Information Geometry and Its Applications: Convex Function and Dually Flat Manifold

2009 ◽  
pp. 75-102 ◽  
Author(s):  
Shun-ichi Amari
Keyword(s):  
Convex Function ◽  
Information Geometry ◽  
Flat Manifold

Information Geometry of U-Boost and Bregman Divergence

2004 ◽  
Vol 16 (7) ◽  
pp. 1437-1481 ◽  
Author(s):  
Noboru Murata ◽  
Takashi Takenouchi ◽  
Takafumi Kanamori ◽  
Shinto Eguchi
Keyword(s):  
Convex Function ◽  
Expectation Maximization ◽  
Convergence Property ◽  
Learning Algorithms ◽  
Expectation Maximization Algorithm ◽  
Information Geometry ◽  
Training Data ◽  
Bregman Divergence ◽  
Learning Machines ◽  
Finite Measures

We aim at an extension of AdaBoost to U-Boost, in the paradigm to build a stronger classification machine from a set of weak learning machines. A geometric understanding of the Bregman divergence defined by a generic convex function U leads to the U-Boost method in the framework of information geometry extended to the space of the finite measures over a label set. We propose two versions of U-Boost learning algorithms by taking account of whether the domain is restricted to the space of probability functions. In the sequential step, we observe that the two adjacent and the initial classifiers are associated with a right triangle in the scale via the Bregman divergence, called the Pythagorean relation. This leads to a mild convergence property of the U-Boost algorithm as seen in the expectation-maximization algorithm. Statistical discussions for consistency and robustness elucidate the properties of the U-Boost methods based on a stochastic assumption for training data.

Extreme-point solutions in Markov decision processes

1983 ◽  
Vol 20 (04) ◽  
pp. 835-842
Author(s):  
David Assaf
Keyword(s):  
Convex Function ◽  
Extreme Point ◽  
Markov Decision Processes ◽  
Convex Functions ◽  
Sufficient Conditions ◽  
Decision Processes ◽  
Markov Decision ◽  
Full Solution

The paper presents sufficient conditions for certain functions to be convex. Functions of this type often appear in Markov decision processes, where their maximum is the solution of the problem. Since a convex function takes its maximum at an extreme point, the conditions may greatly simplify a problem. In some cases a full solution may be obtained after the reduction is made. Some illustrative examples are discussed.

scholarly journals Quantum Orlicz Spaces in Information Geometry

2004 ◽  
Vol 11 (04) ◽  
pp. 359-375 ◽  
Author(s):  
R. F. Streater
Keyword(s):  
Quantum Information ◽  
Tangent Space ◽  
Selfadjoint Operator ◽  
Affine Connection ◽  
Orlicz Spaces ◽  
Trace Class ◽  
Information Geometry ◽  
Young Function ◽  
Affine Structure ◽  
Cotangent Space

Let H0 be a selfadjoint operator such that Tr e−βH0 is of trace class for some β < 1, and let χɛ denote the set of ɛ-bounded forms, i.e., ∥(H0+C)−1/2−ɛX(H0+C)−1/2+ɛ∥ < C for some C > 0. Let χ := Span ∪ɛ∈(0,1/2]χɛ. Let [Formula: see text] denote the underlying set of the quantum information manifold of states of the form ρx = e−H0−X−ψx, X ∈ χ. We show that if Tr e−H0 = 1. 1. the map Φ, [Formula: see text] is a quantum Young function defined on χ 2. The Orlicz space defined by Φ is the tangent space of [Formula: see text] at ρ0; its affine structure is defined by the (+1)-connection of Amari 3. The subset of a ‘hood of ρ0, consisting of p-nearby states (those [Formula: see text] obeying C−1ρ1+p ≤ σ ≤ Cρ1 − p for some C > 1) admits a flat affine connection known as the (−1) connection, and the span of this set is part of the cotangent space of [Formula: see text] 4. These dual structures extend to the completions in the Luxemburg norms.

scholarly journals On Certain Differential Subordination of Harmonic Mean Related to a Linear Function

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 966
Author(s):  
Anna Dobosz ◽  
Piotr Jastrzębski ◽  
Adam Lecko
Keyword(s):  
Convex Function ◽  
Linear Function ◽  
Differential Subordination ◽  
Harmonic Mean ◽  
Best Dominant ◽  
Symmetry Properties ◽  
Differential Subordinations ◽  
Difficult Issue

In this paper we study a certain differential subordination related to the harmonic mean and its symmetry properties, in the case where a dominant is a linear function. In addition to the known general results for the differential subordinations of the harmonic mean in which the dominant was any convex function, one can study such differential subordinations for the selected convex function. In this case, a reasonable and difficult issue is to look for the best dominant or one that is close to it. This paper is devoted to this issue, in which the dominant is a linear function, and the differential subordination of the harmonic mean is a generalization of the Briot–Bouquet differential subordination.

scholarly journals A note on generalized convex functions

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Syed Zaheer Ullah ◽  
Muhammad Adil Khan ◽  
Yu-Ming Chu
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Type Inequality ◽  
Generalized Convex Functions

Abstract In the article, we provide an example for a η-convex function defined on rectangle is not convex, prove that every η-convex function defined on rectangle is coordinate η-convex and its converse is not true in general, define the coordinate $(\eta _{1}, \eta _{2})$(η1,η2)-convex function and establish its Hermite–Hadamard type inequality.

scholarly journals Optimal UAV Circumnavigation Control with Input Saturation Based on Information Geometry

2020 ◽  
Vol 53 (2) ◽  
pp. 2471-2476
Author(s):  
Yangguang Yu ◽  
Xiangke Wang ◽  
Lincheng Shen
Keyword(s):  
Input Saturation ◽  
Information Geometry

scholarly journals Information Geometry for Radar Target Detection with Total Jensen–Bregman Divergence

Entropy ◽  
2018 ◽  
Vol 20 (4) ◽  
pp. 256 ◽  
Author(s):  
Xiaoqiang Hua ◽  
Haiyan Fan ◽  
Yongqiang Cheng ◽  
Hongqiang Wang ◽  
Yuliang Qin
Keyword(s):  
Target Detection ◽  
Information Geometry ◽  
Bregman Divergence ◽  
Radar Target

scholarly journals Estimation of different entropies via Lidstone polynomial using Jensen-type functionals

2020 ◽  
Vol 9 (3) ◽  
pp. 613-631
Author(s):  
Khuram Ali Khan ◽  
Tasadduq Niaz ◽  
Đilda Pečarić ◽  
Josip Pečarić
Keyword(s):  
Convex Function ◽  
Shannon Entropy ◽  
Rényi Divergence ◽  
Lidstone Polynomial

Abstract In this work, some new functional of Jensen-type inequalities are constructed using Shannon entropy, f-divergence, and Rényi divergence, and some estimates are obtained for these new functionals. Also using the Zipf–Mandelbrot law and hybrid Zipf–Mandelbrot law, we investigate some bounds for these new functionals. Furthermore, we generalize these new functionals for m-convex function using Lidstone polynomial.

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