Scalable Rigid-Invariant Distance for Shape Matching and Alignment

Comparing probability distributions from biological images requires metrics that are geometrically grounded and invariant to orientation. Classical optimal transport (OT) distances are sensitive to rotations, while Gromov–Wasserstein (GW) offers invariance but is computationally prohibitive. We introduce **Rigid-Invariant Sliced Wasserstein via Independent Embeddings (RISWIE)**, a scalable pseudometric that achieves rigid invariance by aligning data-adaptive embeddings through optimal signed permutations, at negligible cost. Evaluated on 2D HuBMAP tissue slices and 3D MPI-FAUST meshes, RISWIE attains 95.8\% accuracy with over $10^4\times$ speedup over GW and an AUC of 0.94 for human pose matching. Its optimization also yields explicit axis alignments usable for downstream analysis, making RISWIE a practical and interpretable distance for large-scale geometric data.