Inversion-Free Natural Gradient Descent on Riemannian Manifolds
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917546149543936 |
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| author | Draca, Dario Matsubara, Takuo Tran, Minh-Ngoc |
| author_facet | Draca, Dario Matsubara, Takuo Tran, Minh-Ngoc |
| contents | The natural gradient method is a central tool for statistical optimisation, but its broader application is hindered by the assumption of a Euclidean parameter space, the repeated estimation of the Fisher information matrix (FIM), and the computational cost of its subsequent inversion. This paper proposes an intrinsic, inversion-free natural gradient method for statistical models whose parameters lie on general Riemannian manifolds. Formulating statistical optimisation in this non-Euclidean setting allows for the natural enforcement of parameter constraints, the elimination of non-identifiable parameters, and the exploitation of geodesic convexity. Our algorithm is based on a moving approximation of the inverse FIM, which is maintained directly on the manifold. This approximation is efficiently updated with new score vectors using low-rank matrix identities. We prove almost-sure convergence rates of $O(\log s / s^α)$ for the sequence of iterates, and a similar rate for the approximate FIM. A limited-memory variant with sub-quadratic storage complexity is further proposed for large-scale applications. We demonstrate the efficacy of our method on variational Bayes within the Bures-Wasserstein manifold, normalising flows on the Stiefel manifold, and reduced-rank logistic regression. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_02969 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Inversion-Free Natural Gradient Descent on Riemannian Manifolds Draca, Dario Matsubara, Takuo Tran, Minh-Ngoc Machine Learning Computation Methodology The natural gradient method is a central tool for statistical optimisation, but its broader application is hindered by the assumption of a Euclidean parameter space, the repeated estimation of the Fisher information matrix (FIM), and the computational cost of its subsequent inversion. This paper proposes an intrinsic, inversion-free natural gradient method for statistical models whose parameters lie on general Riemannian manifolds. Formulating statistical optimisation in this non-Euclidean setting allows for the natural enforcement of parameter constraints, the elimination of non-identifiable parameters, and the exploitation of geodesic convexity. Our algorithm is based on a moving approximation of the inverse FIM, which is maintained directly on the manifold. This approximation is efficiently updated with new score vectors using low-rank matrix identities. We prove almost-sure convergence rates of $O(\log s / s^α)$ for the sequence of iterates, and a similar rate for the approximate FIM. A limited-memory variant with sub-quadratic storage complexity is further proposed for large-scale applications. We demonstrate the efficacy of our method on variational Bayes within the Bures-Wasserstein manifold, normalising flows on the Stiefel manifold, and reduced-rank logistic regression. |
| title | Inversion-Free Natural Gradient Descent on Riemannian Manifolds |
| topic | Machine Learning Computation Methodology |
| url | https://arxiv.org/abs/2604.02969 |