Gromov-Wasserstein Distillation for Inductive Multi-View Embedding

Rafael Pereira Eufrazio; Eduardo Fernandes Montesuma; Charles Casimiro Cavalcante

cs.LGSep 30, 2026

Gromov-Wasserstein Distillation for Inductive Multi-View Embedding

Authors: Rafael Pereira Eufrazio, Eduardo Fernandes Montesuma, Charles Casimiro Cavalcante

Organizations: Instituto Federal do Ceará (IFCE) Canindé, Ceará, Brazil · Federal University of Ceará (UFC) Fortaleza, Ceará, Brazil · Sigma Nova Paris, France

Abstract

Gromov-Wasserstein multidimensional scaling (GW-MDS) learns low-dimensional representations from relational data but remains transductive, providing no explicit mapping for unseen samples. We introduce an inductive framework based on barycentric distillation. A GW-MDS teacher learns a latent support and an optimal transport plan from the training data, and barycentric projection converts the resulting coupling into sample-aligned targets. A neural student then learns an explicit out-of-sample mapping, avoiding additional relational-matrix construction and GW optimization at inference. We formulate the approach for single-view data and extend it to Mean-GWMDS and Multi-GWMDS teachers through consensus and selected-projection targets learned by a multi-view student with view-specific encoders. We also investigate a direct neural baseline trained solely with a GW objective. Experiments on synthetic and real-world data using Euclidean, geodesic, and cosine relations show that the distilled models preserve the teacher geometry on unseen samples and consistently outperform direct neural GW training in sample-indexed relational preservation. These results establish barycentric projection as an effective bridge between transductive GW embeddings and inductive neural mappings.

Figures & tables

Geometry	Method	Pearson $r$ $\uparrow$	Spearman $\rho$ $\uparrow$	Trust. $\uparrow$	Stress $\downarrow$
Euclidean	Ind. GW-MDS	0.8869	0.8911	0.9300	0.2003
	Direct neural GW	0.7533	0.7462	0.8387	0.2791
	PCA	0.8959	0.9075	0.9149	0.1971
Geodesic	Ind. GW-MDS	0.9980	0.9974	0.9981	0.0337
	Direct neural GW	0.7142	0.7248	0.9233	0.3905
	PCA	0.7314	0.7601	0.9369	0.3575

Table 1 : Out-of-sample relational preservation on the S-curve. Bold indicates the best result for each geometry.

Panel of Figure 1 — Figure 1 : Single-view results on the S-curve using geodesic dissimilarities.

		ERA5				rMD17–Aspirin
Geometry	Method	$r\uparrow$	$\rho\uparrow$	Trust. $\uparrow$	Stress $\downarrow$	$r\uparrow$	$\rho\uparrow$	Trust. $\uparrow$	Stress $\downarrow$
Euclidean	Ind. Mean-GWMDS	$\mathbf{0.7853}$	$\mathbf{0.7958}$	$\mathbf{0.8975}$	$\mathbf{0.2798}$	$\mathbf{0.4736}$	$\mathbf{0.4635}$	$\mathbf{0.6852}$	$\mathbf{0.3899}$
	Ind. Multi-GWMDS	$0.7470$	$0.7572$	$0.8643$	$0.3137$	$0.4065$	$0.4018$	$0.6783$	$0.4014$
	Concatenated PCA	$0.7790$	$0.7892$	$0.8924$	$0.2916$	$0.4256$	$0.4266$	$0.6761$	$0.4364$
	Direct multi-view GW	$0.4576$	$0.5372$	$0.8590$	$0.4670$	$0.3540$	$0.3552$	$0.6531$	$0.4142$
Geodesic	Ind. Mean-GWMDS	$\mathbf{0.8507}$	$\mathbf{0.8439}$	$\mathbf{0.9373}$	$\mathbf{0.2603}$	$\mathbf{0.4787}$	$\mathbf{0.4701}$	$\mathbf{0.6877}$	$\mathbf{0.3897}$

Table 2 : Out-of-sample relational preservation on ERA5 and rMD17–Aspirin. ERA5 results are uniform averages over four views on

97

held-out locations, whereas rMD17 results are averaged over two views. Bold indicates the best result for each dataset and relational geometry.

Panel of Figure 2 — Figure 2 : Joint training and out-of-sample embeddings for ERA5 under geodesic relations. Training locations are shown in gray, whereas the 97 held-out locations are colored by latitude using a common color scale.

Panel of Figure 3 — Figure 3 : Joint training and out-of-sample embeddings for rMD17–Aspirin under cosine relations. Training conformations are shown in gray, whereas the 200 held-out conformations are colored by normalized potential energy using a common color scale.

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

\mathcal{L}_{\mathrm{distill}}(\theta)=\sum_{i\in\mathcal{I}_{\mathrm{fit}}}\left\|f_{\theta}\!\left(x_{i}^{(1)},\ldots,x_{i}^{(V)}\right)-\widetilde{y}_{i}\right\|_{2}^{2}

Appendix

Algorithm 1 Multi-view barycentric distillation

Geometry	Representation	Split	$r\uparrow$	$\rho\uparrow$	Trust. $\uparrow$	Stress $\downarrow$
Euclidean	Mean-GWMDS teacher	Train	0.7661	0.7640	0.9065	0.2914
	Ind. Mean-GWMDS	Train	0.7661	0.7638	0.9050	0.2915
	Ind. Mean-GWMDS	Test	0.7853	0.7958	0.8975	0.2798
	Multi-GWMDS teacher, selected	Train	0.7009	0.6977	0.8617	0.3388
	Ind. Multi-GWMDS	Train	0.7070	0.7025	0.8653	0.3357
	Ind. Multi-GWMDS	Test	0.7470	0.7572	0.8643	0.3137

Appendix

Table 3 : Teacher-to-student transfer on ERA5. All metrics are uniformly averaged over the four views. Teacher representations are available only for the training observations.

Geometry	Method	Temperature	Dewpoint	Pressure	Precipitation
Euclidean	Ind. Mean-GWMDS	0.8995/0.9057	0.9232/0.9186	0.8981/0.8811	0.4203/0.4777
	Ind. Multi-GWMDS	0.9210/0.9291	0.8239/0.8187	0.8444/0.8400	0.3988/0.4411
Geodesic	Ind. Mean-GWMDS	0.9194/0.9290	0.9045/0.9033	0.8767/0.8565	0.7021/0.6867
	Ind. Multi-GWMDS	0.9393/0.9313	0.7821/0.7950	0.7772/0.7626	0.6436/0.6416
Cosine	Ind. Mean-GWMDS	0.7575/0.7770	0.5337/0.5589	0.2667/0.6706	0.5995/0.6263
	Ind. Multi-GWMDS	0.8967/0.8935	0.2920/0.3448	0.2612/0.6197	0.4821/0.4919

Appendix

Table 4 : View-wise out-of-sample correlations on ERA5. Each entry reports Pearson/Spearman correlation,

r/\rho

Projection	Euclidean	Geodesic	Cosine
View 1: temperature	0.7009	0.7357	0.4550
View 2: dewpoint	0.6948	0.6995	0.3762
View 3: pressure	0.6570	0.6780	0.4010
View 4: precipitation	0.5626	0.6453	0.2993

Appendix

Table 5 : Training-only selection scores for the four Multi-GWMDS barycentric projections. Each score is the mean Pearson correlation across the four ERA5 relational matrices.

	GW objective		Test $r$
Geometry	Teacher	Direct	Ind. Multi	Direct
Euclidean	0.01552	0.01565	0.7470	0.4576
Geodesic	0.01176	0.01169	0.7855	0.3409
Cosine	0.10290	0.10328	0.4830	0.4416

Appendix

Table 6 : Last recorded training objective after

100

outer iterations and out-of-sample Pearson correlation. The objective values are uniformly averaged over the four views.

Stage	Euclidean	Geodesic	Cosine	Mean
Mean-GWMDS teacher	76.45	80.58	82.00	79.68
Multi-GWMDS teacher	238.00	229.60	244.23	237.28
Mean student	1.97	2.18	1.34	1.83
Selected-projection student	1.39	1.23	1.41	1.34
Direct multi-view GW	193.72	192.84	240.32	208.96

Appendix

Table 7 : ERA5 execution times in seconds for a single run with seed

0

. Timings are implementation- and hardware-dependent.

Panel of Figure 4 — Figure 4 : Geodesic teacher representations for ERA5.

Panel of Figure 5 — Figure 5 : Out-of-sample embeddings of the held-out ERA5 locations under geodesic relations.

Panel of Figure 6 — Figure 6 : Optimization trajectories for the geodesic ERA5 experiment.

Geometry	Method	Interatomic-distance view	Force-Gram view
Euclidean	Inductive Mean-GWMDS	0.7294	0.2178
	Inductive Multi-GWMDS	0.7506	0.0624
Geodesic	Inductive Mean-GWMDS	0.7860	0.1715
	Inductive Multi-GWMDS	0.7928	0.0524
Cosine	Inductive Mean-GWMDS	0.7740	0.2743
	Inductive Multi-GWMDS	0.8204	0.0669

Appendix

Table 8 : View-wise test Pearson correlations on rMD17-Aspirin. The reported embeddings are produced by the two inductive teacher–student formulations.

Barycentric projection	$\rho_{1}$	$\rho_{2}$	Mean	$\rho_{1}$	$\rho_{2}$	Mean	$\rho_{1}$	$\rho_{2}$	Mean
	Euclidean			Geodesic			Cosine
Interatomic-distance view	0.7966	0.1318	0.4642	0.8468	0.1057	0.4762	0.8610	0.1695	0.5153
Force-Gram view	0.0926	0.5902	0.3414	0.0682	0.5918	0.3300	0.1195	0.7203	0.4199
Selected-minus-other difference	0.7040	$-0.4584$	0.1228	0.7786	$-0.4861$	0.1462	0.7415	$-0.5508$	0.0954

Appendix

Table 9 : Training-only evaluation of the Multi-GWMDS barycentric projections on rMD17-Aspirin. For each candidate projection,

\rho_{1}

and

\rho_{2}

denote the Pearson correlations with the interatomic-distance and Force-Gram views, respectively. The mean is computed uniformly across the two views and used as the selection score. The selected projection is shown in bold. The last row reports the selected-minus-other difference.

	Euclidean	Geodesic	Cosine
Multi-GWMDS teacher objective	0.035079	0.025891	0.009187
Direct GW objective	0.036273	0.025977	0.009195
Inductive Multi-GWMDS test Pearson	0.4065	0.4226	0.4437
Direct GW test Pearson	0.3540	0.3471	0.2942

Appendix

Table 10 : Final GW objectives and view-averaged test Pearson correlations on rMD17-Aspirin.

Formulation	Optimization stage	Time (s)
Inductive Mean-GWMDS	Mean-GWMDS teacher	323.59
Inductive Mean-GWMDS	Consensus-target student	1.63
Inductive Multi-GWMDS	Multi-GWMDS teacher	607.56
Inductive Multi-GWMDS	Selected-projection student	1.36
Direct multi-view GW	Direct neural optimization	546.63

Appendix

Table 11 : Average execution times on rMD17-Aspirin. Teacher and student times are reported separately to distinguish the initial relational optimization from neural distillation.

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Gromov-Wasserstein Methods for Multi-View Relational Embedding and Clustering

Rafael Pereira Eufrazio, Eduardo Fernandes Montesuma, Charles Casimiro Cavalcante

Instituto Federal de Educação, Ciência e Tecnologia do Ceará, Canindé, Ceará, Brazil · Federal University of Ceara, Fortaleza, Ceará, Brazil · Sigma Nova, Paris, France

Gromov--Wasserstein Multi-View Spherical Latent Space Clustering Distance

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Learning low-dimensional representations from multi-view relational data is challenging when underlying geometries differ across views. We propose Bary-GWMDS, a Gromov-Wasserstein-based method that operates directly on distance matrices to learn a consensus embedding preserving shared relational structure. By leveraging intrinsic distances, the approach naturally handles nonlinear distortions across views. We also introduce Mean-GWMDS-C, a clustering-oriented formulation that averages distance matrices and learns reduced-support representations via a consensus Gromov-Wasserstein transport. Experiments on synthetic and real-world datasets show that the proposed framework yields stable and geometrically meaningful embeddings.

Abstract

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