Dynamical Optimal Transport with $\mathfrak{so}(d)$-Invariance: From Theory to Computation

What the math gives to ML

The paper’s transferable asset is dynamic optimal transport after quotienting out global rigid motions, so displacement caused solely by translation or rotation is not charged as deformation. For Gaussian measures, the quotient geometry reduces to Euclidean distance between vectors of square roots of ordered covariance eigenvalues, providing a cheap differentiable invariant representation. This can become a rotation- and translation-invariant latent-distribution loss or regularizer for embeddings, especially when equivalent coordinate frames should produce identical representations. The strongest first test is to replace covariance matching based on raw matrix entries with the spectrum-based quotient loss and measure robustness under random orthogonal transformations.