Fast convergence of the Expectation Maximization algorithm under a logarithmic Sobolev inequality
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918210691923968 |
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| author | Caprio, Rocco Johansen, Adam M |
| author_facet | Caprio, Rocco Johansen, Adam M |
| contents | We present a new framework for analysing the Expectation Maximization (EM) algorithm. Drawing on recent advances in the theory of gradient flows over Euclidean-Wasserstein spaces, we extend techniques from alternating minimization in Euclidean spaces to the EM algorithm, via its representation as coordinate-wise minimization of the free energy. In so doing, we obtain finite sample error bounds and exponential convergence of the EM algorithm under a natural generalisation of the log-Sobolev inequality. We further show that this framework naturally extends to several variants of EM, offering a unified approach for studying such algorithms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_17949 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fast convergence of the Expectation Maximization algorithm under a logarithmic Sobolev inequality Caprio, Rocco Johansen, Adam M Machine Learning Optimization and Control Statistics Theory Computation We present a new framework for analysing the Expectation Maximization (EM) algorithm. Drawing on recent advances in the theory of gradient flows over Euclidean-Wasserstein spaces, we extend techniques from alternating minimization in Euclidean spaces to the EM algorithm, via its representation as coordinate-wise minimization of the free energy. In so doing, we obtain finite sample error bounds and exponential convergence of the EM algorithm under a natural generalisation of the log-Sobolev inequality. We further show that this framework naturally extends to several variants of EM, offering a unified approach for studying such algorithms. |
| title | Fast convergence of the Expectation Maximization algorithm under a logarithmic Sobolev inequality |
| topic | Machine Learning Optimization and Control Statistics Theory Computation |
| url | https://arxiv.org/abs/2407.17949 |