**Date & Time:** Monday, January 26, 2009, 16:00-17:00.

**Venue:** Room 216

**Title:** Optimization on Manifolds: An Introduction

**Speaker:** Rodolphe Sepulchre, Université de Liège

**Abstract:** This is the first talk in a Lecture Series on Optimization on Manifolds.

**Moe about the series:**

The recent years have witnessed an increasing interest in the development of efficient optimization algorithms defined on manifolds. Applications abound in numerical linear algebra (eigenproblems), statistical analysis (Principal and Independent Component analysis), signal processing (blind source separation, subspace tracking), machine learning (clustering), computer vision (pose estimation), to name a few. Good algorithms result from a combination of insights from differential geometry, optimization and numerical analysis. The purpose of the course is to provide a tutorial introduction to this rich field of applied mathematics with a parsimonious selection of topics in differential geometry and in numerical algebra, and with an illustration of engineering problems where the theory is currently applied. The course will provide the participants with the basic concepts of differential geometry instrumental to algorithmic development. It will illustrate why differential geometry provides a natural foundation for the development of efficient numerical algorithms for many equality-constrained optimization problems. Several well-known optimization techniques, such as steepest descent, conjugate gradients, trust-region and Newton-type methods, are generalized to the manifold setting. A generic development of each of these methods is provided, building upon the geometric material. The participants are then guided through the constructions and computations that turn these geometrically formulated methods into concrete numerical algorithms. The techniques are general and are illustrated on several problems in linear algebra, signal processing, data mining, computer vision, and statistical analysis.

This series is jointly organized with the Interdisciplinary Programme in Systems and Control Engineering, IIT Bombay.