33,045 research outputs found
Entropy, Optimization and Counting
In this paper we study the problem of computing max-entropy distributions
over a discrete set of objects subject to observed marginals. Interest in such
distributions arises due to their applicability in areas such as statistical
physics, economics, biology, information theory, machine learning,
combinatorics and, more recently, approximation algorithms. A key difficulty in
computing max-entropy distributions has been to show that they have
polynomially-sized descriptions. We show that such descriptions exist under
general conditions. Subsequently, we show how algorithms for (approximately)
counting the underlying discrete set can be translated into efficient
algorithms to (approximately) compute max-entropy distributions. In the reverse
direction, we show how access to algorithms that compute max-entropy
distributions can be used to count, which establishes an equivalence between
counting and computing max-entropy distributions
Computation of Microcanonical Entropy at Fixed Magnetization Without Direct Counting
We discuss a method to compute the microcanonical entropy at fixed magnetization without direct counting. Our approach is based on the evaluation of a saddle-point leading to an optimization problem. The method is applied to a benchmark Ising model with simultaneous presence of mean-field and nearest-neighbour interactions for which direct counting is indeed possible, thus allowing a comparison. Moreover, we apply the method to an Ising model with mean-field, nearest-neighbour and next-nearest-neighbour interactions, for which direct counting is not straightforward. For this model, we compare the solution obtained by our method with the one obtained from the formula for the entropy in terms of all correlation functions. This example shows that for general couplings our method is much more convenient than direct counting methods to compute the microcanonical entropy at fixed magnetization
Entropy of unimodular Lattice Triangulations
Triangulations are important objects of study in combinatorics, finite
element simulations and quantum gravity, where its entropy is crucial for many
physical properties. Due to their inherent complex topological structure even
the number of possible triangulations is unknown for large systems. We present
a novel algorithm for an approximate enumeration which is based on calculations
of the density of states using the Wang-Landau flat histogram sampling. For
triangulations on two-dimensional integer lattices we achive excellent
agreement with known exact numbers of small triangulations as well as an
improvement of analytical calculated asymptotics. The entropy density is
consistent with rigorous upper and lower bounds. The presented
numerical scheme can easily be applied to other counting and optimization
problems.Comment: 6 pages, 7 figure
Guessing probability distributions from small samples
We propose a new method for the calculation of the statistical properties, as
e.g. the entropy, of unknown generators of symbolic sequences. The probability
distribution of the elements of a population can be approximated by
the frequencies of a sample provided the sample is long enough so that
each element occurs many times. Our method yields an approximation if this
precondition does not hold. For a given we recalculate the Zipf--ordered
probability distribution by optimization of the parameters of a guessed
distribution. We demonstrate that our method yields reliable results.Comment: 10 pages, uuencoded compressed PostScrip
Estimating the Spectrum in Computed Tomography Via Kullback–Leibler Divergence Constrained Optimization
Purpose
We study the problem of spectrum estimation from transmission data of a known phantom. The goal is to reconstruct an x‐ray spectrum that can accurately model the x‐ray transmission curves and reflects a realistic shape of the typical energy spectra of the CT system. Methods
Spectrum estimation is posed as an optimization problem with x‐ray spectrum as unknown variables, and a Kullback–Leibler (KL)‐divergence constraint is employed to incorporate prior knowledge of the spectrum and enhance numerical stability of the estimation process. The formulated constrained optimization problem is convex and can be solved efficiently by use of the exponentiated‐gradient (EG) algorithm. We demonstrate the effectiveness of the proposed approach on the simulated and experimental data. The comparison to the expectation–maximization (EM) method is also discussed. Results
In simulations, the proposed algorithm is seen to yield x‐ray spectra that closely match the ground truth and represent the attenuation process of x‐ray photons in materials, both included and not included in the estimation process. In experiments, the calculated transmission curve is in good agreement with the measured transmission curve, and the estimated spectra exhibits physically realistic looking shapes. The results further show the comparable performance between the proposed optimization‐based approach and EM. Conclusions
Our formulation of a constrained optimization provides an interpretable and flexible framework for spectrum estimation. Moreover, a KL‐divergence constraint can include a prior spectrum and appears to capture important features of x‐ray spectrum, allowing accurate and robust estimation of x‐ray spectrum in CT imaging
Measurement-induced disturbances and nonclassical correlations of Gaussian states
We study quantum correlations beyond entanglement in two-mode Gaussian states
of continuous variable systems, by means of the measurement-induced disturbance
(MID) and its ameliorated version (AMID). In analogy with the recent studies of
the Gaussian quantum discord, we define a Gaussian AMID by constraining the
optimization to all bi-local Gaussian positive operator valued measurements. We
solve the optimization explicitly for relevant families of states, including
squeezed thermal states. Remarkably, we find that there is a finite subset of
two-mode Gaussian states, comprising pure states, where non-Gaussian
measurements such as photon counting are globally optimal for the AMID and
realize a strictly smaller state disturbance compared to the best Gaussian
measurements. However, for the majority of two--mode Gaussian states the
unoptimized MID provides a loose overestimation of the actual content of
quantum correlations, as evidenced by its comparison with Gaussian discord.
This feature displays strong similarity with the case of two qubits. Upper and
lower bounds for the Gaussian AMID at fixed Gaussian discord are identified. We
further present a comparison between Gaussian AMID and Gaussian entanglement of
formation, and classify families of two-mode states in terms of their Gaussian
AMID, Gaussian discord, and Gaussian entanglement of formation. Our findings
provide a further confirmation of the genuinely quantum nature of general
Gaussian states, yet they reveal that non-Gaussian measurements can play a
crucial role for the optimized extraction and potential exploitation of
classical and nonclassical correlations in Gaussian states.Comment: 16 pages, 5 figures; new results added; to appear in Phys. Rev.
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