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arxiv: 2605.12161 · v1 · submitted 2026-05-12 · 💻 cs.LG · cs.CY· math.MG

Recognition: no theorem link

Fused Gromov-Wasserstein Distance with Feature Selection

Harlin Lee , Ying Yu , Mingxin Li , Ranthony Clark
Authors on Pith no claims yet

Pith reviewed 2026-05-13 05:30 UTC · model grok-4.3

classification 💻 cs.LG cs.CYmath.MG
keywords fused gromov-wassersteinfeature selectionoptimal transportgraph alignmentregularizationalternating minimizationinterpretability
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The pith

Fused Gromov-Wasserstein distances with feature selection incorporate adaptive suppression weights to downweight irrelevant features during alignment.

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

Existing fused Gromov-Wasserstein distances align objects by their structure and features but treat all features the same way. This paper modifies the objective to include adaptive weights that suppress the contribution of noisy or irrelevant features while solving for the alignment. It offers two concrete ways to do this: one adds Lasso or Ridge penalties on the weights, the other constrains the weights to a simplex and allows groupwise selection. The authors prove that these new distances remain bounded by the classical versions and continue to behave like metrics. They also give an alternating minimization procedure to compute them and demonstrate the approach on a redistricting task where it highlights meaningful features.

Core claim

The paper establishes that adaptive feature suppression weights can be integrated into the FGW objective, either via regularization penalties or simplex constraints, to enable selective feature downweighting during alignment. This yields distances that satisfy theoretical bounds with respect to standard FGW and GW distances and maintain metric properties. Computation proceeds via an efficient alternating minimization scheme that alternates between the coupling and the weights.

What carries the argument

Adaptive feature suppression weights optimized jointly with the transport coupling in the FGW objective.

If this is right

  • The modified distances satisfy explicit bounds relative to classical FGW and Gromov-Wasserstein distances.
  • Metric properties are preserved under the regularized and constrained formulations.
  • Feature suppression reveals task-relevant structure in applications such as computational redistricting.
  • An alternating minimization algorithm computes the distances efficiently.

Where Pith is reading between the lines

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

  • The groupwise extensions suggest the method can handle predefined feature groups without separate preprocessing.
  • Interpretability improvements may apply to other structured data comparison tasks that rely on optimal transport.
  • Joint optimization of weights and alignment could reduce sensitivity to feature scaling choices in practice.

Load-bearing premise

That jointly optimizing the alignment plan and the feature suppression weights will not introduce instability or break the metric properties in high-dimensional or correlated feature settings.

What would settle it

A numerical example with high-dimensional correlated features where the computed distance violates the triangle inequality or where the optimization yields degenerate zero weights for all features.

Figures

Figures reproduced from arXiv: 2605.12161 by Harlin Lee, Mingxin Li, Ranthony Clark, Ying Yu.

Figure 1
Figure 1. Figure 1: (a) Graphs matched by fsFGW - Ridge. (b) Feature scores [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: fsFGW performance on toy graphs (d = 100, k = 10) are robust to δ and f choice. confirming that the signal is visible to the transport plan before any suppression is applied. Figure 1c and d shows the recovered suppression weights for Ridge (f = 0.2) and Lasso (f = 0.3). Both modes correctly assign high suppression to differentiating features and low suppression to shared ones. Appendix C shows graph match… view at source ↗
Figure 3
Figure 3. Figure 3: Mean suppression weights w¯r per feature on OGBG-MOLBACE, averaged over all graph pairs. {hybridization, is_aromatic, is_in_ring} are most strongly suppressed across both Lasso and Ridge. Next are {atomic_num, degree, num_Hs}, then {chirality, formal_charge}. num_radical_e is consistently retained. 7 Application to Computational Redistricting In the United States, a redistricting plan partitions a state in… view at source ↗
Figure 4
Figure 4. Figure 4: Hierarchical clustering of plans with GW, FGW, and fsFGW (Lasso). [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Mean suppression weights for fsFGW (Simplex) when comparing Plan 22 and Plan 22ct. [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Suppression weights for fsFGW (Lasso) when comparing Plan 23 and Plan 25. Each row is [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Convergence of fsFGW with f = 0.3 on two graphs (Graphs 20 and 26) from OGBG￾MOLBACE. Top: FGW objective per iteration. Bottom: weight change ∥wk+1 − wk∥. All modes converged in 2-8 iterations in our observations. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: (a) The two independent random geometric graphs [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Suppression weights wr recovered by each mode (n = 40, d = 10, k = 3, δ = 2.0). Differentiating features (red) should have wr = 1; shared (blue) should have wr = 0. Ridge (f = 0.2) gives continuous weights; Lasso (f = 0.3) gives binary weights recovering exactly the k = 3 differentiating features. Simplex concentrates all suppression on the single highest-scoring feature. Groupwise simplex suppresses the e… view at source ↗
Figure 10
Figure 10. Figure 10: Visualization of matchings according to different transport plans [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Agreement between different Ts, calculated as proportion of node mappings that agree between two transport plans. normalized globally to [0, 1] per dimension across the subsample. Per-feature cost matrices Mr are further normalized per pair to [0, 1]. Classification uses an SVM with RBF kernel Kij = exp(−Dij/σ) where σ is the median of nonzero distances, evaluated via 5-fold stratified CV. Clustering uses… view at source ↗
Figure 12
Figure 12. Figure 12: Classification results in Table 1 [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Clustering results in Table 1 [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Mean suppression weights per feature on PROTEINS-FULL (d = 32) for Lasso (left) and Ridge (right) across suppression fractions f ∈ {0.3, 0.5, 0.7}. Errorbars are omitted for readability. subgraph of the precinct adjacency graph induced by its constituent precincts. 29 node features are the precinct-level sociodemographic and environmental indicators described in the next subsection. The structure matrix C… view at source ↗
Figure 15
Figure 15. Figure 15: North Carolina Congressional Redistricting Plans (2020-2025) [PITH_FULL_IMAGE:figures/full_fig_p024_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Hierarchical Clustering of NC Congressional Maps. [PITH_FULL_IMAGE:figures/full_fig_p025_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Top 10 mean suppression weights (± 1 standard deviation) for Plan 22 vs. Plan22ct. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Weights for each district pair compared in Plan 23 vs. Plan25. [PITH_FULL_IMAGE:figures/full_fig_p027_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: fsFGW (Lasso) suppression weight heatmap for Plan 22 vs. Plan 22ct, with features [PITH_FULL_IMAGE:figures/full_fig_p027_19.png] view at source ↗
read the original abstract

Fused Gromov-Wasserstein (FGW) distances provide a principled framework for comparing objects by jointly aligning structure and node features. However, existing FGW formulations treat all features uniformly, which limits interpretability and robustness in high-dimensional settings where many features may be irrelevant or noisy. We introduce FGW distances with feature selection, which incorporate adaptive feature suppression weights into the FGW objective to selectively downweight or suppress differentiating features during alignment. We propose two approaches: (1) regularized FGW with Lasso and Ridge penalties, and (2) FGW with simplex-constrained weights, including groupwise extensions. We analyze the resulting models and establish their key theoretical properties, including bounds relative to classical FGW and Gromov-Wasserstein distances, and metric behavior. An efficient alternating minimization algorithm is developed. Experiments illustrate how feature suppression enhances interpretability and reveals task-relevant structure, with a special application to computational redistricting.

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

2 major / 2 minor

Summary. The manuscript introduces Fused Gromov-Wasserstein distances with feature selection (FGW-FS) by incorporating adaptive feature suppression weights into the FGW objective to downweight or suppress differentiating features during alignment. Two approaches are proposed: regularized FGW using Lasso and Ridge penalties, and FGW with simplex-constrained weights (including groupwise extensions). The authors analyze the models to establish theoretical properties including bounds to classical FGW and Gromov-Wasserstein distances plus metric behavior, develop an efficient alternating minimization algorithm, and present experiments showing enhanced interpretability with an application to computational redistricting.

Significance. If the central claims on preserved metric properties and bounds hold under joint optimization of couplings and data-dependent feature weights, the work would meaningfully extend optimal transport tools for high-dimensional structured data by improving robustness to noisy or irrelevant features while adding interpretability, with potential impact on graph matching and alignment tasks.

major comments (2)
  1. [theoretical analysis section] The section establishing key theoretical properties asserts that the FGW-FS distance satisfies metric axioms and provides bounds to classical FGW/GW, but the joint alternating minimization of the coupling and the feature suppression weights (Lasso/Ridge or simplex) means the effective weights can differ across pairs in a triple. This risks violating triangle inequality, and the manuscript does not provide an explicit proof or counterexample analysis showing when the property is inherited from fixed-weight FGW.
  2. [model formulation and analysis] In the formulation of the regularized and simplex-constrained objectives, the claim that the optimized distance remains a metric with the stated bounds assumes the suppression vector is stable, yet no analysis addresses degeneracy when features are linearly dependent or high-dimensional, where the simplex or Lasso solutions may collapse to trivial suppressions (e.g., all weight on one feature).
minor comments (2)
  1. [experiments] The experimental section would benefit from explicit baseline comparisons (standard FGW without feature selection) and details on hyperparameter selection for lambda and the regularization terms to allow reproduction of the interpretability gains.
  2. [notation and preliminaries] Notation for the feature weights could be unified or clearly distinguished between the Lasso/Ridge variant and the simplex variant to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments, which have identified important points for strengthening the theoretical analysis. We address each major comment below and indicate the corresponding revisions.

read point-by-point responses

  1. Referee: [theoretical analysis section] The section establishing key theoretical properties asserts that the FGW-FS distance satisfies metric axioms and provides bounds to classical FGW/GW, but the joint alternating minimization of the coupling and the feature suppression weights (Lasso/Ridge or simplex) means the effective weights can differ across pairs in a triple. This risks violating triangle inequality, and the manuscript does not provide an explicit proof or counterexample analysis showing when the property is inherited from fixed-weight FGW.

    Authors: We appreciate the referee highlighting this subtlety regarding the triangle inequality. The current analysis provides bounds relative to classical FGW and GW distances and states metric properties, but the joint optimization indeed makes the effective feature weights pair-dependent, and an explicit proof accounting for this was not included. We acknowledge that this requires clarification, as the inheritance from fixed-weight FGW is not automatic. In the revised manuscript, we will expand the theoretical analysis section with either a rigorous proof demonstrating that the metric axioms (including triangle inequality) continue to hold under the joint optimization, or a counterexample showing when the property fails, together with sufficient conditions for the bounds to be preserved. revision: yes

  2. Referee: [model formulation and analysis] In the formulation of the regularized and simplex-constrained objectives, the claim that the optimized distance remains a metric with the stated bounds assumes the suppression vector is stable, yet no analysis addresses degeneracy when features are linearly dependent or high-dimensional, where the simplex or Lasso solutions may collapse to trivial suppressions (e.g., all weight on one feature).

    Authors: The referee correctly notes that the manuscript does not analyze potential degeneracy of the suppression weights. In high-dimensional or linearly dependent feature regimes, the Lasso, Ridge, or simplex solutions can indeed collapse to trivial or unstable suppressions, which could affect the claimed metric properties and bounds. We will revise the model formulation and analysis section to include a dedicated discussion of these degeneracy cases. This will cover conditions under which collapse occurs, strategies for choosing regularization parameters to promote stability, and any adjustments to the theoretical bounds under mild feature independence assumptions. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces FGW with feature selection by adding adaptive suppression weights to the existing FGW objective via two concrete formulations (Lasso/Ridge regularized and simplex-constrained, plus groupwise extensions). It then claims to derive bounds to classical FGW/GW and metric behavior through analysis of the new objectives. No step reduces a claimed prediction, bound, or property to a fitted input or self-definition by construction; the theoretical claims are presented as consequences of the modified objective rather than tautological renamings or self-citations. External references to prior FGW work are standard background and not load-bearing for the new results.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach rests on the standard properties of FGW and optimal transport distances plus the assumption that feature weights can be optimized without breaking those properties.

free parameters (1)
  • regularization strength lambda
    Controls the Lasso or Ridge penalty on feature weights; must be chosen by the user or cross-validation.
axioms (1)
  • standard math Fused Gromov-Wasserstein distance is a valid metric on the space of attributed graphs
    Invoked when claiming the new weighted version remains a metric.

pith-pipeline@v0.9.0 · 5465 in / 1176 out tokens · 68895 ms · 2026-05-13T05:30:28.777502+00:00 · methodology

discussion (0)

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