Hilbert Mixture Wasserstein Distance for Efficient Gaussian Mixture Comparison

Mixture Wasserstein compares Gaussian mixtures by optimizing both Gaussian factor alignment and component matching for every pair, making repeated comparisons costly. We introduce Hilbert Mixture Wasserstein (HMW), a metric built from a reusable ordered representation of each mixture. We encode each Gaussian by its mean and Cholesky factor and sort these vectors in a deterministic Hilbert order. Component weights set the lengths of consecutive subintervals of a unit interval. Overlaps between the two mixtures' subintervals specify a sparse coupling, with costs evaluated from the full feature vectors. Comparison is linear in the prepared representations' combined size, and equally weighted mixtures with a common component count admit exact finite Euclidean embeddings. HMW upper-bounds mixture Wasserstein, and we decompose the squared-distance gap into the costs of fixing factor alignment and component matching. We give conditions for a small gap and show that changes in component order can keep HMW bounded away from zero as mixture Wasserstein tends to zero. Constructed examples distinguish covariance response from component matching, and timing experiments examine scaling with mixture size and dimension. We also examine representation reuse in image retrieval and ordered grouping in density reduction and 3D Gaussian Splatting scene compression.