Bispectral OT: Dataset Comparison using Symmetry-Aware Optimal Transport
This work addresses dataset comparison challenges in machine learning and vision for applications with symmetry-rich data, representing an incremental improvement over existing optimal transport methods.
The paper tackled the problem of optimal transport ignoring intrinsic coherence in symmetry-rich datasets by introducing Bispectral Optimal Transport, which uses the bispectrum to preserve signal structure while removing group action variation. The result showed improved class preservation accuracy over naive feature OT on benchmark datasets with visual symmetries, capturing semantic label structure better.
Optimal transport (OT) is a widely used technique in machine learning, graphics, and vision that aligns two distributions or datasets using their relative geometry. In symmetry-rich settings, however, OT alignments based solely on pairwise geometric distances between raw features can ignore the intrinsic coherence structure of the data. We introduce Bispectral Optimal Transport, a symmetry-aware extension of discrete OT that compares elements using their representation using the bispectrum, a group Fourier invariant that preserves all signal structure while removing only the variation due to group actions. Empirically, we demonstrate that the transport plans computed with Bispectral OT achieve greater class preservation accuracy than naive feature OT on benchmark datasets transformed with visual symmetries, improving the quality of meaningful correspondences that capture the underlying semantic label structure in the dataset while removing nuisance variation not affecting class or content.