Analysis of learning a flow-based generative model from limited sample complexity
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866910568427814912 |
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| author | Cui, Hugo Krzakala, Florent Vanden-Eijnden, Eric Zdeborová, Lenka |
| author_facet | Cui, Hugo Krzakala, Florent Vanden-Eijnden, Eric Zdeborová, Lenka |
| contents | We study the problem of training a flow-based generative model, parametrized by a two-layer autoencoder, to sample from a high-dimensional Gaussian mixture. We provide a sharp end-to-end analysis of the problem. First, we provide a tight closed-form characterization of the learnt velocity field, when parametrized by a shallow denoising auto-encoder trained on a finite number $n$ of samples from the target distribution. Building on this analysis, we provide a sharp description of the corresponding generative flow, which pushes the base Gaussian density forward to an approximation of the target density. In particular, we provide closed-form formulae for the distance between the mean of the generated mixture and the mean of the target mixture, which we show decays as $Θ_n(\frac{1}{n})$. Finally, this rate is shown to be in fact Bayes-optimal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_03575 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Analysis of learning a flow-based generative model from limited sample complexity Cui, Hugo Krzakala, Florent Vanden-Eijnden, Eric Zdeborová, Lenka Machine Learning We study the problem of training a flow-based generative model, parametrized by a two-layer autoencoder, to sample from a high-dimensional Gaussian mixture. We provide a sharp end-to-end analysis of the problem. First, we provide a tight closed-form characterization of the learnt velocity field, when parametrized by a shallow denoising auto-encoder trained on a finite number $n$ of samples from the target distribution. Building on this analysis, we provide a sharp description of the corresponding generative flow, which pushes the base Gaussian density forward to an approximation of the target density. In particular, we provide closed-form formulae for the distance between the mean of the generated mixture and the mean of the target mixture, which we show decays as $Θ_n(\frac{1}{n})$. Finally, this rate is shown to be in fact Bayes-optimal. |
| title | Analysis of learning a flow-based generative model from limited sample complexity |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2310.03575 |