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

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 5.6Download Report PDF
Distribution comparison; Optimal Transport; Special Euclidean group; Shape matching
Abstract:

We introduce SEINT, a novel Special Euclidean group-Invariant (SE(\emph{p})) metric for comparing probability distributions on pp-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 nn with a complexity of just O(nlog⁡n)\mathcal{O}(n\log n) to O(n2)\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.

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Overview

Overall Novelty Assessment

The paper introduces SEINT, a metric for comparing probability distributions on p-dimensional measured Banach spaces with Special Euclidean group invariance. It resides in the 'Polar Transport Discrepancy-Based Approaches' leaf, which contains only two papers total (including this one). This leaf sits within the broader 'SE(p)-Invariant Optimal Transport Metrics' branch, indicating a relatively sparse research direction. The taxonomy shows only four papers across the entire field structure, suggesting this is an emerging area rather than a crowded subfield.

The taxonomy reveals three main branches: SE(p)-invariant metrics (where this work sits), foundational probability theory on Banach spaces, and stochastic integration frameworks. The paper's approach connects most directly to the invariant metrics branch but draws on foundational theory for its measure-theoretic underpinnings. Neighboring work includes operator-theoretic methods and geometric characterizations of infinite-dimensional distributions, though these lack the explicit invariance focus. The taxonomy's scope notes clarify that general probability theory without invariance constraints belongs elsewhere, positioning SEINT as addressing a specialized intersection of optimal transport and geometric symmetry.

Among eighteen candidates examined across three contributions, no refutable prior work was identified. The polar transport discrepancy construction examined two candidates with no overlaps; the SEINT metric with theoretical guarantees examined ten candidates with no refutations; the efficient implementation examined six candidates with no overlaps. This limited search scope—focused on top-K semantic matches—suggests the specific combination of polar transport, distance convolution, and SE(p)-invariance may be novel within the examined literature. However, the small candidate pool (eighteen total) means the analysis covers a narrow slice of potentially relevant work.

Given the sparse taxonomy structure and absence of refutations among examined candidates, the work appears to occupy a relatively unexplored niche. The single sibling paper in the same leaf suggests limited direct competition, though the small search scope (eighteen candidates) leaves open the possibility of relevant work outside the semantic neighborhood. The analysis reflects what was found in a targeted search, not an exhaustive field survey.

Taxonomy

Core-task Taxonomy Papers
4
3
Claimed Contributions
18
Contribution Candidate Papers Compared
0
Refutable Paper

Research Landscape Overview

Core task: Comparing probability distributions on measured Banach spaces with SE(p)-invariance. The field structure reflects three main branches that address different aspects of this challenge. The first branch, SE(p)-Invariant Optimal Transport Metrics, develops specialized distance measures between distributions that respect certain geometric symmetries, with approaches ranging from polar transport discrepancy methods to other invariant formulations. The second branch, Foundational Probability Theory on Banach Spaces, establishes the underlying mathematical framework, drawing on classical references such as Banach Spaces Volume II[1] and Infinite Dimensional Probability[2] to handle measure-theoretic subtleties in infinite dimensions. The third branch, Stochastic Integration Frameworks on Banach Spaces, focuses on constructing integration theories for random processes in these settings, exemplified by work on Levy processes like Levy Noises Banach[3]. Together, these branches provide the theoretical infrastructure and computational tools needed to rigorously compare distributions in high-dimensional or infinite-dimensional spaces while preserving structural invariances. Within the SE(p)-Invariant Optimal Transport Metrics branch, a particularly active line of work explores how to design discrepancy measures that remain stable under specific group actions, balancing computational tractability with theoretical guarantees. Polar Transport Discrepancy[0] sits squarely in this area, proposing a novel metric that leverages polar decompositions to achieve SE(p)-invariance. It shares thematic connections with SEINT[4], which also emphasizes symmetry-preserving transport, though the two works differ in their technical construction and the specific invariance properties they prioritize. The main trade-off across these approaches involves the tension between expressiveness—capturing fine-grained distributional differences—and the computational cost of enforcing invariance constraints. Polar Transport Discrepancy[0] addresses this by exploiting geometric structure in the polar representation, positioning itself as a method that aims for both theoretical rigor and practical applicability in settings where symmetry is a natural requirement.

Claimed Contributions

Polar Transport Discrepancy and Distance-convoluted Polar Transport Discrepancy

The authors propose two novel unsupervised feature extraction methods that produce one-dimensional SE(p)-invariant representations from measures in Banach spaces. These techniques do not require training and serve as the foundation for constructing the SEINT distance metric.

2 retrieved papers
SEINT metric with theoretical guarantees

The authors establish that SEINT satisfies the formal properties of a metric on isometry classes of normed vector spaces, ensuring SE(p)-invariance and enabling distribution comparisons across different spaces. This theoretical foundation distinguishes SEINT from methods that only yield pseudometrics.

10 retrieved papers
Efficient numerical implementation of SEINT

The authors develop a computationally efficient algorithm for SEINT with quadratic complexity in the general case and near-linear complexity when distance matrices have decomposable structure. This efficiency makes SEINT practical for large-scale applications.

6 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Polar Transport Discrepancy and Distance-convoluted Polar Transport Discrepancy

The authors propose two novel unsupervised feature extraction methods that produce one-dimensional SE(p)-invariant representations from measures in Banach spaces. These techniques do not require training and serve as the foundation for constructing the SEINT distance metric.

Contribution

SEINT metric with theoretical guarantees

The authors establish that SEINT satisfies the formal properties of a metric on isometry classes of normed vector spaces, ensuring SE(p)-invariance and enabling distribution comparisons across different spaces. This theoretical foundation distinguishes SEINT from methods that only yield pseudometrics.

Contribution

Efficient numerical implementation of SEINT

The authors develop a computationally efficient algorithm for SEINT with quadratic complexity in the general case and near-linear complexity when distance matrices have decomposable structure. This efficiency makes SEINT practical for large-scale applications.

Table of Contents

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