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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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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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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 ›
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