An introduction to support Vector Machines: and other kernel-based learning methods
An introduction to support Vector Machines: and other kernel-based learning methods
Mean Shift: A Robust Approach Toward Feature Space Analysis
IEEE Transactions on Pattern Analysis and Machine Intelligence
Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond
Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond
Diffusion Kernels on Statistical Manifolds
The Journal of Machine Learning Research
Simultaneous Multiple 3D Motion Estimation via Mode Finding on Lie Groups
ICCV '05 Proceedings of the Tenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1 - Volume 01
Semi-supervised graph clustering: a kernel approach
ICML '05 Proceedings of the 22nd international conference on Machine learning
Nonlinear Mean Shift for Clustering over Analytic Manifolds
CVPR '06 Proceedings of the 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition - Volume 1
Grassmann discriminant analysis: a unifying view on subspace-based learning
Proceedings of the 25th international conference on Machine learning
Nonlinear Mean Shift over Riemannian Manifolds
International Journal of Computer Vision
Statistical Computations on Grassmann and Stiefel Manifolds for Image and Video-Based Recognition
IEEE Transactions on Pattern Analysis and Machine Intelligence
Region covariance: a fast descriptor for detection and classification
ECCV'06 Proceedings of the 9th European conference on Computer Vision - Volume Part II
Graph embedding discriminant analysis on Grassmannian manifolds for improved image set matching
CVPR '11 Proceedings of the 2011 IEEE Conference on Computer Vision and Pattern Recognition
A nonparametric Riemannian framework on tensor field with application to foreground segmentation
ICCV '11 Proceedings of the 2011 International Conference on Computer Vision
Semantic segmentation with second-order pooling
ECCV'12 Proceedings of the 12th European conference on Computer Vision - Volume Part VII
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The original mean shift algorithm [1] on Euclidean spaces (MS) was extended in [2] to operate on general Riemannian manifolds. This extension is extrinsic (Ext-MS) since the mode seeking is performed on the tangent spaces [3], where the underlying curvature is not fully considered (tangent spaces are only valid in a small neighborhood). In [3] was proposed an intrinsic mean shift designed to operate on two particular Riemannian manifolds (IntGS-MS), i.e. Grassmann and Stiefel manifolds (using manifold-dedicated density kernels). It is then natural to ask whether mean shift could be intrinsically extended to work on a large class of manifolds. We propose a novel paradigm to intrinsically reformulate the mean shift on general Riemannian manifolds. This is accomplished by embedding the Riemannian manifold into a Reproducing Kernel Hilbert Space (RKHS) by using a general and mathematically well-founded Riemannian kernel function, i.e. heat kernel [5]. The key issue is that when the data is implicitly mapped to the Hilbert space, the curvature of the manifold is taken into account (i.e. exploits the underlying information of the data). The inherent optimization is then performed on the embedded space. Theoretic analysis and experimental results demonstrate the promise and effectiveness of this novel paradigm.