Algebraic Geometry and Statistical Learning Theory Cambridge Monographs on Applied and Computational Mathematics, Series Number 25 1st Edition

The editorial review of the book "Algebraic Geometry and Statistical Learning Theory" acknowledges its potential to provide an approachable understanding of a complex theoretical topic important for practitioners in the field of machine learning and data science. The reviewer praises the content but criticizes the clarity of the author's transitions between definitions, statements, remarks, and theorems, as well as pointing out issues with the English and typographical errors. They note that parsing through the book can be frustrating, requiring the reader to repeatedly verify each step due to these issues. The reviewer recommends a second edition and suggests that readers with a strong background in measure theory, differential geometry, and abstract algebra, and the right motivation, could benefit from the book. Another review commends the clarity of the book, highlighting its relevance for those interested in the application of algebraic geometry in statistical learning theory. The author's use of real algebraic geometry over algebraically closed fields is Considered advantageous, as it reduces the need for heavy machinery associated with contemporary texts on the subject. The book is said to require readers to have preparation in real and functional analysis and a good background in algebraic geometry, although not necessarily at the level of modern approaches to the subject. The reviewer points out that the book covers important concepts from statistical learning theory and the use of Kullback-Leibler distance, and emphasizes the need for "singular" statistical learning theory in dealing with parameter spaces where the Fisher information matrix is not positive definite. The review also mentions the author's generalization of standard statistical learning Constructions to the case of singular theories, such as the Akaike information criterion and Bayes information criterion. The book's treatment of notions like stochastic complexity and its methods for calculating it are noted, as well as the author's strategies for dealing with the divergence of the maximum likelihood estimator and the failure of asymptotic normality in singular theories.