Geometry and Statistics
Will be published approx. on 6. July 2018
Book
Hardback
450 pages
978-1-4987-6204-5 (ISBN)
Description
As massive and complex datasets are now much more prevalent, it has become necessary to understand the underlying geometric structure of the data for reliable statistical inference. This monograph will give a comprehensive and systematic overview of the role of geometry in statistics. It covers three broad modern topics: data geometry, learning geometry (high dimensional inference), and information geometry. The book includes lots of examples to illustrate the methods, and R code for their implementation. The book should be a useful reference for researchers and graduate students in statistics and machine learning with a good background in mathematics.
More details
Persons
Content
Introduction. Part I: Data Geometry; Chapter 1. Introduction; Chapter 2. Statistics on manifolds: location and spread based inference; Chapter 2.1. Extrinsic statistical inference; Chapter 2.2. Intrinsic statistical inference; Chapter 3. Density estimation on manifolds; Chapter 4. Regression and classification on manifolds; Chapter 4.1. Regression with manifold valued predictors; Chapter 4.2. Regression with manifold valued responses; Chapter 5. Examples and applications; Open Problems. Part II: Learning Geometry; Chapter 1. Introduction; Chapter 2. Basics on Manifold learning; Chapter 3. Main methods for learning; Chapter 3.1. Linear dimension reduction; Chapter 3.2. Non-linear dimension reduction; Chapter 3.3. Spectral methods; Chapter 3.4 Bayesian models; Chapter 4. Discriminate and cluster analysis; Chapter 5. Examples and applications; Open Problems. Part III: Information Geometry; Chapter 1. Introduction; Chapter 2. The geometry of statistical models; Chapter 3. Exponential families and beyond; Chapter 4. Information geometry on stratified spaces; Chapter 5. Information geometry in sampling and learning; Chapter 5.1 Computation: optimization on manifolds; Chapter 5.2 Computation: MCMC on manifolds; Chapter 6. Examples and applications; Open Problems. Appendix A: A review of differential geometry; Appendix B: Basics on Riemannian geometry.