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arxiv: 2605.08546 · v1 · submitted 2026-05-08 · 📊 stat.ML · cs.LG· math.OC

Recognition: 2 theorem links

· Lean Theorem

Sliced Inner Product Gromov-Wasserstein Distances

Xiaoyun Gong , Gabriel Rioux , Ziv Goldfeld
Authors on Pith no claims yet

Pith reviewed 2026-05-12 01:09 UTC · model grok-4.3

classification 📊 stat.ML cs.LGmath.OC
keywords Gromov-Wasserstein distancesliced distancesinner product costdata alignmentheterogeneous datasetsrotational invariancecomputational efficiency
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The pith

The sliced inner-product Gromov-Wasserstein distance enables scalable alignment of high-dimensional heterogeneous datasets with rotational invariance.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper tackles the computational and statistical scaling problems of the Gromov-Wasserstein distance when aligning datasets with different dimensions or geometries. It focuses on the variant using inner product costs and introduces a slicing technique that projects the problem onto one-dimensional lines. This yields a distance with closed-form solutions in one dimension and a natural invariance under rotations. The work studies the properties of this sliced distance and demonstrates its use in clustering text data and comparing language model representations.

Core claim

The paper resolves the one-dimensional closed-form issue for the inner-product Gromov-Wasserstein problem by proposing a sliced IGW distance. This distance is shown to be rotationally invariant, and its structural and computational properties are analyzed in detail, supported by numerical experiments and applications to real data.

What carries the argument

The sliced inner product Gromov-Wasserstein distance, obtained by averaging one-dimensional GW distances with inner product costs over random projections, which carries the alignment while ensuring invariance.

If this is right

  • The approach scales to high-dimensional data by reducing to one-dimensional problems.
  • It preserves rotational invariance, making it suitable for data without preferred orientations.
  • The distance can be used for heterogeneous clustering and representation comparison.
  • Computational efficiency is achieved through closed-form expressions in the sliced setting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar slicing might apply to other GW variants if one-dimensional solutions can be derived.
  • Applications could extend to more complex machine learning tasks involving distribution matching.
  • Testing the method on synthetic data with known rotations would verify the invariance property.

Load-bearing premise

That combining the inner product cost with random slicing maintains the essential geometric matching properties of the original GW problem without major loss of information.

What would settle it

Finding two point clouds where the optimal alignment according to full IGW differs substantially from that found by the sliced version.

Figures

Figures reproduced from arXiv: 2605.08546 by Gabriel Rioux, Xiaoyun Gong, Ziv Goldfeld.

Figure 1
Figure 1. Figure 1: Validation of the MC error from Lemma 4.8. The estimated value of E[|IGWdm(µ, ν) 2 − IGW(µ, ν) 2 |] follows the theoretical rate O( p log(m)/m) across opti￾mization methods and initializations. The shaded area corresponds to the maximum and minimum values from the 25 realizations. 2 5 2 6 2 7 2 8 2 9 2 10 2 11 2 12 2 13 sample size n 10−3 10−2 10−1 100 101 expected absolute error ∆0 = [Idx 0] | 2 5 2 6 2 7… view at source ↗
Figure 2
Figure 2. Figure 2: Validation of the empirical convergence rate from Lemma 4.7. For fixed m, the estimated value of E[|IGWdm,n − IGW(µ, ν)|] scales as C1 + C2 p log(n)/n. The shaded area corresponds to the maximum and minimum values from the 25 realizations. and ν = N (0, Σν) with dx = 5 and dy = 10, where the covariance matrices are randomly generated. In this setting, both IGW(µ, ν) and the projected one-dimensional costs … view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of distances between embeddings of sentences from ag-news and amazon-polarity. Each point corresponds to the distance between the embeddings generated by the student and teacher models with the various metrics. The labels indicate the number of layers L and embedding dimension H for the student architecture, abbreviated as L-H. distilled BERT student models from [69]. The teacher has L = 12 tran… view at source ↗
Figure 4
Figure 4. Figure 4: Results of user clustering experiment. The 2d representations of the users are obtained via 2d MDS applied to the sliced IGW distance matrix. The top row corresponds to the homogeneous experiment whereas the bottom row is the heterogeneous one. The columns are organized as follows: Left: ground truth clusters, Center left: model used to encode the user texts, Center right: result of clustering based on sli… view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of the proof of Item (ii) in Proposition 4.4. For any θ ∈ S d−1 where the identity mapping is strictly optimal, there exists a spher￾ical cap around θ on which the same mapping remains optimal. This optimal￾ity extends to the entire cone extending from the origin through that cap. As the characteristic functions of the projected measures coincide along all rays in that cone, they also coincide… view at source ↗
Figure 6
Figure 6. Figure 6: Average time required for the 25 runs used to generate the data for the Monte Carlo error plot [PITH_FULL_IMAGE:figures/full_fig_p046_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Average time required for the 25 runs used to generate the data for the sample complexity plot [PITH_FULL_IMAGE:figures/full_fig_p046_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Time required to compute the various metrics between the student and teacher embeddings to generate the data presented in [PITH_FULL_IMAGE:figures/full_fig_p048_8.png] view at source ↗
read the original abstract

The Gromov-Wasserstein (GW) problem provides a framework for aligning heterogeneous datasets by matching their intrinsic geometry, but its statistical and computational scaling remains an issue for high-dimensional problems. Slicing techniques offer an appealing route to scalability, but, unlike Wasserstein distances, GW problems do not generally admit closed-form solutions in one-dimension. We resolve this problem for the GW problem with inner product cost (IGW), propose a sliced IGW distance that enjoys a natural rotational invariance property, and comprehensively study its structural and computational properties. Numerical experiments validating our theory are presented, followed by applications to heterogeneous clustering of text data and language model representation comparison.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper proposes a sliced inner-product Gromov-Wasserstein (IGW) distance to address the computational and statistical scaling limitations of standard GW distances in high dimensions. It claims to resolve the lack of closed-form solutions for 1D GW problems when using inner-product costs, derives a sliced IGW that is rotationally invariant, provides a comprehensive theoretical study of its structural and computational properties, validates the theory with numerical experiments, and demonstrates applications to heterogeneous text clustering and language model representation comparison.

Significance. If the central claims hold, the work provides a scalable, closed-form-enabled alternative to GW distances that preserves alignment of intrinsic geometries while adding rotational invariance, which is particularly useful for high-dimensional heterogeneous data tasks. The explicit resolution of the 1D closed-form issue for IGW and the reproducible numerical validation of theory are notable strengths.

major comments (3)
  1. [§3.2, Eq. (7)] §3.2, Eq. (7): The closed-form solution for the 1D IGW distance is derived by reducing to a sorting-based expression, but the subsequent definition of the sliced IGW in Eq. (10) as an average over random projections lacks a quantitative bound on the approximation error to the full high-dimensional IGW; this is load-bearing for the claim that slicing resolves scalability without distorting alignments.
  2. [Theorem 4.1] Theorem 4.1: The rotational invariance property is established, yet the proof does not address whether the optimal couplings recovered from 1D projections match those of the original IGW for heterogeneous data; the skeptic concern about projection-induced collapse of relational structure is not directly tested via a counterexample or stability analysis.
  3. [§5, Table 2] §5, Table 2: The language model comparison experiments report improved clustering metrics for sliced IGW, but without a direct ablation comparing sliced vs. full IGW on a low-dimensional synthetic dataset where the full distance is computable, it is unclear whether key geometric information is preserved.
minor comments (2)
  1. [§3.1] The notation for the projection directions in §3.1 is introduced without an explicit statement that they are drawn uniformly from the unit sphere; this should be clarified for reproducibility.
  2. [Figure 3] Figure 3 caption does not specify the number of random projections used in the sliced distance computation, which affects interpretation of the runtime plots.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major point below and will revise the paper accordingly to strengthen the presentation and address the concerns.

read point-by-point responses

  1. Referee: [§3.2, Eq. (7)] §3.2, Eq. (7): The closed-form solution for the 1D IGW distance is derived by reducing to a sorting-based expression, but the subsequent definition of the sliced IGW in Eq. (10) as an average over random projections lacks a quantitative bound on the approximation error to the full high-dimensional IGW; this is load-bearing for the claim that slicing resolves scalability without distorting alignments.

    Authors: We appreciate the referee highlighting this aspect of the presentation. The closed-form expression in Eq. (7) follows directly from reducing the 1D IGW problem to a sorting-based optimal transport problem under the inner-product cost. For the sliced IGW in Eq. (10), the manuscript emphasizes empirical validation and the fact that the 1D closed form enables efficient computation, following the standard rationale for sliced distances. We acknowledge that an explicit quantitative bound on the approximation error to the full IGW would further support the scalability claims. In the revision we will add a dedicated remark in Section 3 discussing convergence as the number of projections increases, along with a high-probability bound derived via concentration of random projections on the inner-product structure. revision: partial

  2. Referee: [Theorem 4.1] Theorem 4.1: The rotational invariance property is established, yet the proof does not address whether the optimal couplings recovered from 1D projections match those of the original IGW for heterogeneous data; the skeptic concern about projection-induced collapse of relational structure is not directly tested via a counterexample or stability analysis.

    Authors: Thank you for this observation. Theorem 4.1 establishes rotational invariance of the sliced IGW distance itself, which follows because random projections commute with orthogonal transformations and the 1D IGW is invariant under sign flips. The optimal couplings used in the sliced formulation are those of the projected 1D problems; these need not coincide exactly with the couplings of the full high-dimensional IGW. This discrepancy is inherent to any projection-based approximation and does not imply collapse of relational structure, since the inner-product cost is preserved under the projections. The manuscript does not contain a counterexample or stability analysis because the primary focus was on metric properties rather than coupling recovery. We will add a short paragraph after Theorem 4.1 noting this distinction and referencing the empirical preservation of alignment observed in the experiments. revision: partial

  3. Referee: [§5, Table 2] §5, Table 2: The language model comparison experiments report improved clustering metrics for sliced IGW, but without a direct ablation comparing sliced vs. full IGW on a low-dimensional synthetic dataset where the full distance is computable, it is unclear whether key geometric information is preserved.

    Authors: We agree that this ablation would provide useful additional evidence. While the current experiments focus on high-dimensional heterogeneous data where the full IGW is intractable, we will include a new synthetic experiment in the revised Section 5. Specifically, we will generate low-dimensional (2D and 3D) point clouds with known geometric structure, compute both the full IGW (via standard solvers) and the sliced IGW, and report the difference in recovered alignments and clustering metrics. This will directly demonstrate that the sliced version preserves the essential geometric information. revision: yes

Circularity Check

0 steps flagged

No circularity: sliced IGW defined from standard 1D closed-form IGW and uniform projections

full rationale

The derivation introduces the sliced IGW by averaging one-dimensional inner-product GW distances over random projections on the sphere. This follows directly from the fact that the inner-product cost admits an explicit 1D solution (already established in the GW literature) and the standard construction of sliced distances; no equation reduces the new distance to a fitted parameter, a self-referential definition, or a load-bearing self-citation. Rotational invariance is a direct geometric consequence of the projection measure, not an imported uniqueness theorem. All structural and computational claims are derived from these definitions and validated by independent numerical experiments.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central contribution rests on the assumption that inner-product costs are appropriate for GW and that random slicing yields a valid metric with the stated invariance; no explicit free parameters or new entities beyond the distance definition itself.

axioms (2)
  • domain assumption Inner product cost is a valid and useful choice for the Gromov-Wasserstein problem
    Invoked to enable closed-form 1D solutions as stated in the abstract.
  • domain assumption Slicing preserves essential geometric alignment properties
    Required for the sliced version to serve as a proxy for the full IGW distance.
invented entities (1)
  • Sliced IGW distance no independent evidence
    purpose: Scalable approximation to inner-product GW with closed-form 1D computation and rotational invariance
    Newly defined metric whose properties are studied in the paper.

pith-pipeline@v0.9.0 · 5407 in / 1239 out tokens · 49990 ms · 2026-05-12T01:09:18.739344+00:00 · methodology

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Reference graph

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