scholarly journals Isometric Signal Processing under Information Geometric Framework

Entropy ◽  
2019 ◽  
Vol 21 (4) ◽  
pp. 332 ◽  
Author(s):  
Hao Wu ◽  
Yongqiang Cheng ◽  
Hongqiang Wang
Keyword(s):  
Signal Processing ◽  
Probability Distribution ◽  
Geometric Structure ◽  
Information Geometry ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Statistical Manifold ◽  
Intrinsic Parameter ◽  
Geometric Framework ◽  
Necessary And Sufficient

Information geometry is the study of the intrinsic geometric properties of manifolds consisting of a probability distribution and provides a deeper understanding of statistical inference. Based on this discipline, this letter reports on the influence of the signal processing on the geometric structure of the statistical manifold in terms of estimation issues. This letter defines the intrinsic parameter submanifold, which reflects the essential geometric characteristics of the estimation issues. Moreover, the intrinsic parameter submanifold is proven to be a tighter one after signal processing. In addition, the necessary and sufficient condition of invariant signal processing of the geometric structure, i.e., isometric signal processing, is given. Specifically, considering the processing with the linear form, the construction method of linear isometric signal processing is proposed, and its properties are presented in this letter.

A characterization of the exponential distribution

1972 ◽  
Vol 9 (02) ◽  
pp. 457-461 ◽  
Author(s):  
M. Ahsanullah ◽  
M. Rahman
Keyword(s):  
Probability Distribution ◽  
Order Statistics ◽  
Lebesgue Measure ◽  
Random Variable ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Positive Random Variable ◽  
Absolutely Continuous ◽  
Necessary And Sufficient

A necessary and sufficient condition based on order statistics that a positive random variable having an absolutely continuous probability distribution (with respect to Lebesgue measure) will be exponential is given.

A New Algorithm for Testing the Stability of a Polytope: A Geometric Approach for Simplification

1996 ◽  
Vol 118 (3) ◽  
pp. 611-615 ◽  
Author(s):  
Jinsiang Shaw ◽  
Suhada Jayasuriya
Keyword(s):  
Characteristic Equation ◽  
Robust Stability ◽  
Geometric Structure ◽  
Geometric Approach ◽  
Geometric Interpretation ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Edge Theorem

Considered in this paper is the robust stability of a class of systems in which a relevant characteristic equation is a family of polynomials F: f(s, q) = a0(q) + a1(q)s + … + an(q)sn with its coefficients ai(q) depending linearly on q unknown-but-bounded parameters, q = (p1, p2, …, pq)T. It is known that a necessary and sufficient condition for determining the stability of such a family of polynomials is that polynomials at all the exposed edges of the polytope of F in the coefficient space be stable (the edge theorem of Bartlett et al., 1988). The geometric structure of such a family of polynomials is investigated and an approach is given, by which the number of edges of the polytope that need to be checked for stability can be reduced considerably. An example is included to illustrate the benefit of this geometric interpretation.

scholarly journals Necessary and Sufficient Condition for Stability of Generalized Expectation Value

2011 ◽  
Vol 2011 ◽  
pp. 1-7 ◽  
Author(s):  
Aziz El Kaabouchi ◽  
Sumiyoshi Abe
Keyword(s):  
Probability Distribution ◽  
Statistical Mechanics ◽  
Complex Systems ◽  
Nonequilibrium Statistical Mechanics ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Expectation Value ◽  
Necessary And Sufficient ◽  
Small Deformations ◽  
Generalized Expectation

A class of generalized definitions of expectation value is often employed in nonequilibrium statistical mechanics for complex systems. Here, the necessary and sufficient condition is presented for such a class to be stable under small deformations of a given arbitrary probability distribution.

scholarly journals CLASSICALITY WITNESS FOR TWO-QUBIT STATES

2012 ◽  
Vol 10 (03) ◽  
pp. 1250028 ◽  
Author(s):  
JONAS MAZIERO ◽  
ROBERTO M. SERRA
Keyword(s):  
Nuclear Magnetic Resonance ◽  
Probability Distribution ◽  
Quantum Discord ◽  
Orthonormal Basis ◽  
Broad Class ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Classical Correlations ◽  
Necessary And Sufficient ◽  
Qubit States

In the last few years one realized that if the state of a bipartite system can be written as ∑i,j pij|ai〉〈ai| ⊗ |bj〉〈bj|, where {|ai〉} and {|bj〉} form orthonormal basis for the subsystems and {pij} is a probability distribution, then it possesses at most classical correlations. In this article we introduce a nonlinear witness providing a sufficient condition for classicality of correlations (absence of quantum discord) in a broad class of two-qubit systems. Such witness turns out to be necessary and sufficient condition in the case of Bell-diagonal states. We show that the witness introduced here can be readily experimentally implemented in nuclear magnetic resonance setups.

The linear stochastic heat equation with Hermite noise

2019 ◽  
Vol 22 (03) ◽  
pp. 1950022 ◽  
Author(s):  
Meryem Slaoui ◽  
C. A. Tudor
Keyword(s):  
Probability Distribution ◽  
Heat Equation ◽  
Stochastic Heat Equation ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Self Similarity ◽  
Wiener Chaos ◽  
Sample Paths ◽  
Non Gaussian ◽  
Necessary And Sufficient

We analyze the solution to the linear stochastic heat equation driven by a multiparameter Hermite process of order [Formula: see text]. This solution is an element of the [Formula: see text]th Wiener chaos. We discuss various properties of the solution, such as the necessary and sufficient condition for its existence, self-similarity, [Formula: see text]-variation and regularity of its sample paths. We will also focus on the probability distribution of the solution, which is non-Gaussian when [Formula: see text].

scholarly journals On the ME-manifold inn-*g-UFT and its conformal change

1994 ◽  
Vol 17 (1) ◽  
pp. 79-90
Author(s):  
Kyung Tae Chung ◽  
Gwang Sik Eun
Keyword(s):  
Riemannian Manifold ◽  
Geometric Structure ◽  
Tensor Field ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Unique Existence ◽  
Conformal Change ◽  
Necessary And Sufficient

An Einstein's connection which takes the form (3.1) is called an ME-connection. A generalizedn-dimensional Riemannian manifoldXnon which the differential geometric structure is imposed by a tensor field*gλνthrough a unique ME-connection subject to the conditions of Agreement (4.1) is called*g-ME-manifold and we denote it by*g-MEXn. The purpose of the present paper is to introduce this new concept of*g-MEXnand investigate its properties. In this paper, we first prove a necessary and sufficient condition for the unique existence of ME-connection inXn, and derive a surveyable tensorial representation of the ME-connection. In the second, we investigate the conformal change of*g-MEXnand present a useful tensorial representation of the conformal change of the ME-connection.

A characterization of the exponential distribution

1972 ◽  
Vol 9 (2) ◽  
pp. 457-461 ◽  
Author(s):  
M. Ahsanullah ◽  
M. Rahman
Keyword(s):  
Probability Distribution ◽  
Order Statistics ◽  
Lebesgue Measure ◽  
Random Variable ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Positive Random Variable ◽  
Absolutely Continuous ◽  
Necessary And Sufficient

A necessary and sufficient condition based on order statistics that a positive random variable having an absolutely continuous probability distribution (with respect to Lebesgue measure) will be exponential is given.

scholarly journals Convergence in Distribution for Subset Counts Between Random Sets

10.37236/1812 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Dudley Stark
Keyword(s):  
Probability Distribution ◽  
Rates Of Convergence ◽  
Stein’S Method ◽  
Random Sets ◽  
Higher Moments ◽  
Sufficient Condition ◽  
Convergence In Distribution ◽  
Stein's Method ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

Erdős posed the problem of how many random subsets need to be chosen from a set of $n$ elements, each element appearing in each subset with probability $p=1/2$, in order that at least one subset is contained in another. Rényi answered this question, but could not determine the limiting probability distribution for the number of subset counts because the higher moments diverge to infinity. The model considered by Rényi with $p$ arbitrary is denoted by ${\cal P}(m,n,p)$, where $m$ is the number of random subsets chosen. We give a necessary and sufficient condition on $p(n)$ and $m(n)$ for subset counts to be asymptotically Poisson and find rates of convergence using Stein's method. We discuss how Poisson limits can be shown for other statistics of ${\cal P}(m,n,p)$.

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