AUTM flow: atomic unrestricted time machine for monotonic normalizing flows

Difeng Cai, Yuliang Ji, Huan He, Qiang Ye, Yuanzhe Xi
Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence, PMLR 180:266-274, 2022.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v180-cai22a, title = {AUTM flow: atomic unrestricted time machine for monotonic normalizing flows}, author = {Cai, Difeng and Ji, Yuliang and He, Huan and Ye, Qiang and Xi, Yuanzhe}, booktitle = {Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence}, pages = {266--274}, year = {2022}, editor = {Cussens, James and Zhang, Kun}, volume = {180}, series = {Proceedings of Machine Learning Research}, month = {01--05 Aug}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v180/cai22a/cai22a.pdf}, url = {https://proceedings.mlr.press/v180/cai22a.html}, 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.} }
Endnote
%0 Conference Paper %T AUTM flow: atomic unrestricted time machine for monotonic normalizing flows %A Difeng Cai %A Yuliang Ji %A Huan He %A Qiang Ye %A Yuanzhe Xi %B Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2022 %E James Cussens %E Kun Zhang %F pmlr-v180-cai22a %I PMLR %P 266--274 %U https://proceedings.mlr.press/v180/cai22a.html %V 180 %X 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.
APA
Cai, D., Ji, Y., He, H., Ye, Q. & Xi, Y.. (2022). AUTM flow: atomic unrestricted time machine for monotonic normalizing flows. Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research 180:266-274 Available from https://proceedings.mlr.press/v180/cai22a.html.

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