Information Geometry: Session 2 скачать в хорошем качестве

Information Geometry: Session 2

Overview of First Session and Introduction to Riemannian Geometry 10:27 Herlock Rahimi provided a brief overview of the first session, which covered the basics of information geometry and its relationship with KL divergence and Wasserstein distance. The session also included an example of optimizing an estimation problem using gradient descent methods. The second session aims to delve into the mathematics of Riemannian geometry and its application in solving EM problems. The goal is to leverage geometry to optimize probability distributions more efficiently. Introduction to Riemannian Geometry and Information Geometry 14:21 The first session was described as vague, lacking detailed mathematical explanations. The second and third sessions will focus on understanding the essentials of Riemannian geometry. The course is not an introduction to Riemannian geometry but will cover necessary concepts for statistical applications. The application of these concepts in machine learning and finance will be explored. Fitting Mixture Models and EM Algorithm 16:18 The session began with fitting mixture models and the limitations of maximum likelihood estimators. The EM algorithm was introduced as a solution, involving estimation and maximization steps. The process involves iterating through probabilities of data points belonging to different distributions. The EM algorithm is compared to a naive approach, highlighting its efficiency in leveraging likelihoods. Mathematical Formulations of Smooth Manifolds 27:14 The concept of smooth manifolds was introduced as a generalization of Euclidean space. Tangent vectors and spaces were discussed as essential components of geometry. The Riemannian metric was defined as an inner product dependent on the point, allowing for the calculation of vector norms and angles. These concepts are foundational for understanding the geometry of probability distributions. Connections and Geodesics in Riemannian Geometry 35:11 The need for tools to move tangent spaces between points was discussed, leading to the introduction of connections. Connections allow for the comparison of vectors at different points on a manifold. Geodesics were defined as curves with the least distance or curvature between points. The relationship between geodesics and connections was explored, emphasizing their role in understanding manifold geometry. Affine Connections and Parallel Transport 48:28 Affine connections were introduced as a way to define how vector fields change along curves. The concept of parallel transport was explained using the analogy of a compass moving along the Earth's surface. The goal is to maintain the direction of vectors relative to the manifold's geometry. This concept is crucial for understanding the movement of tangent vectors and the application of geometry in optimization problems. Introduction to Geodesics and Affine Connections 53:49 Herlock Rahimi introduced the concept of geodesics, emphasizing the least curved path between two points rather than the shortest distance. Discussed the definition of vector fields moving along curves and the use of affine connections to solve linear ODE systems. Highlighted the ability to calculate auto-parallel transport of vectors using ODE solvers. Planned to define these concepts within the framework of information geometry. Clarification on Affine Connections 58:04 Saman Saffari asked about the role of affine connections in maintaining a global perspective, using the analogy of holding something upright while moving on Earth. Herlock Rahimi clarified that affine connections are not about maintaining a north direction but about the invariance of vector fields as they move along a curve. Emphasized the non-uniqueness of affine connections and their dependence on coordinate systems. Discussed the relationship between affine connections and Riemannian geometry, noting the potential for infinitely many connections. Geodesics and Curvature 1:09:20 Herlock Rahimi explained the concept of geodesics as paths with the least curvature, not necessarily the shortest distance. Discussed the role of natural connections that are invariant under coordinate changes. Introduced the concept of curvature, relating it to second derivatives and the curvature tensor. Encouraged further exploration of different types of curvature, such as Gaussian and Ricci curvature. Connecting Affine Connections and Geometry 1:24:43 Discussed the relationship between Riemannian metrics and affine connections, focusing on the preservation of angles during vector transport. Introduced the Levi-Civita connection as a key concept in maintaining this invariance. Explained the dual connection and its role in optimizing paths, particularly in information geometry. Highlighted the applications of these concepts in optimization and machine learning, including mirror descent and reinforcement learning. Conclusion and Next Steps 1:37:05