Bispectral OT: Dataset Comparison using Symmetry-Aware Optimal Transport

2025-09-24

AI Summary

Traditional optimal transport (OT) aligns distributions based solely on raw feature-space geometry, failing to account for intrinsic structural symmetries prevalent in many data domains. Method: We propose Bispectrum OT, the first discrete OT framework that incorporates group-theoretic Fourier invariants—specifically, the bispectrum—as symmetry-invariant geometric features to define the transport cost. By unifying representation theory and OT theory, our method constructs a symmetry-aware ground metric and computes an optimal transport plan that preserves semantic consistency. Contribution/Results: Evaluated on visual symmetry benchmark datasets, Bispectrum OT significantly improves class-preservation accuracy over standard OT baselines. It yields more robust and interpretable semantic correspondences, demonstrating superior alignment under symmetric transformations while maintaining computational tractability within the discrete OT paradigm.

Technology Category

• Computer Vision: Visual Reasoning & Symbolic Representations
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

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.

Problem

• Addresses limitations of optimal transport in symmetry-rich datasets
• Improves alignment quality by preserving intrinsic coherence structure
• Removes nuisance variation while maintaining semantic label relationships

Innovation

• Symmetry-aware optimal transport extension
• Uses bispectrum group Fourier invariant
• Removes nuisance variation preserving semantics