Cycle Consistent Probability Divergences Across Different Spaces
This addresses a foundational challenge in statistical inference and machine learning for applications requiring meaningful correspondences across disparate data domains, offering a theoretical framework for existing methods.
The paper tackles the problem of measuring discrepancies between probability distributions on different spaces by proposing a novel unbalanced Monge optimal transport formulation that encodes consistent bidirectional maps, showing it generalizes cycle-consistent GANs and leads to a kernelized version called GMMD with proven convergence rates.
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.