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

We introduce SEINT, a novel Special Euclidean group-Invariant (SE(\emph{p})) metric for comparing probability distributions on $p$-dimensional measured Banach spaces. Existing SE(\emph{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(\emph{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(\emph{p})-invariance, SEINT also supports cross-space distribution comparison. Computationally, SEINT aligns two samples of size $n$ with a complexity of just $\mathcal{O}(n\log n)$ to $\mathcal{O}(n^2)$. Extensive experiments validate its advantages: As a robust metric, it outperforms or matches existing SE(\emph{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.