Information Geometry and Its Applications (Applied Mathematical Sciences Book 194) by Shun-ichi Amari (PDF)

Description

This is the first comprehensive book on information geometry, written by the founder of the field. It begins with an elementary introduction to dualistic geometry and proceeds to a wide range of applications, covering information science, engineering, and neuroscience. It consists of four parts, which on the whole can be read independently. A manifold with a divergence function is first introduced, leading directly to dualistic structure, the heart of information geometry. This part (Part I) can be apprehended without any knowledge of differential geometry. An intuitive explanation of modern differential geometry then follows in Part II, although the book is for the most part understandable without modern differential geometry. Information geometry of statistical inference, including time series analysis and semiparametric estimation (the Neyman–Scott problem), is demonstrated concisely in Part III. Applications addressed in Part IV include hot current topics in machine learning, signal processing, optimization, and neural networks. The book is interdisciplinary, connecting mathematics, information sciences, physics, and neurosciences, inviting readers to a new world of information and geometry. This book is highly recommended to graduate students and researchers who seek new mathematical methods and tools useful in their own fields.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I really like this book, and I would love to give it five stars. As I have some background in differential geometry, and currently work with large sets of data, I find the marriage between machine learning and differential geometry very appealing. Having said this, the resulting “information geometry” seems to have very little new insight to offer. Standard concepts and methods of statistics / machine learning are dressed up and reformulated in the language of differential geometry (with some effort), leading to new vocabulary but no new practically applicable methods and algorithms. Deep concepts of geometry such as that of the curvature or geodesic do not seem to translate into equally powerful concepts for data analysis. The book makes a wonderful reading for someone who enjoys the aesthetics of mathematics but do not expect to find in it anything that would help you with number crunching.

⭐This book is an updated review of the field by a main founder of Information Geometry.From the previous literature “Methods of Information Geometry,” extensive update has been made.Far more references are added as sections / chapters, e.g. many recent results from after 2010 are cited and comprehensively explained.This book should be the new standard as an introductory textbook for the field.# remarks:## Pythagorean theoremThere is a mistake in the statement (and proof) of theorem 1.2 (Generalized Pythagorean Theoreom).The duality of the two geodesics are reversed in the statement (which you can confirm by proving the theorem).That is, the dual geodesic PQ should be the geodesic (nabla), and the geodesic QR should be the dual geodesic (nabla-*)A correct statement would be:When triangle P Q R is orthogonal such that the geodesic connecting P and Q is orthogonal to the dual geodesic connecting Q and R,D_psi[P:R] = D_psi[P:Q] + D_psi[Q:R]In the proof, (1.114) is the wrong part.The dual version of the theorem (where PQ is dual geodesic and QR is the primal geodesic) is also wrong accordingly.## Asymptotic properties of MLEThe part is not written precisely in the book (so the formulas don’t make rigorous sense). I recommend reading Amari (1985) that is frequently cited in those sections if you want to follow the equations.

⭐On the bottom of page 10 he states a that a Riemannian metric is a positive definite matrix. It needs to be symmetric too. He then states that the inner product w.r.t. to this metric is the distance. He then completely ignores geodesics 2 pages later and misses the entire point of manifold by declaring “straight” lines to be linear functions. He doesn’t even get what derivatives are on manifolds are, let alone what the dual is.I have other problems with this book in some later chapters. I figured that he’s got a quasi-metric and you can probably get the theorems you need out of just that. The chapter on neural network singularities is also wrong. The group actions (he doesn’t indicate they are a group) are discrete and nice enough that the quotient space is a smooth manifold with 0 curvature. Aside from a removable 0-weight output vector (which is irrelevant to the theorems), there are no singularities.I don’t know if there’s redeemable information in this book, but the geometry I’ve seen is very wrong.Citations: John M Lee, Introduction to Smooth Manifolds, pages 132, 274, and 282

⭐Useful summary of an unfamiliar but increasingly important topic in modelling and optimisation

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