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Kernel Methods on the Riemannian Manifold of Symmetric Positive Definite Matrices [article]

Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash Harandi
2014 arXiv   pre-print
To encode the geometry of the manifold in the mapping, we introduce a family of provably positive definite kernels on the Riemannian manifold of SPD matrices.  ...  Symmetric Positive Definite (SPD) matrices have become popular to encode image information.  ...  Symmetric positive definite (SPD) matrices are another class of entities lying on a Riemannian manifold.  ... 
arXiv:1412.4172v1 fatcat:3efk2lu2f5d3bpcefzdn7n6wyu

Kernel Methods on the Riemannian Manifold of Symmetric Positive Definite Matrices

Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash Harandi
2013 2013 IEEE Conference on Computer Vision and Pattern Recognition  
To encode the geometry of the manifold in the mapping, we introduce a family of provably positive definite kernels on the Riemannian manifold of SPD matrices.  ...  Symmetric Positive Definite (SPD) matrices have become popular to encode image information.  ...  Symmetric positive definite (SPD) matrices are another class of entities lying on a Riemannian manifold.  ... 
doi:10.1109/cvpr.2013.17 dblp:conf/cvpr/JayasumanaHSLH13 fatcat:44tkt4gltfelpnbzb5u3xutqgm

Kernel-based classification for brain connectivity graphs on the Riemannian manifold of positive definite matrices

Luca Dodero, Ha Quang Minh, Marco San Biagio, Vittorio Murino, Diego Sona
2015 2015 IEEE 12th International Symposium on Biomedical Imaging (ISBI)  
In this paper, we propose a mathematical framework based on Riemannian geometry and kernel methods that can be applied to connectivity matrices for the classification task.  ...  The empirical results obtained clearly show the superior performance of our approach compared with baseline methods, demonstrating the advantages of our manifold framework and its potential for other applications  ...  BACKGROUND The Riemannian manifold of SPD matrices Let Sym ++ n denote the set of symmetric, positive definite (SPD) matrices of size n × n, that is the set of all symmetric n × n matrices W such that  ... 
doi:10.1109/isbi.2015.7163812 dblp:conf/isbi/DoderoMSMS15 fatcat:4eph2lewkbbg3a5ate7ujjhupa

Combining Multiple Manifold-Valued Descriptors for Improved Object Recognition

Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash Harandi
2013 2013 International Conference on Digital Image Computing: Techniques and Applications (DICTA)  
To this end, we propose a kernel on the n-dimensional unit sphere and prove its positive definiteness.  ...  We propose a feature combination and selection method that optimally combines descriptors lying on different manifolds while respecting the Riemannian geometry of each underlying manifold.  ...  The Riemannian manifold of SPD matrices Symmetric Positive Definite (SPD) matrices are characterized by the property that their eigenvalues are positive.  ... 
doi:10.1109/dicta.2013.6691493 dblp:conf/dicta/JayasumanaHSLH13 fatcat:4cx6g37vb5c5bnad6h7pn36dhu

Sparse Coding and Dictionary Learning for Symmetric Positive Definite Matrices: A Kernel Approach [chapter]

Mehrtash T. Harandi, Conrad Sanderson, Richard Hartley, Brian C. Lovell
2012 Lecture Notes in Computer Science  
This paper tackles the problem of sparse coding and dictionary learning in the space of symmetric positive definite matrices, which form a Riemannian manifold.  ...  With the aid of the recently introduced Stein kernel (related to a symmetric version of Bregman matrix divergence), we propose to perform sparse coding by embedding Riemannian manifolds into reproducing  ...  NICTA is funded by the Australian Government as represented by the Department of Broadband, Communications and the Digital Economy, as well as by the Australian Research Council through the ICT Centre  ... 
doi:10.1007/978-3-642-33709-3_16 fatcat:iaakzcnfsjgvfjyo5m3ve7vr7e

A Fast Graph Kernel Based Classification Method for Wireless Link Scheduling on Riemannian Manifold [article]

Rashed Shelim, Ahmed S. Ibrahim
2021 arXiv   pre-print
) Laplacian matrices which are symmetric positive definite (SPD) one.  ...  In this paper, we propose a novel graph kernel method for the wireless link scheduling problem in device-to-device (D2D) networks on Riemannian manifold.  ...  Let, Sym ++ n denote the set of all symmetric positive definite matrices of size n × n.  ... 
arXiv:2106.13707v1 fatcat:rrjzn3bfzzaq3gofhwp2psvew4

Kernel Methods on Riemannian Manifolds with Gaussian RBF Kernels

Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash Harandi
2015 IEEE Transactions on Pattern Analysis and Machine Intelligence  
We then use the proposed framework to identify positive definite kernels on two specific manifolds commonly encountered in computer vision: the Riemannian manifold of symmetric positive definite matrices  ...  the help of such positive definite Gaussian kernels.  ...  The authors would like thank Bob Williamson for useful discussions.  ... 
doi:10.1109/tpami.2015.2414422 pmid:26539851 fatcat:zyremymijjbghcozmcet7ev474

Kernel analysis over Riemannian manifolds for visual recognition of actions, pedestrians and textures

Mehrtash T. Harandi, Conrad Sanderson, Arnold Wiliem, Brian C. Lovell
2012 2012 IEEE Workshop on the Applications of Computer Vision (WACV)  
Instead of using tangent spaces, we propose embedding into the Reproducing Kernel Hilbert Space by introducing a Riemannian pseudo kernel.  ...  A convenient way of analysing Riemannian manifolds is to embed them in Euclidean spaces, with the embedding typically obtained by flattening the manifold via tangent spaces.  ...  Performance comparison on Sequences 1 through 3 of the ETHZ dataset, in terms of Cumulative Matching Characteristic curves.  ... 
doi:10.1109/wacv.2012.6163005 dblp:conf/wacv/HarandiSWL12 fatcat:pxqe4xpwr5cono7is7omhi6xqm

Towards Generalized and Efficient Metric Learning on Riemannian Manifold

Pengfei Zhu, Hao Cheng, Qinghua Hu, Qilong Wang, Changqing Zhang
2018 Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence  
For widely used symmetric positive definite (SPD) manifold and Grassmann manifold, most of existing metric learning methods are designed for one manifold, and are not straightforward for the other one.  ...  By minimizing the geodesic distance of similar pairs and the interpoint geodesic distance of dissimilar ones on nonlinear manifolds, the proposed RMML is optimized by computing the geodesic mean between  ...  Acknowledgments This work was supported by the National Natural Science Foundation of China under Grants 61502332 and 61732011, Natural Science Foundation of Tianjin Under Grants 17JCZDJC30800.  ... 
doi:10.24963/ijcai.2018/449 dblp:conf/ijcai/ZhuCHWZ18 fatcat:uwsplhmkwjffppb2pihcrksawi

Parallel Transport on the Cone Manifold of SPD Matrices for Domain Adaptation

Or Yair, Mirela Ben-Chen, Ronen Talmon
2019 IEEE Transactions on Signal Processing  
We propose to view the data through the lens of covariance matrices and present a method for domain adaptation using parallel transport on the cone manifold of symmetric positive-definite matrices.  ...  We provide rigorous analysis using Riemanninan geometry, illuminating the theoretical guarantees and benefits of the presented method.  ...  One prominent case is when positive semi-definite kernel functions are used as features instead of the covariance matrices.  ... 
doi:10.1109/tsp.2019.2894801 fatcat:2h4u4queljfvvmvnqzpmmve3cu

Filter bank riemannian-based kernel support vector machine for motor imagery decoding

Yueqi Zhang, Jiaming Chen, L. Nguyen
2022 ITM Web of Conferences  
To solve this problem, the Filter Bank Riemannian-based Kernel Support Vector Machine (FBRK-SVM) method that combines the filter bank structure and Riemannian-based kernel was proposed.  ...  Motor Imagery-based BCI (MI-BCI) is one of the most important paradigms in the BCI field.  ...  is a symmetric positive definite matrix on a Riemannian manifold, is the tangent vector on tangent space ࣮ ࣧ, and ‫)ڄ(݈݉݃‬ denotes the logarithm of a matrix.  ... 
doi:10.1051/itmconf/20224702013 fatcat:d7pidntu4rfivko6cskdsigvxm

Component SPD matrices: A low-dimensional discriminative data descriptor for image set classification

Kai-Xuan Chen, Xiao-Jun Wu
2018 Computational Visual Media  
Here, the Riemannian kernel is shown to satisfy the Mercer's theorem, so our proposed CSPD matrix is symmetric and positive definite and also lies on a Riemannian manifold.  ...  In the domain of pattern recognition, using the SPD (Symmetric Positive Definite) matrices to represent data and taking the metrics of resulting Riemannian manifold into account have been widely used for  ...  Note that as Eq. ( 6 ) shows that the log-Euclidean kernel guarantees positive definiteness of the Riemannian kernel, the CSPD matrix also lies on the SPD manifold.  ... 
doi:10.1007/s41095-018-0119-7 fatcat:2cklljsii5cu5jidstn2xwwlhm

Density estimation and modeling on symmetric spaces [article]

Didong Li, Yulong Lu, Emmanuel Chevallier, David B. Dunson
2020 arXiv   pre-print
We illustrate the theory and practical utility of the proposed approach on the space of positive definite matrices.  ...  With these new kernels on symmetric spaces, we also consider the problem of density estimation.  ...  Examples of locally symmetric spaces Space of positive definite matrices The space of all positive definite matrices arises in a variety of applications.  ... 
arXiv:2009.01983v3 fatcat:l62dohofzvf4pmnlljgqmct3t4

Dimensionality reduction based on distance preservation to local mean for symmetric positive definite matrices and its application in brain–computer interfaces

Alireza Davoudi, Saeed Shiry Ghidary, Khadijeh Sadatnejad
2017 Journal of Neural Engineering  
In this paper, we propose a nonlinear dimensionality reduction algorithm for the manifold of Symmetric Positive Definite (SPD) matrices that considers the geometry of SPD matrices and provides a low dimensional  ...  representation of the manifold with high class discrimination.  ...  Covariance matrices lie in the space of Symmetric Positive Definite (SPD) matrices, which can be formulated as a Riemannian manifold.  ... 
doi:10.1088/1741-2552/aa61bb pmid:28220764 fatcat:nz3ritn5efg4rbnwepem27e25u

Riemannian kernel based Nyström method for approximate infinite-dimensional covariance descriptors with application to image set classification [article]

Kai-Xuan Chen, Xiao-Jun Wu, Rui Wang, Josef Kittler
2018 arXiv   pre-print
We start by modeling the images via CovDs, which lie on the Riemannian manifold spanned by SPD (Symmetric Positive Definite) matrices.  ...  We propose a novel framework for representing image sets by approximating infinite-dimensional CovDs in the paradigm of the Nystr\"om method based on a Riemannian kernel.  ...  The CovDs are in the form of SPD(Symmetric Positive Definite) matrices which lie on a non-linear manifold known as the SPD manifold [7, 8] .  ... 
arXiv:1806.06177v1 fatcat:bgb7laeulrcrbjg6py2jtm7eli
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