Classification on Riemannian Manifolds

author: Fatih M. Porikli, MERL - Mitsubishi Electric Research Laboratories
published: Sept. 13, 2010,   recorded: August 2010,   views: 11204


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A large number of natural phenomena can be formulated as inference on differentiable manifolds. More specifically in computer vision, such underlying notions emerge in feature selection, pose estimation, structure from motion, appearance tracking, and shape embedding. Unlike the uniform Euclidean space, differentiable manifolds exhibit local homeomorphism, thus, the differential geometry is applicable only within local tangent spaces. This prevents incorporation of conventional methods that require vector norms into the classification problems on manifolds where distances are defined through the curves of minimal length connecting two points. Recently we introduced a region covariance descriptor that exhibits a Riemannian manifold structure on positive definite matrices. By imposing weak classifiers on tangent spaces and establishing weighted sums via Karcher means, we bootstrap an ensemble of boosted classifiers with logistic loss functions. In this manner, we do not need to flatten the manifold or discover its topology. We demonstrate the new manifold classifiers on human detection and face recognition problems.

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Reviews and comments:

Comment1 Joris Bierkens, May 14, 2013 at 5:37 p.m.:

Please take care. The presenter is sloppy in some 'details'. E.g. the manifold of positive definite matrices is _not_ a Lie group, with the usual definition of matrix multiplication. (If you multiply two symmetric matrices, these are typically not symmetric, unless they commute.)

The manifold of positive definite matrices can perhaps be made into a Lie group, e.g. by defining group multiplication as
A \circ B := A^{1/2} B A^{1/2}, but the presenter is not specific about this.

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