropy inference.Comment: 39 pages; 8 figure
The monomer-dimer problem and moment Lyapunov exponents of homogeneous Gaussian random fields
We consider an "elastic" version of the statistical mechanical monomer-dimer
problem on the n-dimensional integer lattice. Our setting includes the
classical "rigid" formulation as a special case and extends it by allowing each
dimer to consist of particles at arbitrarily distant sites of the lattice, with
the energy of interaction between the particles in a dimer depending on their
relative position. We reduce the free energy of the elastic dimer-monomer (EDM)
system per lattice site in the thermodynamic limit to the moment Lyapunov
exponent (MLE) of a homogeneous Gaussian random field (GRF) whose mean value
and covariance function are the Boltzmann factors associated with the monomer
energy and dimer potential. In particular, the classical monomer-dimer problem
becomes related to the MLE of a moving average GRF. We outline an approach to
recursive computation of the partition function for "Manhattan" EDM systems
where the dimer potential is a weighted l1-distance and the auxiliary GRF is a
Markov random field of Pickard type which behaves in space like autoregressive
processes do in time. For one-dimensional Manhattan EDM systems, we compute the
MLE of the resulting Gaussian Markov chain as the largest eigenvalue of a
compact transfer operator on a Hilbert space which is related to the
annihilation and creation operators of the quantum harmonic oscillator and also
recast it as the eigenvalue problem for a pantograph functional-differential
equation.Comment: 24 pages, 4 figures, submitted on 14 October 2011 to a special issue
of DCDS-
Correlated topographic analysis: estimating an ordering of correlated components
Abstract This paper describes a novel method, which we call correlated topographic analysis (CTA), to estimate non-Gaussian components and their ordering (topography). The method is inspired by a central motivation of recent variants of independent component analysis (ICA), namely, to make use of the residual statistical dependency which ICA cannot remove. We assume that components nearby on the topographic arrangement have both linear and energy correlations, while far-away components are statistically independent. We use these dependencies to fix the ordering of the components. We start by proposing the generative model for the components. Then, we derive an approximation of the likelihood based on the model. Furthermore, since gradient methods tend to get stuck in local optima, we propose a three-step optimization method which dramatically improves topographic estimation. Using simulated data, we show that CTA estimates an ordering of the components and generalizes a previous method in terms of topography estimation. Finally, to demonstrate that CTA is widely applicable, we learn topographic representations for three kinds of real data: natural images, outputs of simulated complex cells and text data