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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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“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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Lu, K., Wang, H., Zhang, H., & Wang, L. (2023) · IEEE Transactions on Automatic Control · doi:10.1109/tac.2023.3327319 · Cited by 31
Efficiently escaping saddle points on manifoldsPreprint
Criscitiello, C., & Boumal, N. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1906.04321 · Cited by 31
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
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