Dimensionality Reduction of SPD Data Based on Riemannian Manifold Tangent Spaces and Isometry
Symmetric positive definite (SPD) data have become a hot topic in machine learning. Instead of a linear Euclidean space, SPD data generally lie on a nonlinear Riemannian manifold. To get over the problems caused by the high data dimensionality, dimensionality reduction (DR) is a key subject for SPD...
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MDPI AG
2021-08-01
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| author | Wenxu Gao Zhengming Ma Weichao Gan Shuyu Liu |
| author_facet | Wenxu Gao Zhengming Ma Weichao Gan Shuyu Liu |
| author_sort | Wenxu Gao |
| collection | DOAJ |
| description | Symmetric positive definite (SPD) data have become a hot topic in machine learning. Instead of a linear Euclidean space, SPD data generally lie on a nonlinear Riemannian manifold. To get over the problems caused by the high data dimensionality, dimensionality reduction (DR) is a key subject for SPD data, where bilinear transformation plays a vital role. Because linear operations are not supported in nonlinear spaces such as Riemannian manifolds, directly performing Euclidean DR methods on SPD matrices is inadequate and difficult in complex models and optimization. An SPD data DR method based on Riemannian manifold tangent spaces and global isometry (RMTSISOM-SPDDR) is proposed in this research. The main contributions are listed: (1) Any Riemannian manifold tangent space is a Hilbert space isomorphic to a Euclidean space. Particularly for SPD manifolds, tangent spaces consist of symmetric matrices, which can greatly preserve the form and attributes of original SPD data. For this reason, RMTSISOM-SPDDR transfers the bilinear transformation from manifolds to tangent spaces. (2) By <i>log</i> transformation, original SPD data are mapped to the tangent space at the identity matrix under the affine invariant Riemannian metric (AIRM). In this way, the geodesic distance between original data and the identity matrix is equal to the Euclidean distance between corresponding tangent vector and the origin. (3) The bilinear transformation is further determined by the isometric criterion guaranteeing the geodesic distance on high-dimensional SPD manifold as close as possible to the Euclidean distance in the tangent space of low-dimensional SPD manifold. Then, we use it for the DR of original SPD data. Experiments on five commonly used datasets show that RMTSISOM-SPDDR is superior to five advanced SPD data DR algorithms. |
| first_indexed | 2024-03-10T07:42:31Z |
| format | Article |
| id | doaj.art-6035689fa3624ac8bc3674890157b019 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T07:42:31Z |
| publishDate | 2021-08-01 |
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| series | Entropy |
| spelling | doaj.art-6035689fa3624ac8bc3674890157b0192023-11-22T12:56:49ZengMDPI AGEntropy1099-43002021-08-01239111710.3390/e23091117Dimensionality Reduction of SPD Data Based on Riemannian Manifold Tangent Spaces and IsometryWenxu Gao0Zhengming Ma1Weichao Gan2Shuyu Liu3School of Electronics and Information Technology, Sun Yat-Sen University, Guangzhou 510006, ChinaSchool of Electronics and Information Technology, Sun Yat-Sen University, Guangzhou 510006, ChinaSchool of Electronics and Information Technology, Sun Yat-Sen University, Guangzhou 510006, ChinaPublic Experimental Teaching Center, Sun Yat-sen University, Guangzhou 510006, ChinaSymmetric positive definite (SPD) data have become a hot topic in machine learning. Instead of a linear Euclidean space, SPD data generally lie on a nonlinear Riemannian manifold. To get over the problems caused by the high data dimensionality, dimensionality reduction (DR) is a key subject for SPD data, where bilinear transformation plays a vital role. Because linear operations are not supported in nonlinear spaces such as Riemannian manifolds, directly performing Euclidean DR methods on SPD matrices is inadequate and difficult in complex models and optimization. An SPD data DR method based on Riemannian manifold tangent spaces and global isometry (RMTSISOM-SPDDR) is proposed in this research. The main contributions are listed: (1) Any Riemannian manifold tangent space is a Hilbert space isomorphic to a Euclidean space. Particularly for SPD manifolds, tangent spaces consist of symmetric matrices, which can greatly preserve the form and attributes of original SPD data. For this reason, RMTSISOM-SPDDR transfers the bilinear transformation from manifolds to tangent spaces. (2) By <i>log</i> transformation, original SPD data are mapped to the tangent space at the identity matrix under the affine invariant Riemannian metric (AIRM). In this way, the geodesic distance between original data and the identity matrix is equal to the Euclidean distance between corresponding tangent vector and the origin. (3) The bilinear transformation is further determined by the isometric criterion guaranteeing the geodesic distance on high-dimensional SPD manifold as close as possible to the Euclidean distance in the tangent space of low-dimensional SPD manifold. Then, we use it for the DR of original SPD data. Experiments on five commonly used datasets show that RMTSISOM-SPDDR is superior to five advanced SPD data DR algorithms.https://www.mdpi.com/1099-4300/23/9/1117dimensionality reductiontangent spacesymmetric positivedefinite matricesisometry |
| spellingShingle | Wenxu Gao Zhengming Ma Weichao Gan Shuyu Liu Dimensionality Reduction of SPD Data Based on Riemannian Manifold Tangent Spaces and Isometry Entropy dimensionality reduction tangent space symmetric positive definite matrices isometry |
| title | Dimensionality Reduction of SPD Data Based on Riemannian Manifold Tangent Spaces and Isometry |
| title_full | Dimensionality Reduction of SPD Data Based on Riemannian Manifold Tangent Spaces and Isometry |
| title_fullStr | Dimensionality Reduction of SPD Data Based on Riemannian Manifold Tangent Spaces and Isometry |
| title_full_unstemmed | Dimensionality Reduction of SPD Data Based on Riemannian Manifold Tangent Spaces and Isometry |
| title_short | Dimensionality Reduction of SPD Data Based on Riemannian Manifold Tangent Spaces and Isometry |
| title_sort | dimensionality reduction of spd data based on riemannian manifold tangent spaces and isometry |
| topic | dimensionality reduction tangent space symmetric positive definite matrices isometry |
| url | https://www.mdpi.com/1099-4300/23/9/1117 |
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