Cycle Consistent Probability Divergences Across Different Spaces

Zhengxin Zhang, Youssef Mroueh, Ziv Goldfeld, Bharath K. Sriperumbudur

Research output: Contribution to journalConference articlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)7257-7285
Number of pages29
JournalProceedings of Machine Learning Research
StatePublished - 2022
Event25th International Conference on Artificial Intelligence and Statistics, AISTATS 2022 - Virtual, Online, Spain
Duration: Mar 28 2022Mar 30 2022

All Science Journal Classification (ASJC) codes

  • Artificial Intelligence
  • Software
  • Control and Systems Engineering
  • Statistics and Probability

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