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Mixed-Curvature Tree-Sliced Wasserstein Distance

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
mixed curvature spacesliced optimal transport
Abstract:

Mixed-curvature spaces have emerged as a powerful alternative to their Euclidean counterpart, enabling data representations better aligned with the intrinsic structure of complex datasets. However, comparing probability distributions in such spaces remains underdeveloped: existing measures such as KL divergence and Wasserstein either rely on strong assumptions on distributions or incur high computational costs. The Sliced-Wasserstein (SW) framework provides an alternative approach for constructing distributional distances; however, its reliance on one-dimensional projections limits its ability to capture the geometry of the ambient space. To address this limitation, the Tree-Sliced Wasserstein (TSW) framework employs tree structures as a richer projected space. Motivated by the intuition that such a space is particularly suitable for representing the geometric properties of mixed-curvature manifolds, we introduce the Mixed-Curvature Tree-Sliced Wasserstein (MC-TSW), a novel discrepancy measure that is computationally efficient while faithfully capturing both the topological and geometric structures of mixed-curvature spaces. Specifically, we introduce an adaptation of tree systems and Radon transform to mixed-curvature spaces, which yields a closed-form solution for the optimal transport problem on the tree system. We further provide theoretical analysis on the properties of the Radon transform and the MC-TSW distance. Experimental results demonstrate that MC-TSW improves distributional comparisons over product-space-based distance and line-based baselines, and that mixed-curvature representations consistently outperform constant-curvature alternatives, highlighting their importance for modeling complex datasets.

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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.
If you have any questions, please contact: mingzhang23@m.fudan.edu.cn

Overview

Overall Novelty Assessment

The paper introduces MC-TSW, a novel distance measure for comparing probability distributions in mixed-curvature spaces by adapting tree-sliced Wasserstein frameworks to non-Euclidean geometries. Within the taxonomy, it occupies the 'Tree-Sliced and Projection-Based Wasserstein Distances' leaf under 'Optimal Transport and Wasserstein-Based Methods'. Notably, this leaf contains no sibling papers in the current taxonomy, suggesting this specific research direction—combining tree-based projections with mixed-curvature optimal transport—is relatively sparse. The broader parent category addresses computational challenges in Wasserstein distances for curved spaces, but the tree-sliced approach appears underexplored in this geometric context.

The taxonomy reveals neighboring work in divergence-based methods (Fisher-Rao and KL divergence on manifolds) and wrapped distributions (exponential map-based constructions on symmetric spaces), which provide alternative frameworks for distribution comparison and modeling in curved geometries. The paper's approach diverges from these by prioritizing computational efficiency through projection structures rather than direct divergence computation or distribution wrapping. Curvature-based representation learning methods (anomaly detection, knowledge graph embeddings) occupy a separate branch focused on embedding design rather than distribution comparison, highlighting that MC-TSW addresses a complementary measurement problem once representations are established in mixed-curvature spaces.

Among the three contributions analyzed, the first two show no clear refutation across limited candidates examined: 'Mixed-Curvature Tree Systems and Radon Transform' examined six candidates with zero refutable matches, while 'MCTSW Distance' examined ten candidates, also with zero refutable. The third contribution, 'Theoretical Analysis of Radon Transform and MCTSW Properties', examined ten candidates and found three potentially refutable matches, suggesting some theoretical properties may overlap with existing work. The total search scope covered twenty-six candidates, indicating these assessments reflect a focused semantic search rather than exhaustive coverage of all optimal transport or tree-sliced literature.

Based on the limited search scope of twenty-six semantically related candidates, the work appears to occupy a novel intersection of tree-sliced methods and mixed-curvature geometry, with the sparse taxonomy leaf and low refutation rates supporting this impression. However, the analysis does not cover the full breadth of optimal transport literature or recent advances in hyperbolic/spherical geometry that may exist outside the top-K semantic matches examined. The theoretical contribution shows more overlap with prior work than the methodological innovations.

Taxonomy

Core-task Taxonomy Papers
10
3
Claimed Contributions
26
Contribution Candidate Papers Compared
3
Refutable Paper

Research Landscape Overview

Core task: Comparing probability distributions in mixed-curvature spaces. The field addresses how to measure similarity or divergence between distributions when the underlying geometry combines flat, positively curved (spherical), and negatively curved (hyperbolic) regions. The taxonomy reveals several complementary perspectives: Optimal Transport and Wasserstein-Based Methods focus on geometric distance metrics that respect the manifold structure; Divergence-Based Distribution Comparison emphasizes information-theoretic measures adapted to non-Euclidean settings; Wrapped and Exponential Map-Based Distributions (e.g., Exponential-Wrapped Distributions[2]) construct probabilistic models by mapping Euclidean distributions onto curved spaces; Curvature-Based Representation Learning (including works like Hyperbolic VAE[4] and Curved Geometric Networks[5]) learns embeddings where curvature captures hierarchical or relational structure; Curvature Analysis in Stochastic and Physical Systems (such as Chaos Mixed Curvature[7]) studies how curvature influences dynamical behavior; and Theoretical Foundations of Probability Spaces and Curvature (e.g., Probability Curvature Mixtures[10]) provides rigorous mathematical underpinnings for these geometric probability frameworks. A central tension across branches is balancing computational tractability with geometric fidelity: optimal transport methods offer principled distance measures but can be expensive in high dimensions, while divergence-based approaches (like KL-Aware Dimensionality Reduction[6]) may scale better but require careful adaptation to curved geometries. Mixed-Curvature Tree-Sliced[0] sits within the Optimal Transport and Wasserstein-Based Methods branch, specifically under Tree-Sliced and Projection-Based Wasserstein Distances. This positioning suggests it addresses computational challenges by projecting distributions onto lower-dimensional tree structures, enabling efficient Wasserstein distance computation in mixed-curvature settings. Compared to general curvature-based representation learning methods like Curved Geometric Networks[5], which focus on embedding design, Mixed-Curvature Tree-Sliced[0] emphasizes the comparison step itself, providing a scalable metric for distributions already residing in or mapped to mixed-curvature spaces. This complements probabilistic modeling approaches such as Exponential-Wrapped Distributions[2] by offering a practical tool to evaluate and compare the resulting distributions.

Claimed Contributions

Mixed-Curvature Tree Systems and Radon Transform

The authors construct tree metric systems within mixed-curvature spaces by defining geodesic rays and tree structures that combine heterogeneous geometries. They develop a Radon-type transform operator on these tree systems to enable efficient distributional comparison across different curvature components.

6 retrieved papers
Mixed-Curvature Tree-Sliced Wasserstein (MCTSW) Distance

The authors introduce MCTSW, a novel discrepancy measure for comparing probability distributions in mixed-curvature spaces. This distance leverages tree structures to transport mass across heterogeneous geometric components while admitting closed-form solutions for optimal transport on the tree system.

10 retrieved papers
Theoretical Analysis of Radon Transform and MCTSW Properties

The authors establish theoretical properties of their proposed Radon transform and MCTSW distance, including well-definedness, linearity, injectivity of the transform, and metricity of the distance measure.

10 retrieved papers
Can Refute

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

Mixed-Curvature Tree Systems and Radon Transform

The authors construct tree metric systems within mixed-curvature spaces by defining geodesic rays and tree structures that combine heterogeneous geometries. They develop a Radon-type transform operator on these tree systems to enable efficient distributional comparison across different curvature components.

Contribution

Mixed-Curvature Tree-Sliced Wasserstein (MCTSW) Distance

The authors introduce MCTSW, a novel discrepancy measure for comparing probability distributions in mixed-curvature spaces. This distance leverages tree structures to transport mass across heterogeneous geometric components while admitting closed-form solutions for optimal transport on the tree system.

Contribution

Theoretical Analysis of Radon Transform and MCTSW Properties

The authors establish theoretical properties of their proposed Radon transform and MCTSW distance, including well-definedness, linearity, injectivity of the transform, and metricity of the distance measure.

Table of Contents