Block Relaxation Algorithms in Statistics -- Part I
Project
1.
Preface
2.
Introduction
2.1.
Some History
2.2.
Optimization Methods
3.
Block Relaxation
3.1.
Introduction
3.2.
Definition
3.3.
First Examples
3.3.1.
Two-block Least Squares
3.3.2.
Multiple-block Least Squares
3.4.
Generalized Block Relaxation
3.4.1.
Rasch Model
3.4.2.
Nonlinear Least Squares
3.5.
Block Order
3.5.1.
Projecting Blocks
3.6.
Rate of Convergence
3.6.1.
LU-form
3.6.2.
Product Form
3.6.3.
Block Optimization Methods
3.6.4.
Block Newton Methods
3.6.5.
Constrained Problems
3.7.
Additional Examples
3.7.1.
Canonical Correlation
3.7.2.
Singular Value Decomposition
3.7.3.
Optimal Scaling with LINEALS
3.7.4.
Multinormal Maximum Likelihood
3.7.5.
Array Multinormals
3.7.6.
Rasch, Once More
3.8.
Counterexamples
3.8.1.
Convergence to a Saddle
3.8.2.
Convergence to Incorrect Solution
3.8.3.
Nonconvergence and Cycling
3.8.4.
Sublinear Convergence
4.
Coordinate Descent
4.1.
Introduction
4.2.
Rate of Convergence
4.3.
Examples
4.3.1.
The Cartesian Folium
4.3.2.
A Family of Quadratics
4.3.3.
Loglinear Models
4.3.4.
Rayleigh Quotient
4.3.5.
Squared Distance Scaling
4.3.6.
Least Squares Factor Analysis
5.
Alternating Least Squares
5.1.
Introduction
5.2.
Close Relatives
5.2.1.
ALSOS
5.2.2.
ACE
5.2.3.
NIPALS and PLS
5.3.
Rate of Convergence
5.4.
Examples
5.4.1.
Homogeneity Analysis
5.4.2.
Bilinear Fitting and Fixed Rank Approximation
5.4.3.
Multilinear Fitting
5.4.4.
MCR-ALS
5.4.5.
Scaling and Splitting
6.
Augmentation and Decomposition Methods
6.1.
Introduction
6.2.
Definition
6.3.
Rate of Convergence
6.4.
Half-Quadratic Methods
6.5.
Examples
6.5.1.
Yates Augmentation
6.5.2.
Optimal Scaling with ORDINALS
6.5.3.
Least Squares Factor Analysis
6.5.4.
Squared Distance Scaling
6.5.5.
Linear Mixed Model
6.6.
Decomposition Methods
6.6.1.
Quadratic Form on a Sphere
6.6.2.
Multidimensional Unfolding
7.
Notation
8.
Bibliography
9.
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Block Relaxation Algorithms in Statistics -- Part I
I.3.8: Some Counterexamples