Stochastic Gradient Descent on Riemannian Manifolds
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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.
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An analytic closed-form expression for the exponential map in the n-dimensional Poincaré ball is computed, allowing full Riemannian optimization in the Poincaré ball, as opposed to the approximate optimization method used by Nickel & Kiela (2017).
“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
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Cites
A Stochastic Approximation Method
Robbins, H., & Monro, S. (1951) · The Annals of Mathematical Statistics · doi:10.1214/aoms/1177729586 · Cited by 9,643 · see citing works
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 2,773 · see citing works
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 · see citing works
Boyd, S., Ghosh, A., Prabhakar, B., & Shah, D. (2006) · IEEE Transactions on Information Theory · doi:10.1109/tit.2006.874516 · Cited by 2,439 · see citing works
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 · see citing works
Faraut, J., & Korányi, A. (1994) · doi:10.1093/oso/9780198534778.001.0001 · Cited by 1,659 · see citing works
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 · see citing works
Analysis of recursive stochastic algorithms
Ljung, L. (1977) · IEEE Transactions on Automatic Control · doi:10.1109/tac.1977.1101561 · Cited by 1,485 · see citing works
Principal components, minor components, and linear neural networks
Oja, E. (1992) · Neural Networks · doi:10.1016/s0893-6080(05)80089-9 · Cited by 826 · see citing works
Candès, E. J., & Plan, Y. (2011) · IEEE Transactions on Information Theory · doi:10.1109/tit.2011.2111771 · Cited by 602 · see citing works
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 · see citing works
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 · see citing works
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 · see citing works
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 · see citing works
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 214 · see citing works
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 · see citing works
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 · see citing works
Self-citationFisk, D. L. (1965) · Transactions of the American Mathematical Society · doi:10.1090/s0002-9947-1965-0192542-5 · Cited by 84 · see citing works
Albert Benveniste, A., Goursat, M., & Ruget, G. (1980) · IEEE Transactions on Automatic Control · doi:10.1109/tac.1980.1102497 · Cited by 79 · see citing works
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 · see citing works
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 · see citing works
Self-citationOnline learning in the embedded manifold of low-rank matrices
Shalit, U., Weinshall, D., & Chechik, G. (2012) · Cited by 51 · see citing works
Linear Regression under Fixed-Rank Constraints: A Riemannian Approach
Meyer, G., Bonnabel, S., & Sepulchre, R. J. (2011) · ORBi (University of Liège) · Cited by 50 · see citing works
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 · see citing works
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 · see citing works
Average consensus on Riemannian manifolds with bounded curvature
Tron, R., Afsari, B., & Vidal, R. (2011) · doi:10.1109/cdc.2011.6160965 · Cited by 24 · see citing works
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 · see citing works
Self-citation
Flagged first, then by citations
Research works printed but not linked: 10
- [1][1] P.A. Absil, R. Mahony, and R. Sepulchre. Optimization Algorithms on Matrix Manifolds . Princeton University Press, 2007. -
- [3][3] S.I. Amari. Natural gradient works efficiently in learning . Neural Computation, MIT Press, 1998. -
- [5][5] F. Barbaresco. Innovative tools for radar signal processing based on cartan's geometry of symmetric positive-definite matrices and information geometry. In IEEE Radar Conference , 2008. -
- [9][9] L. Bottou. Online Algorithms and Stochastic Approximations . Online Learning and Neural Networks, Edited by David Saad, Cambridge University Press, 1998. -
- [16][16] Nicholas J. Higham. Matrix nearness problems and applications . Oxford University Press, 1989. -
- [21][21] G. Meyer. Geometric optimization algorithms for linear regression on fixed-rank matrices . PhD thesis, University of Li` ege, 2011. -
- [25][25] E. Oja. Subspace methods of pattern recognition. Research Studies Press, 1983. -
- [27][27] B. O'Neill. Semi-Riemaniann geometry . Pure and applied mathematics. Academic Press Inc., New York, 1983. -
- [34][34] S.T. Smith. Covariance, subspace, and intrinsic Cramer-Rao bounds. IEEE Trans. on Signal Processing , 53(5):1610-1629, 2005. -
- [37][37] Ratsch-G. Tsuda, K. and M. Warmuth. Matrix exponentiated gradient updates for online learning and bregman projection. Journal of Machine Learning Research , 36(6):9951018, 2005.
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“The exponential map is important, for instance, when performing gradient-descent over parameters lying in a manifold (Bonnabel, 2013).”
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Wensing, P. M., & Slotine, J.-J. E. (2020) · PLoS ONE · doi:10.1371/journal.pone.0236661 · Cited by 41 · see citing works
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Zhu, H. (2020) · IEEE Transactions on Computational Social Systems · doi:10.1109/tcss.2020.3006381 · Cited by 41 · see citing works
Online Dictionary Learning on Symmetric Positive Definite Manifolds with Vision Applications
Zhang, S., Kasiviswanathan, S. P., Yuen, P. C., & Harandi, M. T. (2015) · Proceedings of the AAAI Conference on Artificial Intelligence · doi:10.1609/aaai.v29i1.9595 · Cited by 40 · see citing works
Mixed-curvature Variational AutoencodersPreprint
Skopek, O., Ganea, O.-E., & Bécigneul, G. (2019) · arXiv (Cornell University) · doi:10.3929/ethz-b-000372387 · Cited by 40 · see citing works
On approximating the Riemannian 1-center
Arnaudon, M., & Nielsen, F. (2012) · Computational Geometry · doi:10.1016/j.comgeo.2012.04.007 · Cited by 39 · see citing works
Accelerated gradient algorithm for RBF neural network
Han, H.-G., Ma, M.-L., & Qiao, J. (2021) · Neurocomputing · doi:10.1016/j.neucom.2021.02.009 · Cited by 39 · see citing works
Han, H.-G., Ma, M.-L., Yang, H., & Qiao, J. (2021) · Neurocomputing · doi:10.1016/j.neucom.2021.10.065 · Cited by 39 · see citing works
Riemannian Continuous Normalizing FlowsPreprint
Mathieu, É., & Nickel, M. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2006.10605 · Cited by 39 · see citing works
A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifold
Fei, Y., Liu, Y., Jia, C., Zheng-yu, L., Wei, X. H., & Chen, M. (2024) · ACM Computing Surveys · doi:10.1145/3708498 · Cited by 39 · see citing works
Gradient descent algorithms for Bures-Wasserstein barycentersPreprint
Chewi, S., Maunu, T., Rigollet, P., & Stromme, A. J. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2001.01700 · Cited by 38 · see citing works
Hyperbolic Neural Collaborative Recommender
Li, A., Yang, B. X., Huo, H., Chen, H., Xu, G., & Wang, Z. (2022) · IEEE Transactions on Knowledge and Data Engineering · doi:10.1109/tkde.2022.3221386 · Cited by 36 · see citing works
Trivializations for Gradient-Based Optimization on ManifoldsPreprint
Lezcano-Casado, M. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1909.09501 · Cited by 36 · see citing works
Mixed-curvature knowledge-enhanced graph contrastive learning for recommendation
Zhang, Y., Zhu, J., Chen, R., Liao, W., Wang, Y., & Zhou, W. G. (2023) · Expert Systems with Applications · doi:10.1016/j.eswa.2023.121569 · Cited by 35 · see citing works
Hyperbolic Graph Learning for Social Recommendation
Yang, Y., Wu, L., Zhang, K., Hong, R., Zhou, H., Zhang, Z., Zhou, J., & Wang, M. (2023) · IEEE Transactions on Knowledge and Data Engineering · doi:10.1109/tkde.2023.3343402 · Cited by 35 · see citing works
Faster First-Order Methods for Stochastic Non-Convex Optimization on Riemannian Manifolds
Zhou, P., Yuan, X., Yan, S., & Feng, J. (2019) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2019.2933841 · Cited by 35 · see citing works
Hyperbolic Temporal Knowledge Graph Embeddings with Relational and Time Curvatures
Montella, S., Rojas-Barahona, L. M., & Heinecke, J. (2021) · Findings of the Association for Computational Linguistics: ACL (Findings of ACL) · doi:10.18653/v1/2021.findings-acl.292 · Cited by 35 · see citing works
Optimization on Submanifolds of Convolution Kernels in CNNsPreprint
Özay, M., & Okatani, T. (2016) · arXiv (Cornell University) · doi:10.48550/arxiv.1610.07008 · Cited by 34 · see citing works
Learning Feature Interactions with Lorentzian Factorization Machine
Xu, C., & Wu, M. F. (2020) · Proceedings of the AAAI Conference on Artificial Intelligence · doi:10.1609/aaai.v34i04.6119 · Cited by 33 · see citing works
Hyperbolic Knowledge Graph Embeddings for Knowledge Base Completion
Kolyvakis, P., Kalousis, A., & Kiritsis, D. (2020) · Lecture notes in computer science · doi:10.1007/978-3-030-49461-2_12 · Cited by 33 · see citing works
Optimizing quantum circuits with Riemannian gradient flow
Wiersema, R., & Killoran, N. (2023) · Physical Review A · doi:10.1103/physreva.107.062421 · Cited by 34 · see citing works
Adaptive regularization with cubics on manifolds
Agarwal, N., Boumal, N., Bullins, B., & Cartis, C. (2020) · Mathematical Programming · doi:10.1007/s10107-020-01505-1 · Cited by 32 · see citing works
Performance of Hyperbolic Geometry Models on Top-N Recommendation Tasks
Mirvakhabova, L., Frolov, E., Khrulkov, V., Oseledets, I., & Tuzhilin, A. (2020) · ACM Conference on Recommender Systems (RecSys) · doi:10.1145/3383313.3412219 · Cited by 32 · see citing works
Riemannian conjugate gradient methods with inverse retraction
Zhu, X., & Sato, H. (2020) · Computational Optimization and Applications · doi:10.1007/s10589-020-00219-6 · Cited by 32 · see citing works
Asymptotic optimality in stochastic optimization
Duchi, J. C., & Ruan, F. (2021) · The Annals of Statistics · doi:10.1214/19-aos1831 · Cited by 32 · see citing works
Riemannian adaptive stochastic gradient algorithms on matrix manifoldsPreprint
Kasai, H., Jawanpuria, P., & Mishra, B. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1902.01144 · Cited by 32 · see citing works
Convergence in High Probability of Distributed Stochastic Gradient Descent Algorithms
Lu, K., Wang, H., Zhang, H., & Wang, L. (2023) · IEEE Transactions on Automatic Control · doi:10.1109/tac.2023.3327319 · Cited by 31 · see citing works
Efficiently escaping saddle points on manifoldsPreprint
Criscitiello, C., & Boumal, N. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1906.04321 · Cited by 31 · see citing works
- 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 · 2012: 3
- 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 › Computer Science › Computer Networks and Communications › 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
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