Block Relaxation Algorithms in Statistics -- Part II
Project
1.
Majorization Methods
1.1.
Introduction
1.2.
Definitions
1.2.1.
Majorization at a Point
1.2.2.
Majorization on a Set
1.2.3.
Majorization Algorithms
1.2.4.
Alternative Definitions
1.3.
Close Relatives
1.3.1.
Concave-Convex Procedure
1.3.2.
Generalized Weiszfeld Methods
1.3.3.
The EM Algorithm
1.3.4.
The Lower-Bound Principle
1.4.
Some Results
1.4.1.
Rate of Convergence
1.4.2.
Univariate and Separable Functions
1.4.3.
Differentiable Functions
1.4.4.
Composition
1.4.5.
Majorization Duality
1.4.6.
Necessary Conditions by Majorization
1.4.7.
Majorizing Constraints
1.4.8.
Majorizing Value Functions
1.5.
Some Examples
1.5.1.
The Reciprocal
1.5.2.
Cubics and Quartics
1.5.3.
Normal PDF and CDF
1.5.4.
Logistic PDF and CDF
2.
Inequalities for Majorization
2.1.
Introduction
2.2.
The AMGM Inequality
2.2.1.
Absolute Values
2.2.2.
Gini Mean Difference
2.2.3.
Location Problems
2.2.4.
The Lasso and the Bridge
2.3.
The Cauchy-Schwartz Inequality
2.3.1.
Rayleigh Quotient
2.3.2.
The Majorization Method for MDS
2.4.
Young's Inequality
2.4.1.
Support Vector Machines
3.
Using Convexity
3.1.
Introduction
3.2.
Jensen's Inequality
3.2.1.
Tomography
3.2.2.
Sums and Integrals
3.3.
The EM Algorithm
4.
Tangential Majorization
4.1.
Concave Functions
4.1.1.
Majorizing and Minorizing the Logarithm
4.1.2.
Aspects of Correlation Matrices
4.1.3.
Partially Observed Linear Systems
4.1.4.
Gpower
4.2.
Broadening the Scope
4.2.1.
Differences of Convex Functions
4.2.2.
Convexifiable Functions
4.2.3.
Piecewise Linear Majorization
5.
Quadratic Majorization
5.1.
Introduction
5.2.
Existence
5.3.
Convergence
5.4.
Bounding Second Derivatives
5.4.1.
Normal Density and Distribution
5.4.2.
Nondiagonal Weights in Least Squares
5.4.3.
Quadratic on a Sphere
5.4.4.
Gifi Goes Logistic
5.4.5.
A Matrix Example
5.4.6.
Gauss-Newton Majorization
5.4.7.
Margnal Functions
6.
Higher Order Majorization
6.1.
Introduction
6.2.
Mean Value Majorization
6.3.
Taylor Majorization
6.3.1.
Second Order
6.3.2.
Higher Order
6.4.
Nestorov Majorization
6.5.
Examples
6.5.1.
Revisiting the Reciprocal
6.5.2.
Logit
6.5.3.
Probit
7.
Sharp Majorization
7.1.
Introduction
7.2.
Comparing Majorizations
7.3.
Sharp Quadratic Majorization
7.3.1.
Existence
7.3.2.
Two-Point Majorization
7.3.3.
Even and Odd Functions
7.4.
Sharp Piecewise Linear
7.5.
Examples
7.5.1.
The Cosine
7.5.2.
The Rasch Model
7.5.3.
Logits
7.5.4.
Probits
8.
Local and Localized Majorization
8.1.
Introduction
8.2.
Majorization in the Neighborhood
8.2.1.
Cartesian Folium
8.2.2.
Univariate Cubics
8.2.3.
Majorization on a Sphere
8.2.4.
Majorization on a Hyperrectangle
8.3.
Proximal Point Majorization
8.4.
Sublevel Majorization
8.5.
Dinkelbach Majorization
9.
Notation
10.
Bibliography
11.
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Block Relaxation Algorithms in Statistics -- Part II
Majorization on a Hyperrectangle
Here we discuss the work of
Mönnigmann [2011]
and others.