## Abstract

Nonlinear monotone transformations are used extensively in normalizing flows to construct invertible triangular mappings from simple distributions to complex ones. In existing literature, monotonicity is usually enforced by restricting function classes or model parameters and the inverse transformation is often approximated by root-finding algorithms as a closed-form inverse is unavailable. In this paper, we introduce a new integral-based approach termed: Atomic Unrestricted Time Machine (AUTM), equipped with unrestricted integrands and easy-to-compute explicit inverse. AUTM offers a versatile and efficient way to the design of normalizing flows with explicit inverse and unrestricted function classes or parameters. Theoretically, we present a constructive proof that AUTM is universal: all monotonic normalizing flows can be viewed as limits of AUTM flows. We provide a concrete example to show how to approximate any given monotonic normalizing flow using AUTM flows with guaranteed convergence. The result implies that AUTM can be used to transform an existing flow into a new one equipped with explicit inverse and unrestricted parameters. The performance of the new approach is evaluated on high dimensional density estimation, variational inference and image generation. Experiments demonstrate superior speed and memory efficiency of AUTM.

Original language | English |
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Title of host publication | Proceedings of the 38th Conference on Uncertainty in Artificial Intelligence, UAI 2022 |

Pages | 266-274 |

Number of pages | 9 |

ISBN (Electronic) | 9781713863298 |

State | Published - 2022 |

Event | 38th Conference on Uncertainty in Artificial Intelligence, UAI 2022 - Eindhoven, Netherlands Duration: Aug 1 2022 → Aug 5 2022 |

### Publication series

Name | Proceedings of the 38th Conference on Uncertainty in Artificial Intelligence, UAI 2022 |
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### Conference

Conference | 38th Conference on Uncertainty in Artificial Intelligence, UAI 2022 |
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Country/Territory | Netherlands |

City | Eindhoven |

Period | 8/1/22 → 8/5/22 |

### Bibliographical note

Funding Information:D. Cai and Y. Xi are supported by NSF awards OAC 2003720 and DMS 2038118. Y. Xi is also supported by DMS 2208412. Q. Ye and H. He are supported by NSF awards DMS 1821144 and IIS 1838200, respectively.

Publisher Copyright:

© 2022 Proceedings of the 38th Conference on Uncertainty in Artificial Intelligence, UAI 2022. All right reserved.

## ASJC Scopus subject areas

- Artificial Intelligence