Constructive approximate transport maps with normalizing flows
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916901732483072 |
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| author | Álvarez-López, Antonio Geshkovski, Borjan Ruiz-Balet, Domènec |
| author_facet | Álvarez-López, Antonio Geshkovski, Borjan Ruiz-Balet, Domènec |
| contents | We study an approximate controllability problem for the continuity equation and its application to constructing transport maps with normalizing flows. Specifically, we construct time-dependent controls $θ=(w, a, b)$ in the vector field $x\mapsto w(a^\top x + b)_+$ to approximately transport a known base density $ρ_{\mathrm{B}}$ to a target density $ρ_*$. The approximation error is measured in relative entropy, and $θ$ are constructed piecewise constant, with bounds on the number of switches being provided. Our main result relies on an assumption on the relative tail decay of $ρ_*$ and $ρ_{\mathrm{B}}$, and provides hints on characterizing the reachable space of the continuity equation in relative entropy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_19366 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Constructive approximate transport maps with normalizing flows Álvarez-López, Antonio Geshkovski, Borjan Ruiz-Balet, Domènec Optimization and Control Machine Learning We study an approximate controllability problem for the continuity equation and its application to constructing transport maps with normalizing flows. Specifically, we construct time-dependent controls $θ=(w, a, b)$ in the vector field $x\mapsto w(a^\top x + b)_+$ to approximately transport a known base density $ρ_{\mathrm{B}}$ to a target density $ρ_*$. The approximation error is measured in relative entropy, and $θ$ are constructed piecewise constant, with bounds on the number of switches being provided. Our main result relies on an assumption on the relative tail decay of $ρ_*$ and $ρ_{\mathrm{B}}$, and provides hints on characterizing the reachable space of the continuity equation in relative entropy. |
| title | Constructive approximate transport maps with normalizing flows |
| topic | Optimization and Control Machine Learning |
| url | https://arxiv.org/abs/2412.19366 |