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Algorithmic Advances in Riemannian Geometry and Applications
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This book presents a selection of the most recent algorithmic advances in Riemanniangeometry in the context of machine learning, statistics, optimization, computervision, and related fields. The unifying theme of the different chapters in the bookis the exploitation of the geometry of data using the mathematical machinery ofRiemannian geometry. As demonstrated by all the chapters in the book, when the datais intrinsically non-Euclidean, the utilization of this geometrical information can leadto better algorithms that can capture more accurately the structures inherent in thedata, leading ultimately to better empirical performance. This book is not intended tobe an encyclopedic compilation of the applications of Riemannian geometry. Instead, itfocuses on several important research directions that are currently actively pursued byresearchers in the field. These include statistical modeling and analysis on manifolds,optimization on manifolds, Riemannian manifolds and kernel methods, and dictionarylearning and sparse coding on manifolds. Examples of applications include novel algorithmsfor Monte Carlo sampling and Gaussian Mixture Model fitting, 3D brain image analysis,image classification, action recognition, and motion tracking.
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