Lecture: Information Geometry and Machine Learning (MM25)

Language: English or German, as the audience requests.

Content: Information geometry is a mathematical framework that enables to understand in a coherent way data models, probabilistic inference and parameter estimation in machine learning. It provides an essential basis for advancing research in a systematic way and, in the long run, for a better understanding of the design of networks for data analysis.

The lecture (German: Aufbaukurs) is devoted to students of (or: interested in) mathematics in order to prepare research-oriented master theses.

Organisation: The lecture is organized into four parts: (i) differential geometry, (ii) information geometry, (iii) smooth convex analysis, (iv) applications to machine learning. Understanding the interplay between these parts is a major learning target.

(i) Differential geometry: Smooth manifolds and submanifolds; vector-, covector and tensor fields, Riemannian metrics, affine connections, geodesics, torsion and curvature.

(ii) Information geometry: Measures on finite sets, Fisher-Rao metric, alpha-connections, divergence (contrast) functions, information projections, graphical models, statistical models, statistical manifolds.

(iii) Convex analysis: classes of smooth convex functions, conjugation, optimality and duality.

(iv) Application to machine learning: Case studies from the literature.

Prerequisites: Good knowledge is required of the basic math courses on analysis and linear algebra, respectively. Elementary probability theory will be only used. Some prior knowledge of differential geometry would be useful but is not mandatory. The lecture will be self-contained in this respect.

Registration: If you wish to attend the lecture and the exercises, please sign up using MÜSLI.

Lecture Notes

Exercise Sheets