Projective Power Entropy and Maximum Tsallis Entropy Distributions

Eguchi, Shinto; Komori, Osamu; Kato, Shogo

doi:10.3390/e13101746

Cited by 28 publications

(32 citation statements)

References 25 publications

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“…Definition 1 [Fujisawa and Eguchi (2008), Cichocki and Amari (2010), Eguchi, Komori and Kato (2011)]. For f, g ∈ M, define the γ-divergence D γ (· ·) and γ-cross entropy C γ (· ·) as follows:…”

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confidence: 99%

$\gamma$-SUP: A clustering algorithm for cryo-electron microscopy images of asymmetric particles

Chen¹,

Hsieh²,

Hung³

et al. 2014

Ann. Appl. Stat.

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Cryo-electron microscopy (cryo-EM) has recently emerged as a powerful tool for obtaining three-dimensional (3D) structures of biological macromolecules in native states. A minimum cryo-EM image data set for deriving a meaningful reconstruction is comprised of thousands of randomly orientated projections of identical particles photographed with a small number of electrons. The computation of 3D structure from 2D projections requires clustering, which aims to enhance the signal to noise ratio in each view by grouping similarly oriented images. Nevertheless, the prevailing clustering techniques are often compromised by three characteristics of cryo-EM data: high noise content, high dimensionality and large number of clusters. Moreover, since clustering requires registering images of similar orientation into the same pixel coordinates by 2D alignment, it is desired that the clustering algorithm can label misaligned images as outliers. Herein, we introduce a clustering algorithm γ-SUP to model the data with a q-Gaussian mixture and adopt the minimum γ-divergence for estimation, and then use a self-updating procedure to obtain the numerical solution. We apply γ-SUP to the cryo-EM images of two benchmark macromolecules, RNA polymerase II and ribosome. In the former case, simulated images were chosen to decouple clustering from alignment to demonstrate γ-SUP is more robust to misalignment outliers than the existing clustering methods used in the cryo-EM community. In the latter case, the clustering of real cryo-EM data by our γ-SUP method eliminates noise in many views to reveal true structure features of ribosome at the projection level.

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confidence: 99%

$\gamma$-SUP: A clustering algorithm for cryo-electron microscopy images of asymmetric particles

Chen¹,

Hsieh²,

Hung³

et al. 2014

Ann. Appl. Stat.

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“…It is shown that classes of nonlinear N-dimensional Fokker-Planck equations are connected to a single entropic form and emphasis is given to the class of equations associated to Tsallis entropy, in both cases of the standard and generalized definitions for the internal energy. Finally, Eguchi, Komori and Kato [10] discuss a one-parameter family of generalized cross entropy between two distributions with the power index, called the projective power entropy which is reduced to the Tsallis entropy if two distributions are taken to be equal. They investigate statistical and probabilistic properties associated with the projective power entropy, including a characterization problem of which conditions uniquely determine the projective power entropy up to the power index.…”

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confidence: 99%

Special Issue: Tsallis Entropy

Anastasiadis

2012

Entropy

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One of the crucial properties of the Boltzmann-Gibbs entropy in the context of classical thermodynamics is extensivity, namely proportionality with the number of elements of the system. The Boltzmann-Gibbs entropy satisfies this prescription if the subsystems are statistically (quasi-) independent, or typically if the correlations within the system are essentially local. In such cases the energy of the system is typically extensive and the entropy is additive. In general, however, the situation is not of this type and correlations may be far from negligible at all scales. Tsallis in 1988 introduced an entropic expression characterized by an index q which leads to a non-extensive statistics. Tsallis entropy, Sq, is the basis of the so called non-extensive statistical mechanics, which generalizes the Boltzmann-Gibbs theory. Tsallis statistics have found applications in a wide range of phenomena in diverse disciplines such as physics, chemistry, biology, medicine, economics, geophysics, etc. The focus of this special issue of Entropy was to solicit contributions that apply Tsallis entropy in various scientific fields.This special issue consists of nine regular papers, covering various aspects and applications of Tsallis non-additive entropy, and an invited review paper written by Tsallis [1]. In this review, the following aspects of Tsallis entropy are discussed: (i) Additivity versus extensivity; (ii) Probability distributions that constitute attractors in the sense of Central Limit Theorems; (iii) The analysis of paradigmatic low-dimensional nonlinear dynamical systems near the edge of chaos; and (iv) The analysis of paradigmatic long-range-interacting many-body classical Hamiltonian systems. Finally, recent as well as typical predictions, verifications and applications of these concepts in natural, artificial, and social systems, as shown through theoretical, experimental, observational and computational results are presented.In their paper, Zhang and Wu [2] propose a global multi-level thresholding method for image segmentation by applying the Tsallis entropy, as a general information theory entropy formalism, and using an artificial bee colony algorithm. They demonstrate that Tsallis entropy is superior to traditional maximum entropy thresholding, based on Shannon entropy, and that the artificial bee colony is more rapid than either genetic algorithm or particle swarm optimization. Vila, Bardera, Feixas and Sbert [3] investigate the application of three different Tsallis-based generalizations of mutual information to OPEN ACCESS

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“…See [35,36] for the detailed discussion [37,38] for the discussion on group invariance. Thus, if β > 0, then the maximum β-power entropy distribution has a compact support…”

Section: Maximum Entropy Distributionmentioning

confidence: 99%

Duality of Maximum Entropy and Minimum Divergence

Eguchi

Komori

Ohara

2014

Entropy

Self Cite

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Abstract:We discuss a special class of generalized divergence measures by the use of generator functions. Any divergence measure in the class is separated into the difference between cross and diagonal entropy. The diagonal entropy measure in the class associates with a model of maximum entropy distributions; the divergence measure leads to statistical estimation via minimization, for arbitrarily giving a statistical model. The dualistic relationship between the maximum entropy model and the minimum divergence estimation is explored in the framework of information geometry. The model of maximum entropy distributions is characterized to be totally geodesic with respect to the linear connection associated with the divergence. A natural extension for the classical theory for the maximum likelihood method under the maximum entropy model in terms of the Boltzmann-Gibbs-Shannon entropy is given. We discuss the duality in detail for Tsallis entropy as a typical example.

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Projective Power Entropy and Maximum Tsallis Entropy Distributions

Cited by 28 publications

References 25 publications

$\gamma$-SUP: A clustering algorithm for cryo-electron microscopy images of asymmetric particles

$\gamma$-SUP: A clustering algorithm for cryo-electron microscopy images of asymmetric particles

Special Issue: Tsallis Entropy

Duality of Maximum Entropy and Minimum Divergence

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