Differential geometry with extreme eigenvalues in the positive semidefinite cone
Differential geometric approaches to the analysis and processing of data in the form of symmetric positive definite (SPD) matrices have had notable successful applications to numerous fields including computer vision, medical imaging, and machine learning. The dominant geometric paradigm for such...
| Main Authors: | , , , |
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| Other Authors: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://hdl.handle.net/10356/179733 |
| _version_ | 1826116579735109632 |
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| author | Mostajerant, Cyrus da Costa, Nathaël Van Goffrier, Graham Sepulchres, Rodolphe |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Mostajerant, Cyrus da Costa, Nathaël Van Goffrier, Graham Sepulchres, Rodolphe |
| author_sort | Mostajerant, Cyrus |
| collection | NTU |
| description | Differential geometric approaches to the analysis and processing of data in
the form of symmetric positive definite (SPD) matrices have had notable
successful applications to numerous fields including computer vision, medical
imaging, and machine learning. The dominant geometric paradigm for such
applications has consisted of a few Riemannian geometries associated with
spectral computations that are costly at high scale and in high dimensions. We
present a route to a scalable geometric framework for the analysis and
processing of SPD-valued data based on the efficient computation of extreme
generalized eigenvalues through the Hilbert and Thompson geometries of the
semidefinite cone. We explore a particular geodesic space structure based on
Thompson geometry in detail and establish several properties associated with
this structure. Furthermore, we define a novel iterative mean of SPD matrices
based on this geometry and prove its existence and uniqueness for a given
finite collection of points. Finally, we state and prove a number of desirable
properties that are satisfied by this mean. |
| first_indexed | 2024-10-01T04:13:46Z |
| format | Journal Article |
| id | ntu-10356/179733 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T04:13:46Z |
| publishDate | 2024 |
| record_format | dspace |
| spelling | ntu-10356/1797332024-08-20T01:42:27Z Differential geometry with extreme eigenvalues in the positive semidefinite cone Mostajerant, Cyrus da Costa, Nathaël Van Goffrier, Graham Sepulchres, Rodolphe School of Physical and Mathematical Sciences Mathematical Sciences Affine-invariance Convex cones Differential geometric approaches to the analysis and processing of data in the form of symmetric positive definite (SPD) matrices have had notable successful applications to numerous fields including computer vision, medical imaging, and machine learning. The dominant geometric paradigm for such applications has consisted of a few Riemannian geometries associated with spectral computations that are costly at high scale and in high dimensions. We present a route to a scalable geometric framework for the analysis and processing of SPD-valued data based on the efficient computation of extreme generalized eigenvalues through the Hilbert and Thompson geometries of the semidefinite cone. We explore a particular geodesic space structure based on Thompson geometry in detail and establish several properties associated with this structure. Furthermore, we define a novel iterative mean of SPD matrices based on this geometry and prove its existence and uniqueness for a given finite collection of points. Finally, we state and prove a number of desirable properties that are satisfied by this mean. Nanyang Technological University The first author was supported by a Presidential Postdoctoral Fellowship at Nanyang Technological University (NTU Singapore) and an Early Career Research Fellowship at the University of Cambridge. The third author was supported by the UCL Centre for Doctoral Training in Data Intensive Science funded by STFC and by an Overseas Research Scholarship from UCL. The research leading to these results has also received funding from the European Research Council under the Advanced ERC grant agreement SpikyControl 101054323. 2024-08-20T01:41:49Z 2024-08-20T01:41:49Z 2023 Journal Article Mostajerant, C., da Costa, N., Van Goffrier, G. & Sepulchres, R. (2023). Differential geometry with extreme eigenvalues in the positive semidefinite cone. SIAM Journal On Matrix Analysis and Applications, 45(2), 1089-1113. https://dx.doi.org/10.1137/23M1563906 0895-4798 https://hdl.handle.net/10356/179733 10.1137/23M1563906 2-s2.0-85196645302 2 45 1089 1113 en SIAM Journal on Matrix Analysis and Applications © 2024 Society for Industrial and Applied Mathematics. All rights reserved. |
| spellingShingle | Mathematical Sciences Affine-invariance Convex cones Mostajerant, Cyrus da Costa, Nathaël Van Goffrier, Graham Sepulchres, Rodolphe Differential geometry with extreme eigenvalues in the positive semidefinite cone |
| title | Differential geometry with extreme eigenvalues in the positive semidefinite cone |
| title_full | Differential geometry with extreme eigenvalues in the positive semidefinite cone |
| title_fullStr | Differential geometry with extreme eigenvalues in the positive semidefinite cone |
| title_full_unstemmed | Differential geometry with extreme eigenvalues in the positive semidefinite cone |
| title_short | Differential geometry with extreme eigenvalues in the positive semidefinite cone |
| title_sort | differential geometry with extreme eigenvalues in the positive semidefinite cone |
| topic | Mathematical Sciences Affine-invariance Convex cones |
| url | https://hdl.handle.net/10356/179733 |
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