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A Differential Geometric Approach to the Geometric Mean of Symmetric Positive-Definite Matrices
Abstract
In this paper we introduce metric-based means for the space of positive-definite matrices. The mean associated with the Euclidean metric of the ambient space is the usual arithmetic mean. The mean associated with the Riemannian metric corresponds to the geometric mean. We discuss some invariance properties of the Riemannian mean and we use differential geometric tools to give a characterization of this mean.
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Cites
Golub, G. (2012) · doi:10.56021/9781421407944 · Cited by 34,505
Riemannian center of mass and mollifier smoothing
Karcher, H. (1977) · Communications on Pure and Applied Mathematics · doi:10.1002/cpa.3160300502 · Cited by 995
Cited by
Log-Euclidean metrics for fast and simple calculus on diffusion tensors
Arsigny, V., Fillard, P., Pennec, X., & Ayache, N. (2006) · Magnetic Resonance in Medicine · doi:10.1002/mrm.20965 · Cited by 1,113
Multiclass Brain–Computer Interface Classification by Riemannian Geometry
Barachant, A., Bonnet, S., Congedo, M., & Jutten, C. (2012) · IEEE Transactions on Biomedical Engineering · doi:10.1109/tbme.2011.2172210 · Cited by 878
Geometric Means in a Novel Vector Space Structure on Symmetric Positive‐Definite Matrices
Arsigny, V., Fillard, P., Pennec, X., & Ayache, N. (2007) · SIAM Journal on Matrix Analysis and Applications · doi:10.1137/050637996 · Cited by 819
Classification of covariance matrices using a Riemannian-based kernel for BCI applications
Barachant, A., Bonnet, S., Congedo, M., & Jutten, C. (2013) · Neurocomputing · doi:10.1016/j.neucom.2012.12.039 · Cited by 493
Riemannian geometry for EEG-based brain-computer interfaces; a primer and a review
Congedo, M., Barachant, A., & Bhatia, R. (2017) · Brain-Computer Interfaces · doi:10.1080/2326263x.2017.1297192 · Cited by 456
Metadata
- Authors
- Moakher, M.
- Venue
- SIAM Journal on Matrix Analysis and Applications
- Published
- 2005-01
- Volume
- 26
- Issue
- 3
- Pages
- 735-747
- ISSN
- 0895-4798 (ISSN-L)
- 1095-7162
- OpenAlex
- W2032236594
- Cited by
- 702
- References
- 2
- FWCI
- 27.00
- Type
- journal article
- Language
- en
- Open access
- closed
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Bibliographic data: Semantic Scholar, OpenAlex, Crossref.
Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
Semantic Scholar’s data is described in: Kinney, R., et al. (2023). The Semantic Scholar Open Data Platform. arXiv:2301.10140.
