TY - JOUR
T1 - Cycle Consistent Probability Divergences Across Different Spaces
AU - Zhang, Zhengxin
AU - Mroueh, Youssef
AU - Goldfeld, Ziv
AU - Sriperumbudur, Bharath K.
N1 - Funding Information:
We thank the anonymous reviewers and the area chair, whose feedback helped improve this paper. Z. Goldfeld is supported by the NSF CRII grant CCF-1947801, the 2020 IBM Academic Award, and the NSF CAREER Award CCF-2046018.
Publisher Copyright:
Copyright © 2022 by the author(s)
PY - 2022
Y1 - 2022
N2 - Discrepancy measures between probability distributions are at the core of statistical inference and machine learning. In many applications, distributions of interest are supported on different spaces, and yet a meaningful correspondence between data points is desired. Motivated to explicitly encode consistent bidirectional maps into the discrepancy measure, this work proposes a novel unbalanced Monge optimal transport formulation for matching, up to isometries, distributions on different spaces. Our formulation arises as a principled relaxation of the Gromov-Haussdroff distance between metric spaces, and employs two cycle-consistent maps that push forward each distribution onto the other. We study structural properties of the proposed discrepancy and, in particular, show that it captures the popular cycle-consistent generative adversarial network (GAN) framework as a special case, thereby providing the theory to explain it. Motivated by computational efficiency, we then kernelize the discrepancy and restrict the mappings to parametric function classes. The resulting kernelized version is coined the generalized maximum mean discrepancy (GMMD). Convergence rates for empirical estimation of GMMD are studied and experiments to support our theory are provided.
AB - Discrepancy measures between probability distributions are at the core of statistical inference and machine learning. In many applications, distributions of interest are supported on different spaces, and yet a meaningful correspondence between data points is desired. Motivated to explicitly encode consistent bidirectional maps into the discrepancy measure, this work proposes a novel unbalanced Monge optimal transport formulation for matching, up to isometries, distributions on different spaces. Our formulation arises as a principled relaxation of the Gromov-Haussdroff distance between metric spaces, and employs two cycle-consistent maps that push forward each distribution onto the other. We study structural properties of the proposed discrepancy and, in particular, show that it captures the popular cycle-consistent generative adversarial network (GAN) framework as a special case, thereby providing the theory to explain it. Motivated by computational efficiency, we then kernelize the discrepancy and restrict the mappings to parametric function classes. The resulting kernelized version is coined the generalized maximum mean discrepancy (GMMD). Convergence rates for empirical estimation of GMMD are studied and experiments to support our theory are provided.
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M3 - Conference article
AN - SCOPUS:85163117021
SN - 2640-3498
VL - 151
SP - 7257
EP - 7285
JO - Proceedings of Machine Learning Research
JF - Proceedings of Machine Learning Research
T2 - 25th International Conference on Artificial Intelligence and Statistics, AISTATS 2022
Y2 - 28 March 2022 through 30 March 2022
ER -