The DOI matches its Crossref record; 27 of the 27 references it cites were found in a registry or an index.
Stochastic Gradient Descent on Riemannian Manifolds
Open accessAbstract
Stochastic gradient descent is a simple approach to find the local minima of a cost function whose evaluations are corrupted by noise. In this paper, we develop a procedure extending stochastic gradient descent algorithms to the case where the function is defined on a Riemannian manifold. We prove that, as in the Euclidian case, the gradient descent algorithm converges to a critical point of the cost function. The algorithm has numerous potential applications, and is illustrated here by four examples. In particular a novel gossip algorithm on the set of covariance matrices is derived and tested numerically.
Overview
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The cited source says what the claim attributes to it.
In the citing paper
“We also compute an analytic closed-form expression for the exponential map in the n -dimensional Poincar´ e ball, allowing us to perform full Riemannian optimization (Bonnabel, 2013) in the Poincar´ e ball, as opposed to the approximate optimization method used by (Nickel & Kiela, 2017).”
From Hyperbolic Entailment Cones for Learning Hierarchical Embeddings (2018) · cites this paper as Bonnabel, 2013
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Citations
Cites
A Stochastic Approximation Method
Robbins, H., & Monro, S. (1951) · The Annals of Mathematical Statistics · doi:10.1214/aoms/1177729586 · Cited by 9,643
The Geometry of Algorithms with Orthogonality Constraints
Edelman, A. S., Arias, T. A., & Smith, S. T. (1998) · SIAM Journal on Matrix Analysis and Applications · doi:10.1137/s0895479895290954 · Cited by 2,741
Stability of multiagent systems with time-dependent communication links
Moreau, L. (2005) · IEEE Transactions on Automatic Control · doi:10.1109/tac.2004.841888 · Cited by 2,667
Boyd, S., Ghosh, A., Prabhakar, B., & Shah, D. (2006) · IEEE Transactions on Information Theory · doi:10.1109/tit.2006.874516 · Cited by 2,439
Distributed asynchronous deterministic and stochastic gradient optimization algorithms
Tsitsiklis, J., Bertsekas, D., & Athans, M. (1986) · IEEE Transactions on Automatic Control · doi:10.1109/tac.1986.1104412 · Cited by 1,967
Faraut, J., & Korányi, A. (1994) · doi:10.1093/oso/9780198534778.001.0001 · Cited by 1,659
A Riemannian Framework for Tensor Computing
Pennec, X., Fillard, P., & Ayache, N. (2005) · International Journal of Computer Vision · doi:10.1007/s11263-005-3222-z · Cited by 1,553
Analysis of recursive stochastic algorithms
Ljung, L. (1977) · IEEE Transactions on Automatic Control · doi:10.1109/tac.1977.1101561 · Cited by 1,485
Principal components, minor components, and linear neural networks
Oja, E. (1992) · Neural Networks · doi:10.1016/s0893-6080(05)80089-9 · Cited by 826
Candès, E. J., & Plan, Y. (2011) · IEEE Transactions on Information Theory · doi:10.1109/tit.2011.2111771 · Cited by 602
Stabilization of Planar Collective Motion With Limited Communication
Sepulchre, R. J., Paley, D. A., & Leonard, N. E. (2008) · IEEE Transactions on Automatic Control · doi:10.1109/tac.2008.919857 · Cited by 455
A Riemannian interpolation inequality à la Borell, Brascamp and Lieb
Cordero–Erausquin, D., McCann, R. J., & Schmuckenschläger, M. (2001) · Inventiones mathematicae · doi:10.1007/s002220100160 · Cited by 285
Riemannian $L^{p}$ center of mass: Existence, uniqueness, and convexity
Afsari, B. (2010) · Proceedings of the American Mathematical Society · doi:10.1090/s0002-9939-2010-10541-5 · Cited by 282
Covariance, subspace, and intrinsic Crame/spl acute/r-Rao bounds
Smith, S. T. (2005) · IEEE Transactions on Signal Processing · doi:10.1109/tsp.2005.845428 · Cited by 245
Low-Rank Optimization on the Cone of Positive Semidefinite Matrices
Journée, M., Bach, F., Absil, P.-A., & Sepulchre, R. J. (2010) · SIAM Journal on Optimization · doi:10.1137/080731359 · Cited by 239
Learning low-rank kernel matrices
Kulis, B., Sustik, M. A., & Dhillon, I. S. (2006) · International Conference on Machine Learning · doi:10.1145/1143844.1143908 · Cited by 185
Coordinated Motion Design on Lie Groups
Sarlette, A., Bonnabel, S., & Sepulchre, R. J. (2010) · IEEE Transactions on Automatic Control · doi:10.1109/tac.2010.2042003 · Cited by 116
Self-citationFisk, D. L. (1965) · Transactions of the American Mathematical Society · doi:10.1090/s0002-9947-1965-0192542-5 · Cited by 84
Albert Benveniste, A., Goursat, M., & Ruget, G. (1980) · IEEE Transactions on Automatic Control · doi:10.1109/tac.1980.1102497 · Cited by 79
Four Encounters with System Identification
Lennart Ljung, A., Hjalmarsson, H., & Ohlsson, H. (2011) · European Journal of Control · doi:10.3166/ejc.17.449-471 · Cited by 65
Regression on fixed-rank positive semidefinite matrices: a Riemannian approachPreprint
Meyer, G., Bonnabel, S., & Sepulchre, R. J. (2010) · arXiv (Cornell University) · doi:10.48550/arxiv.1006.1288 · Cited by 61
Self-citationOnline learning in the embedded manifold of low-rank matrices
Shalit, U., Weinshall, D., & Chechik, G. (2012) · doi:10.5555/2188385.2188399 · Cited by 53
Linear Regression under Fixed-Rank Constraints: A Riemannian Approach
Meyer, G., Bonnabel, S., & Sepulchre, R. J. (2011) · ORBi (University of Liège) · Cited by 50
Self-citationStochastic algorithms for computing means of probability measures
Arnaudon, M., Dombry, C., Phan, A., & Yang, L. (2011) · Stochastic Processes and their Applications · doi:10.1016/j.spa.2011.12.011 · Cited by 48
Global synchronization on the circle
Sarlette, A., Tuna, S. E., Blondel, V. D., & Sepulchre, R. J. (2008) · IFAC Proceedings Volumes · doi:10.3182/20080706-5-kr-1001.01527 · Cited by 34
Average consensus on Riemannian manifolds with bounded curvature
Tron, R., Afsari, B., & Vidal, R. (2011) · doi:10.1109/cdc.2011.6160965 · Cited by 24
Adaptive filtering for estimation of a low-rank positive semidefinite matrix
Bonnabel, S., Meyer, G., & Sepulchre, R. J. (2010) · Open Repository and Bibliography (University of Liège) · Cited by 4
Self-citation
Research works printed but not linked: 10
- tsuda, 2005, Matrix exponentiated gradient updates for online learning and Bregman projection, J Mach Learn Res, 36, 995
- higham, 1989, Matrix Nearness Problems and Applications
- o'neill, 1983, Semi-Riemaniann Geometry
- amari, 1998, Natural Gradient Works Efficiently in Learning
- barbaresco, 2008, Innovative tools for radar signal processing based on Cartan's geometry of symmetric positive-definite matrices and information geometry, Proc IEEE Radar Conf RADAR'08, 1
- bonnabel, 2011, Convergence des m�thodes de gradient stochastique sur les vari�t�s riemanniennes, Proc XIII i�me Colloque GRETSI Bordeaux
- bottou, 1998, Online Algorithms and Stochastic Approximations
- absil, 2007, Optimization Algorithms on Matrix Manifolds
- meyer, 2011, Geometric optimization algorithms for linear regression on fixed-rank matrices
- oja, 1983, Subspace Methods of Pattern Recognition
Cited by
Sorted by citations
Poincaré Embeddings for Learning Hierarchical Representations
Nickel, M., & Kiela, D. (2017) · arXiv · doi:10.48550/arxiv.1705.08039 · Cited by 667
Riemannian Approaches in Brain-Computer Interfaces: A Review
Yger, F., Berar, M., & Lotte, F. (2017) · IEEE Transactions on Neural Systems and Rehabilitation Engineering · doi:10.1109/tnsre.2016.2627016 · Cited by 420
A Riemannian Network for SPD Matrix Learning
Huang, Z., & Gool, L. V. (2017) · Proceedings of the AAAI Conference on Artificial Intelligence · doi:10.1609/aaai.v31i1.10866 · Cited by 472
Transfer Learning: A Riemannian Geometry framework with applications to Brain-Computer Interfaces
Zanini, P., Congedo, M., Jutten, C., Said, S., & Berthoumieu, Y. (2018) · IEEE Transactions on Biomedical Engineering · doi:10.1109/tbme.2017.2742541 · Cited by 412
Low-Dimensional Hyperbolic Knowledge Graph Embeddings
Chami, I., Wolf, A., Juan, D.-C., Sala, F., Ravi, S., & Ré, C. (2020) · Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics · doi:10.18653/v1/2020.acl-main.617 · Cited by 384
Metadata
- Authors
- Bonnabel, S.
- Affiliations
- Centre de Robotique (FR)
- Corresponding author
- Bonnabel, Silvere
- Venue
- IEEE Transactions on Automatic Control
- Published
- 2013-09
- Volume
- 58
- Issue
- 9
- Pages
- 2217-2229
- ISSN
- 0018-9286 (ISSN-L)
- 1558-2523
- OpenAlex
- W2138674039
- Cited by
- 540
- References
- 27
- Citations by year
- 2026: 33 · 2025: 52 · 2024: 56 · 2023: 51 · 2022: 52 · 2021: 80 · 2020: 64 · 2019: 53 · 2018: 26 · 2017: 24 · 2016: 22 · 2015: 12 · 2014: 8 · 2013: 4
- FWCI
- 16.96
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- IEEE Transactions on Automatic Control (publishedVersion)
- arXiv (Cornell University) (publishedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Stochastic Gradient Optimization Techniques
- Physical Sciences › Engineering › Control and Systems Engineering › Distributed Control Multi-Agent Systems
- Physical Sciences › Mathematics › Numerical Analysis › Advanced Optimization Algorithms Research
- Keywords
- stochastic gradient descent, Riemannian manifolds, gossip algorithms, covariance matrix, distributed optimization, Riemannian optimization
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
