Tree-sliced Sobolev IPM

ICLR 2026 Conference SubmissionAnonymous Authors

OpenReview Score: 6.0 Download Report PDF

sliced optimal transportsobolev ipmtree-sliced wasserstein distancetree wasserstein distance

Recent work shows Tree-Sliced Optimal Transport to be an efficient and more expressive alternative to Sliced Wasserstein (SW), improving downstream performance. Tree-sliced metrics compare probability distributions by projecting measures onto tree metric spaces; a central example is the Tree-Sliced Wasserstein (TSW) distance, which applies the $1$ -Wasserstein metric after projection. However, computing tree-based $p$ -Wasserstein for general $p$ is costly, largely confining practical use to $p=1$ . This restriction is a significant bottleneck, as higher-order metrics ( $p > 1$ ) are preferred in gradient-based learning for their more favorable optimization landscapes. In this work, we revisit Sobolev integral probability metrics (IPM) on trees to obtain a practical generalization of TSW. Building on the insight that a suitably regularized Sobolev IPM admits a closed-form expression, we introduce TS-Sobolev, a tree-sliced metric that aggregates regularized Sobolev IPMs over random tree systems and remains tractable for all $p\ge1$ ; for $p>1$ , TS-Sobolev has the same computational complexity as TSW at $p=1$ . Notably, at $p=1$ it recovers TSW exactly. Consequently, TS-Sobolev serves as a drop-in replacement for TSW in practical applications, with an additional flexibility in changing $p$ . Furthermore, we extend this framework to define a corresponding metric for probability measures on hyperspheres. Experiments on Euclidean and spherical datasets show that TS-Sobolev and its spherical variant improve downstream performance in gradient flows, self-supervised learning, generative modeling, and text topic modeling over recent SW and TSW variants.

Abstract:

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.

NOTE that some papers exist in multiple, slightly different versions (e.g., with different titles or URLs). The system may retrieve several versions of the same underlying work. The current automated pipeline does not reliably align or distinguish these cases, so human reviewers will need to disambiguate them manually.

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Overview

Overall Novelty Assessment

The paper introduces TS-Sobolev, a tree-sliced metric that generalizes Tree-Sliced Wasserstein (TSW) by aggregating regularized Sobolev integral probability metrics over random tree systems for all p≥1. Within the taxonomy, it occupies the 'Tree-Sliced Sobolev IPM' leaf under 'Tree-Sliced Integral Probability Metrics Beyond Wasserstein'. Notably, this leaf contains only the original paper itself—no sibling papers exist in this specific category. This positioning suggests the work explores a relatively sparse research direction within the broader tree-sliced metrics landscape, which comprises twelve papers across multiple branches.

The taxonomy reveals that most prior work concentrates on Wasserstein-based tree-sliced methods (seven papers across three leaves) and computational approximations (three papers). The original paper's branch—'Tree-Sliced Integral Probability Metrics Beyond Wasserstein'—contains only two leaves: the original paper's leaf and 'Max-Sliced Bures Distance'. Neighboring branches include geometric extensions like spherical projections and projection optimization on structured domains. The scope note explicitly excludes Wasserstein-based methods, positioning TS-Sobolev as exploring function-space structures (Sobolev norms) rather than purely geometric or computational variants.

Among thirteen candidates examined, no contributions were clearly refuted by prior work. The core TS-Sobolev contribution examined one candidate with zero refutable matches; the spherical variant (STS-Sobolev) examined ten candidates, again with zero refutations; the closed-form regularized Sobolev IPM examined two candidates without refutation. This limited search scope—thirteen papers total—suggests the analysis captures immediate semantic neighbors but cannot claim exhaustive coverage. The absence of refutations across all contributions, combined with the sparse taxonomy leaf, indicates the work may occupy genuinely underexplored territory within tree-sliced metrics.

Based on the top-thirteen semantic matches and taxonomy structure, the work appears to introduce a novel direction by bridging Sobolev IPMs with tree-slicing frameworks. The sparse taxonomy leaf and zero refutations across contributions suggest substantive novelty, though the limited search scope means potentially relevant work in broader IPM or Sobolev metric literature may exist outside this analysis. The contribution's distinctiveness lies in enabling tractable higher-order metrics (p>1) while recovering TSW at p=1, addressing a recognized bottleneck in gradient-based learning applications.

Taxonomy

Core-task Taxonomy Papers

Claimed Contributions

Contribution Candidate Papers Compared

Refutable Paper

Research Landscape Overview

Core task: Comparing probability distributions using tree-sliced integral probability metrics. The field centers on efficiently measuring distances between probability distributions by projecting them onto tree structures rather than one-dimensional lines. The taxonomy reveals four main branches: foundational work on tree-sliced Wasserstein distances and their theoretical extensions; generalizations beyond Wasserstein to broader integral probability metrics; computational methods for tree-based optimal transport; and diverse applications ranging from machine learning to domain adaptation. Early contributions like Tree-sliced Wasserstein distance[4] and Fixed support tree-sliced Wasserstein[5] established the core methodology, while recent extensions such as Spherical tree-sliced Wasserstein distance[1] and Tree-Sliced Wasserstein Distance with[2] explore geometric variants. Computational branches include fast approximation schemes like Approximating 1-wasserstein distance with[7] and Fast tree variants of[8], alongside specialized tools such as Fast Stealthily Biased Sampling[3]. A particularly active line of work focuses on extending tree-slicing beyond classical Wasserstein metrics to richer function spaces and alternative divergence measures. Tree-sliced Sobolev IPM[0] exemplifies this direction by incorporating Sobolev norms into the tree-slicing framework, offering smoother regularization properties compared to standard Wasserstein approaches. This contrasts with purely geometric extensions like Spherical tree-sliced Wasserstein distance[1], which adapts tree-slicing to non-Euclidean manifolds, or computational accelerations such as Fast Stealthily Biased Sampling[3] that prioritize efficiency over metric generalization. The original paper sits within the branch exploring integral probability metrics beyond Wasserstein, addressing how tree-based projections can leverage function space structures to capture distributional differences that classical tree-Wasserstein distances might miss, while maintaining computational tractability through hierarchical decomposition.

Claimed Contributions

Tree-Sliced Sobolev IPM (TS-Sobolev)

1 retrieved paper

The authors propose TS-Sobolev, a novel tree-sliced distance metric for probability measures in Euclidean spaces that generalizes Tree-Sliced Wasserstein by using regularized Sobolev integral probability metrics. This metric is computationally tractable for any order p≥1, overcoming the limitation of TSW which is restricted to p=1, while maintaining the same computational complexity.

1 retrieved paper

Spherical Tree-Sliced Sobolev IPM (STS-Sobolev)

10 retrieved papers

The authors extend the TS-Sobolev framework to define STS-Sobolev, a metric for comparing probability measures on hyperspheres. This extension uses spherical tree systems and the spherical Radon transform to enable tree-sliced Sobolev IPM computations in the spherical setting.

10 retrieved papers

Closed-form regularized Sobolev IPM on trees

2 retrieved papers

The authors leverage a regularized Sobolev integral probability metric that admits a closed-form solution on tree metric spaces for any order p≥1. This theoretical insight enables the practical computation of higher-order tree-sliced metrics, which was previously intractable.

2 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Within the taxonomy built over the current TopK core-task papers, the original paper is assigned to a leaf with no direct siblings and no cousin branches under the same grandparent topic. In this retrieved landscape, it appears structurally isolated, which is one partial signal of novelty, but still constrained by search coverage and taxonomy granularity.

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution