Deep Generative Learning via Schrödinger Bridge

Gefei Wang, Yuling Jiao, Qian Xu, Yang Wang, Can Yang
Proceedings of the 38th International Conference on Machine Learning, PMLR 139:10794-10804, 2021.

Abstract

We propose to learn a generative model via entropy interpolation with a Schr{ö}dinger Bridge. The generative learning task can be formulated as interpolating between a reference distribution and a target distribution based on the Kullback-Leibler divergence. At the population level, this entropy interpolation is characterized via an SDE on [0,1] with a time-varying drift term. At the sample level, we derive our Schr{ö}dinger Bridge algorithm by plugging the drift term estimated by a deep score estimator and a deep density ratio estimator into the Euler-Maruyama method. Under some mild smoothness assumptions of the target distribution, we prove the consistency of both the score estimator and the density ratio estimator, and then establish the consistency of the proposed Schr{ö}dinger Bridge approach. Our theoretical results guarantee that the distribution learned by our approach converges to the target distribution. Experimental results on multimodal synthetic data and benchmark data support our theoretical findings and indicate that the generative model via Schr{ö}dinger Bridge is comparable with state-of-the-art GANs, suggesting a new formulation of generative learning. We demonstrate its usefulness in image interpolation and image inpainting.

Cite this Paper

BibTeX
@InProceedings{pmlr-v139-wang21l, title = {Deep Generative Learning via Schr{ö}dinger Bridge}, author = {Wang, Gefei and Jiao, Yuling and Xu, Qian and Wang, Yang and Yang, Can}, booktitle = {Proceedings of the 38th International Conference on Machine Learning}, pages = {10794--10804}, year = {2021}, editor = {Meila, Marina and Zhang, Tong}, volume = {139}, series = {Proceedings of Machine Learning Research}, month = {18--24 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v139/wang21l/wang21l.pdf}, url = {https://proceedings.mlr.press/v139/wang21l.html}, abstract = {We propose to learn a generative model via entropy interpolation with a Schr{ö}dinger Bridge. The generative learning task can be formulated as interpolating between a reference distribution and a target distribution based on the Kullback-Leibler divergence. At the population level, this entropy interpolation is characterized via an SDE on [0,1] with a time-varying drift term. At the sample level, we derive our Schr{ö}dinger Bridge algorithm by plugging the drift term estimated by a deep score estimator and a deep density ratio estimator into the Euler-Maruyama method. Under some mild smoothness assumptions of the target distribution, we prove the consistency of both the score estimator and the density ratio estimator, and then establish the consistency of the proposed Schr{ö}dinger Bridge approach. Our theoretical results guarantee that the distribution learned by our approach converges to the target distribution. Experimental results on multimodal synthetic data and benchmark data support our theoretical findings and indicate that the generative model via Schr{ö}dinger Bridge is comparable with state-of-the-art GANs, suggesting a new formulation of generative learning. We demonstrate its usefulness in image interpolation and image inpainting.} }
Endnote
%0 Conference Paper %T Deep Generative Learning via Schrödinger Bridge %A Gefei Wang %A Yuling Jiao %A Qian Xu %A Yang Wang %A Can Yang %B Proceedings of the 38th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2021 %E Marina Meila %E Tong Zhang %F pmlr-v139-wang21l %I PMLR %P 10794--10804 %U https://proceedings.mlr.press/v139/wang21l.html %V 139 %X We propose to learn a generative model via entropy interpolation with a Schr{ö}dinger Bridge. The generative learning task can be formulated as interpolating between a reference distribution and a target distribution based on the Kullback-Leibler divergence. At the population level, this entropy interpolation is characterized via an SDE on [0,1] with a time-varying drift term. At the sample level, we derive our Schr{ö}dinger Bridge algorithm by plugging the drift term estimated by a deep score estimator and a deep density ratio estimator into the Euler-Maruyama method. Under some mild smoothness assumptions of the target distribution, we prove the consistency of both the score estimator and the density ratio estimator, and then establish the consistency of the proposed Schr{ö}dinger Bridge approach. Our theoretical results guarantee that the distribution learned by our approach converges to the target distribution. Experimental results on multimodal synthetic data and benchmark data support our theoretical findings and indicate that the generative model via Schr{ö}dinger Bridge is comparable with state-of-the-art GANs, suggesting a new formulation of generative learning. We demonstrate its usefulness in image interpolation and image inpainting.
APA
Wang, G., Jiao, Y., Xu, Q., Wang, Y. & Yang, C.. (2021). Deep Generative Learning via Schrödinger Bridge. Proceedings of the 38th International Conference on Machine Learning, in Proceedings of Machine Learning Research 139:10794-10804 Available from https://proceedings.mlr.press/v139/wang21l.html.

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