Moment data can be analytically completed

Gavriliadis, P. N.; Athanassoulis, G. A.

doi:10.1016/j.probengmech.2003.07.001

Cited by 11 publications

(4 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, the first few moments are only sufficient if the class of functions in which the reconstruction is sought is restricted appropriately. It has been mentioned in the literature that the most of the information about the measure is contained in the first few moments, and the higher-order ones providing only little additional information [18,27,22]. Since the moment matrix has sufficient information about the distribution, a distance between moment matrices can be calculated to measure a distance between two spectral distributions.…”

Section: Remark That the Spectral Distribution Includes Information Nmentioning

confidence: 99%

See 1 more Smart Citation

Comparing large-scale graphs based on quantum probability theory

Choi

Lee

Shen

et al. 2019

Applied Mathematics and Computation

View full text Add to dashboard Cite

In this paper, a new measurement to compare two large-scale graphs based on the theory of quantum probability is proposed. An explicit form for the spectral distribution of the corresponding adjacency matrix of a graph is established. Our proposed distance between two graphs is defined as the distance between the corresponding moment matrices of their spectral distributions. It is shown that the spectral distributions of their adjacency matrices in a vector state includes information not only about their eigenvalues, but also about the corresponding eigenvectors. Moreover, we prove that such distance is graph invariant and sub-structure invariant. Examples with various graphs are given, and distances between graphs with few vertices are checked. Computational results for real large-scale networks show that its accuracy is better than any existing methods and time cost is extensively cheap. (Yuanming Shi ) 1 This article is partially based on preliminary results published in the proceeding of IEEE International Symposium on Information Theory from June 17 to 22, 2018.

show abstract

Section: Remark That the Spectral Distribution Includes Information Nmentioning

confidence: 99%

“…We consider comparing two graphs G ). However, n is relatively small, say 4 or 5, in practical problems because most of the information about a distribution is contained in the first few moments [18,27,22]. Thus the time complexity and space complexity of proposed method are both O(|E|).…”

Section: Complexitymentioning

confidence: 99%

Comparing large-scale graphs based on quantum probability theory

Choi

Lee

Shen

et al. 2019

Applied Mathematics and Computation

View full text Add to dashboard Cite

show abstract

“…II). (It has been conjectured that "most of the information defining a compactly supported [probability distribution function] is usually contained in its first few moments" [12][13][14].) As a complementary exercise, we similarly analyze-but with considerably less severe computational demands-the Hilbert-Schmidt probability distribution over the determinant |ρ| itself (sec.…”

Section: Introductionmentioning

confidence: 99%

Moments of the Hilbert-Schmidt probability distributions over determinants of real two-qubit density matrices and of their partial transposes

Slater¹

2010

Preprint

View full text Add to dashboard Cite

The nonnegativity of the determinant of the partial transpose of a two-qubit (4×4) density matrix (ρ) is both a necessary and sufficient condition for the separability of ρ. While the determinant of ρ itself is restricted to the interval [0, 1 256 ], the determinant of the partial transpose (|ρ P T |) can range over [− 1 16 , 1 256 ], with negative values corresponding to entangled states. We report here the exact values of the first nine moments of the probability distribution of |ρ P T | over this interval, with respect to the Hilbert-Schmidt (metric volume element) measure on the nine-dimensional convex set of real two-qubit density matrices. Rational functions C 2j (m), yielding the coefficients of the 2j-th power of even polynomials occurring at intermediate steps in our derivation of the m-th moment, emerge. These functions possess poles at finite series of consecutive half-integers (m = − 3 2 , − 1 2 , . . . , 2j−1 2 ), and certain (trivial) roots at finite series of consecutive natural numbers (m = 0, 1, . . .). Additionally, the (nontrivial) dominant roots of C 2j (m) approach the same halfinteger values (m = 2j−1 2 , 2j−3 2 , . . .), as j increases. The first two moments (mean and variance) found-when employed in the one-sided Chebyshev inequality-give an upper bound of 30397 34749 ≈ 0.874759 on the separability probability of real two-qubit density matrices. We are able to report general formulas for the m-th moment of the Hilbert-Schmidt probability distribution of |ρ| over [0, 1 256 ], in the real, complex and quaternionic two-qubit cases.

show abstract

“…The a priori characterization of the (moment) data space can be achieved by using appropriate restrictions on the moment data, ensuring that the given moment sequence, does belong to the appropriate moment space [9]. The characterization of the solution space is mainly controlled by the choice of an appropriate representation for the function to be recovered [5,29,2,11].…”

Section: Introductionmentioning

confidence: 99%

Moment information for probability distributions, without solving the moment problem, II: Main-mass, tails and shape approximation

Gavriliadis

Athanassoulis

2009

Journal of Computational and Applied Mathematics

Self Cite

View full text Add to dashboard Cite

a b s t r a c tHow much information does a small number of moments carry about the unknown distribution function? Is it possible to explicitly obtain from these moments some useful information, e.g., about the support, the modality, the general shape, or the tails of a distribution, without going into a detailed numerical solution of the moment problem? In this, previous and subsequent papers, clear and easy to implement answers will be given to some questions of this type. First, the question of how to distinguish between the main-mass interval and the tail regions, in the case we know only a number of moments of the target distribution function, will be addressed. The answer to this question is based on a version of the Chebyshev-Stieltjes-Markov inequality, which provides us with upper and lower, moment-based, bounds for the target distribution. Then, exploiting existing asymptotic results in the main-mass region, an explicit, moment-based approximation of the target probability density function is provided. Although the latter cannot be considered, in general, as a satisfactory solution, it can always serve as an initial approximation in any iterative scheme for the numerical solution of the moment problem. Numerical results illustrating all the theoretical statements are also presented.

show abstract

Moment data can be analytically completed

Cited by 11 publications

References 17 publications

Comparing large-scale graphs based on quantum probability theory

Comparing large-scale graphs based on quantum probability theory

Moments of the Hilbert-Schmidt probability distributions over determinants of real two-qubit density matrices and of their partial transposes

Moment information for probability distributions, without solving the moment problem, II: Main-mass, tails and shape approximation

Contact Info

Product

Resources

About