An Efficient SE(p)-Invariant Transport Metric Driven by Polar Transport Discrepancy-based Representation

We introduce SEINT, a novel Special Euclidean group-Invariant (SE(p)) metric for comparing probability distributions on p-dimensional measured Banach spaces. Existing SE(p)-invariant alignment methods often face high computational costs or lack metric guarantees. To overcome these limitations, we develop a polar transport discrepancy combined with distance convolution to extract SE(p)-invariant representations. These representations are then used to compute the alignment between two distributions via optimal transport. Theoretically, we prove that SEINT is a well-defined metric on the space of isometry classes of normed vector spaces. Beyond its inherent SE(p)-invariance, SEINT also supports cross-space distribution comparison. Computationally, SEINT aligns two samples of size n with a complexity of just O(n log n) to O(n 2 ). Extensive experiments validate its advantages: As a robust metric, it outperforms or matches existing SE(p)-invariant methods in classification and cross-space tasks under isometries. As a regularizer, it greatly enhances molecular generation performance across both pre-training and fine-tuning tasks, achieving state-of-the-art (SOTA) results on key benchmarks. The code is available at https://github.com/junyilin559/SEINT .