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Constant-Curvature Sliced Gromov-Wasserstein for Heterogeneous Cross-Curvature Alignment

Shanglin Li, Wenjing Lu, Muyang Li, Nicu Sebe, Ziheng Chen

arXiv:2610.07218v1cs.LG

TL;DR

Mixed-curvature models lack an explicit mechanism for geometric consistency across heterogeneous spaces, and comparing distributions across those spaces remains an open problem. CCSGW extends sliced Gromov–Wasserstein with geodesic-based spherical projections; the paper reports consistent gains across graph and multimodal tasks. Its analysis connects the objective to intrinsic geometric discrepancy, while the discussion identifies extension beyond constant-curvature product spaces as future work.

  • Problem

    Mixed-curvature models typically learn component spaces independently, without an explicit mechanism for geometric consistency, while distribution comparison across such heterogeneous spaces remains unexplored.

    Source: §1 · p. 2

  • Method

    CCSGW extends sliced Gromov–Wasserstein to heterogeneous constant-curvature spaces and introduces geodesic-based one-dimensional projections for spherical spaces.

    Source: §1 · p. 2

  • Results

    CCSGW consistently improves performance across graph anomaly detection, node classification, and multimodal learning; CurvGAD gains up to 5.11 percentage points in AUROC on Elliptic.

    Source: §5 · p. 9, §4.2 · p. 9

  • Takeaways & Limitations

    Minimizing CCSGW controls intrinsic Gromov–Wasserstein discrepancy up to projection error, providing a theoretical connection to distribution-level geometric consistency.

    Source: §5 · p. 9

    The framework is developed for constant-curvature product spaces; extending it to more general heterogeneous manifolds remains future work.

    Source: §5 · p. 10

Abstract

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Recent advances in representation learning have highlighted the utility of constant-curvature models, such as hyperbolic and spherical spaces, for modeling complex data. Mixed-curvature models further enhance this by integrating multiple constant-curvature components. However, these models typically learn each component space independently because spaces with different curvatures are inherently heterogeneous and lack a unified metric. Consequently, they lack explicit mechanisms to enforce geometric consistency across various spaces. Moreover, the problem of comparing probability distributions across mixed-curvature spaces remains unexplored. To compare distributions on heterogeneous spaces, Gromov-Wasserstein (GW) distances provide a principled framework by aligning their intra-space geometries. Building on this, we propose constant-curvature sliced Gromov-Wasserstein (CCSGW), a novel divergence for aligning distributions supported on heterogeneous constant-curvature spaces. We first introduce the missing geodesic-based one-dimensional projections for spherical spaces, and then extend sliced GW to constant-curvature spaces, enabling efficient and principled comparison across manifolds with different curvatures. This formulation preserves intrinsic geometric relationships while avoiding the high computational cost. We provide theoretical analysis showing that CCSGW controls intrinsic geometric discrepancy across heterogeneous spaces, promoting distribution-level geometric consistency. By integrating CCSGW into existing mixed-curvature learning tasks, including graph anomaly detection, graph node classification, and multimodal learning, we observe consistent performance gains across diverse settings.

1 INTRODUCTION

Mixed-curvature models address data with varied geometric structure, but independently learned components lack an explicit way to maintain consistency across spaces. CCSGW compares distributions across constant-curvature spaces, with theoretical support and reported performance gains across several learning tasks.

1 INTRODUCTION

  • Mixed-curvature components are typically learned independently, leaving distances across different-curvature spaces incomparable and without an explicit consistency mechanism.The paper notes that this mismatch can distort global relational structure and potentially degrade downstream performance.Source: §1 · p. 2
  • CCSGW extends sliced Gromov–Wasserstein to compare distributions across heterogeneous constant-curvature spaces, using geodesic-based one-dimensional projections for spherical spaces.The framework completes one-dimensional slicing across hyperbolic, Euclidean, and spherical geometries.Source: §1 · pp. 2–3
  • Theoretical analysis links CCSGW to intrinsic Gromov–Wasserstein discrepancy up to projection error, supporting approximate distance-preserving alignment for most sample pairs.The paper presents this as a justification for distribution-level geometric consistency.Source: §1 · pp. 2–3
  • CCSGW integration consistently improves performance across graph anomaly detection, node classification, and multimodal learning.The introduction reports superior performance across these settings.Source: §1 · p. 3

2 PRELIMINARIES

The preliminaries define constant- and mixed-curvature spaces and review geodesic geometry. They then motivate sliced comparisons through efficient one-dimensional Wasserstein and Gromov–Wasserstein calculations, including geodesic-based slicing on curved spaces.

2.1 RIEMANNIAN MANIFOLD

  • Hyperbolic, spherical, and Euclidean spaces have negative, positive, and zero curvature, respectively, while a mixed-curvature space is a Cartesian product of constant-curvature manifolds.The component manifolds have their own curvatures and dimensions.Source: §2.1 · p. 3
  • Geodesics are minimum-length curves between points; on geodesically complete manifolds, a geodesic exists between every pair, and tangent vectors can be parallel-transported between endpoints.The passage also notes that in this setting geodesics can be extended to geodesic lines.Source: §2.1 · p. 3

2.2 THE SLICED-WASSERSTEIN

  • Sliced Wasserstein projects high-dimensional distributions onto one-dimensional subspaces, approximating the integral with L directions at O(L n log n) complexity.The one-dimensional distance can be computed by sorting samples.Source: §2.2 · pp. 3–4
  • Gromov–Wasserstein matches distributions by preserving their within-distribution pairwise distances, allowing comparison across different spaces or dimensions.Sliced Gromov–Wasserstein reduces this comparison to one-dimensional GW problems, with O(L n log n) overall complexity for L directions.Source: §2.2 · p. 4
  • On curved spaces, Wasserstein costs use geodesic distance, and hyperbolic slicing projects points onto geodesics through the origin and represents them by signed geodesic coordinates.The passage describes geodesics as the curved-space counterpart of one-dimensional subspaces.Source: §2.2 · p. 4

3 METHODS

The methods introduce spherical geodesic projections, define CCSGW across constant-curvature spaces, and analyze when its sliced discrepancy controls intrinsic geometric mismatch.

3.1 SPHERE GEODESIC PROJECTIONS

  • Spherical projections map points to signed geodesic coordinates on the real line, avoiding circular optimal transport and shift alignment.The paper derives projection onto a fixed spherical geodesic and a closed-form projection solution; equatorial and degenerate cases are handled in the appendix.Source: §3.1 · pp. 4–5

3.2 CONSTANT-CURVATURE SLICED GROMOV–WASSERSTEIN

  • CCSGW compares distributions across different curvatures by projecting each onto one-dimensional geodesic coordinates and evaluating sliced GW with shared directions after dimension alignment.The definition reduces to Euclidean SGW when both curvatures are zero; with equal dimensions, zero-padding is the identity.Source: §3.2 · p. 5

3.3 THEORETICAL ANALYSIS

  • Small CCSGW, projection displacement, and matching error imply approximate distance preservation for most sample pairs; spherical guarantees require bounded support small enough to prevent projection wraparound.The intrinsic GW bound includes projection errors from both spaces, and the guarantee is informative in the near-isometric regime.Source: §3.3 · pp. 5–6

4 EXPERIMENTS

The experiments evaluate CCSGW as an alignment objective in mixed-curvature models, first visualizing cross-curvature alignment and then testing three downstream tasks. Across those tasks, CCSGW improves the mean performance of all three backbones.

4 EXPERIMENTS

  • The experiments incorporate CCSGW as an additional training objective to evaluate geometric consistency across heterogeneous constant-curvature spaces.The study first uses a controlled simulation, then evaluates mixed-curvature backbones on downstream tasks.Source: §4 · p. 6

4.2 MAIN EXPERIMENTS

  • CCSGW improves CurvGAD AUROC by up to 5.11 percentage points on Elliptic.This is the reported maximum gain for graph anomaly detection.Source: Table 1 · p. 7, §4.2 · p. 9

4.1 SIMULATIONS

  • The simulation optimizes a Euclidean line-shaped point cloud against a fixed ring-shaped hyperbolic target using CCSGW.As optimization proceeds, the Euclidean points form a circular structure and the CCSGW discrepancy decreases.Source: §4.1 · p. 7, Figure 2 · p. 7

4.2 MAIN EXPERIMENTS

  • CUSP weighted F1 rises by 1.02 percentage points on Cora and 1.34 on Chameleon with CCSGW.The gains are reported for one homophilic and one heterophilic dataset.Source: Table 2 · p. 8, §4.2 · p. 9
  • The main experiments add CCSGW alignment to CurvGAD, CUSP, and EEG-MoCE while retaining each backbone’s stated training protocol.The evaluations cover graph anomaly detection, homophilic and heterophilic node classification, and EEG emotion and cognitive tasks.Source: §4.2 · pp. 8–9
  • For EEG-based multimodal learning, CCSGW increases balanced accuracy by 0.69, 1.42, and 1.61 percentage points on emotion recognition, N-back, and word generation, respectively.Source: §4.2 · p. 9

5 DISCUSSION

The discussion presents CCSGW as a geometry-aware divergence whose minimization controls intrinsic GW discrepancy up to projection error. Its effectiveness is limited by sampled geodesic projections, and the appendix material describes spherical geodesic projection and its optimal parameter.

5 DISCUSSION

  • CCSGW minimization controls intrinsic GW discrepancy up to projection error, connecting its sliced objective to distribution-level geometric consistency.The paper reports consistent performance improvements across graph anomaly detection, node classification, and multimodal learning.Source: §5 · p. 9
  • CCSGW’s effectiveness depends on sampled geodesic projections, which may not fully preserve the geometry of the original high-dimensional manifolds.The discussion proposes subspace alignment as a direction for preserving richer geometric information.Source: §5 · pp. 9–10

1. Geometric Setting

  • The spherical projection setup considers a positive-curvature sphere and defines a geodesic from a starting point and direction via the exponential map.The projection parameter is tied to geodesic distance on the sphere.Source: §1 · p. 16

2. Theorem (Closed-form Geodesic Projection Parameter)

  • The spherical projection theorem characterizes an optimal projection parameter case by case; in a special orthogonality case, every parameter is optimal.The generic case holds for almost every point outside a measure-zero set.Source: §2 · pp. 16, 18

B THEORETICAL ANALYSIS PROOFS

The theoretical setup defines intrinsic GW distortion between distributions on ambient constant-curvature spaces and describes geodesic projections used in the sliced comparison.

B.1 SETTING

  • The setup assumes each distribution is supported in a closed ball of finite radius, with an additional condition for spherical components.The supplied passage does not specify the additional spherical condition.Source: §B.1 · p. 19
  • For a coupling between the distributions, distortion is defined using their ambient geodesic distances; squared GW is the infimum of this distortion.The infimum is attained for compact supports.Source: §B.1 · p. 19
  • A geodesic projection maps a point to its closest point on a geodesic from the component origin, and the sliced discrepancy averages over independent directions.The setup defines the direction-pair quantity before taking its expectation.Source: §B.1 · p. 19

B.2 PROOF OF THM. 3.1

The proof establishes projection properties for constant-curvature spaces, then uses projection-induced distortion bounds and averaging over directions to prove the theorem. Spherical projections are unique under the stated radius condition.

B.2 PROOF OF THM. 3.1

  • For every direction, projections are unique and their signed coordinates are bounded in magnitude by the radius R.The proof establishes these properties for nonpositive curvature and for spherical spaces within the stated radius condition.Source: §B.2 · p. 19
  • For spherical spaces, projection onto span{o, v} is well defined and unique when R < π/(2k), and projected coordinate differences equal shortest great-circle distances.The proof uses the great-circle projection formula and bounds projected coordinates by R.Source: §B.2 · p. 19
  • The triangle inequality bounds the discrepancy between projected and original pairwise distances, and an L2 Minkowski step transfers this control to δX(v).The same argument is stated to apply to Y.Source: §B.2 · p. 20
  • The proof averages the right-hand-side bound over projection directions and invokes Theorem B.3 to obtain Theorem 3.1.The left-hand side of Eq. (18) is independent of the directions being averaged.Source: §B.2 · p. 20

B.3 PROOF OF THM. 3.2

The proof bounds the distortion of a measurable map using coupling distortion and map-to-target residuals, then uses pairwise separation and a tail bound to control dilation deviation.

B.3 PROOF OF THM. 3.2

  • The triangle inequality and an L2 Minkowski argument decompose map distortion into coupling distortion plus two map-to-target residual terms.The two residual terms have norm ρπ(g), while the coupling term has norm dis(π).Source: §B.3 · p. 20
  • For distinct source pairs, the separation condition gives dX(x, x′) ≥ r; the resulting norm bound is at most B.The proof obtains this bound using Lemma B.5 with π = π⋆ and Theorem 3.1.Source: §B.3 · p. 21
  • Chebyshev’s inequality bounds the probability of dilation deviation exceeding t = B/(r√ϵ) using the second-moment bound.The resulting tail bound is expressed as at most ϵ.Source: §B.3 · p. 21

C EXPERIMENT DETAILS

The experiments describe how CCSGW is added to graph anomaly detection, graph node classification, and EEG-based multimodal learning protocols. They specify datasets, evaluation measures, and alignment configurations for each task.

C.1 GRAPH ANOMALY DETECTION

  • On five GADBench datasets, CCSGW is added to CurvGAD using 256 sampled nodes, 32 directions, and a shared alignment weight λ = 10.Test AUROC is reported as the mean and 95% confidence interval at the best validation epoch over ten splits.Source: §C.1 · p. 22

C.2 GRAPH NODE CLASSIFICATION

  • On three homophilic and two heterophilic datasets, CCSGW is added to CUSP with 256 or 512 sampled nodes, 32 directions, and λ = 1.Weighted F1 is reported at the best validation epoch over ten random 60/20/20 splits; λ is warmed up over 50 epochs.Source: §C.2 · p. 22

C.3 EEG-BASED MULTIMODAL LEARNING

  • For EEG-MoCE, CCSGW aligns each non-reference modality’s embedding distribution with a reference hyperbolic modality using averaged one-dimensional GW discrepancies.Evaluation uses subject-wise 10-fold cross-validation on EAV (42 participants) and Cognitive (26 participants), reporting fold-averaged test balanced accuracy.Source: §C.3 · pp. 22–23

D.1 HYPERPARAMETER SENSITIVITY

The sensitivity study finds CCSGW robust across tested alignment weights, direction counts, and sample sizes. Redrawing directions each epoch supports insensitivity to the number of directions, while node subsampling reduces cost without sacrificing accuracy.

D.1 HYPERPARAMETER SENSITIVITY

  • Every tested configuration except one stays within the reference confidence interval, and gains remain positive across two orders of magnitude of λ and a sixteenfold range of L.The exception is the all-node configuration, which yields the only negative gain, on Cora.Source: §D.1 · p. 23
  • Redrawing directions each epoch averages the approximation error of a small L over training, explaining the observed insensitivity to L.The study varies λ, L, and node sample size n around dataset-specific reference configurations.Source: §D.1 · p. 23
  • Subsamples of 128 to 1024 nodes perform comparably, whereas using all nodes gives the smallest gain on both datasets and the only negative gain, on Cora.The authors conclude that subsampling reduces CCSGW cost without sacrificing accuracy.Source: §D.1 · p. 23

D.2 COMPARISON WITH SGW AND GW

CCSGW outperforms SGW across all five datasets and is on par with GW, while the sliced losses are substantially cheaper at larger sample sizes. The comparison isolates geodesic slicing from linear slicing and contrasts both with full GW.

D.2 COMPARISON WITH SGW AND GW

  • CCSGW replaces SGW’s ambient linear projections with geodesic slicing, while GW compares geodesic distance matrices without slicing.The comparison replaces CCSGW in the CUSP objective with SGW or GW while keeping other settings fixed.Source: §D.2 · pp. 23–24
  • CCSGW gains +0.70 mean weighted F1 over CUSP versus +0.28 for SGW across five datasets, and improves over SGW on all five.Table 4 compares alignment losses on top of CUSP and reports the mean gain over CUSP in its last column.Source: §D.2 · p. 24, Table 4 · p. 24
  • CCSGW is better than GW on the three homophilic datasets and slightly worse on Chameleon and Actor.The comparison uses test weighted F1 across five datasets.Source: §D.2 · p. 24, Table 4 · p. 24

D.3 COMPUTATIONAL COST

  • At larger sample sizes, CCSGW and SGW remain in the millisecond range, while GW is one to two orders of magnitude slower and runs out of memory at n = 8192.Figure 4 measures one forward-and-backward alignment-loss evaluation on a single A100 GPU; its caption notes GW runs out of memory beyond n = 4096.Source: §D.3 · p. 25, Figure 4 · p. 25
  • For experimental subsamples n ≤512, CCSGW adds only a small fraction of a training epoch’s cost, and geodesic projections cost essentially the same as SGW’s linear projections.CCSGW and SGW have per-component-pair complexity O(L n log n), compared with O(n^2) memory and time per Sinkhorn iteration for entropic GW.Source: §D.3 · p. 25

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