Newton's Algorithm as a Gradient Flow: A Geometric Framework for Recursive Mixture Estimation
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866913032955756544 |
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| author | Flores, Bernardo |
| author_facet | Flores, Bernardo |
| contents | Bayesian nonparametric mixture models provide a flexible framework for data analysis but are often hindered by the computational expense of traditional inference methods like MCMC. A fast, recursive algorithm proposed by Newton (2002) offers a practical alternative, yet its formal connection to Bayesian inference and its theoretical properties remain only partially understood. This paper reveals a new geometric interpretation of this class of predictive recursions. We demonstrate that Newton's recursion is a discrete-time approximation of a gradient flow on the space of probability measures governed by the Fisher-Rao geometry, providing the first rigorous dynamical characterisation of this family of estimators. This geometric perspective provides a principled theoretical foundation for studying these recursions: it clarifies their convergence behaviour, situates them within the variational Bayes literature, and yields a systematic basis for generalisation by modifying the underlying geometry and discretisation. In contrast to approaches that construct gradient flows from a prescribed variational objective, this work proceeds in the reverse direction: beginning from an existing recursive estimator and uncovering the variational problem it implicitly solves, it opens a pathway for the systematic analysis and extension of a broad class of sequential Bayesian estimators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_13341 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Newton's Algorithm as a Gradient Flow: A Geometric Framework for Recursive Mixture Estimation Flores, Bernardo Methodology Bayesian nonparametric mixture models provide a flexible framework for data analysis but are often hindered by the computational expense of traditional inference methods like MCMC. A fast, recursive algorithm proposed by Newton (2002) offers a practical alternative, yet its formal connection to Bayesian inference and its theoretical properties remain only partially understood. This paper reveals a new geometric interpretation of this class of predictive recursions. We demonstrate that Newton's recursion is a discrete-time approximation of a gradient flow on the space of probability measures governed by the Fisher-Rao geometry, providing the first rigorous dynamical characterisation of this family of estimators. This geometric perspective provides a principled theoretical foundation for studying these recursions: it clarifies their convergence behaviour, situates them within the variational Bayes literature, and yields a systematic basis for generalisation by modifying the underlying geometry and discretisation. In contrast to approaches that construct gradient flows from a prescribed variational objective, this work proceeds in the reverse direction: beginning from an existing recursive estimator and uncovering the variational problem it implicitly solves, it opens a pathway for the systematic analysis and extension of a broad class of sequential Bayesian estimators. |
| title | Newton's Algorithm as a Gradient Flow: A Geometric Framework for Recursive Mixture Estimation |
| topic | Methodology |
| url | https://arxiv.org/abs/2604.13341 |