Semi-Relaxed Gromov-Wasserstein (srGW)

• Semi-relaxed Gromov-Wasserstein (srGW) is an asymmetric optimal transport framework that relaxes only the target marginal to learn latent structures in graphs and metric spaces.
• It reformulates matching problems into continuous optimization over one-sided constrained transport, enabling sparse, near-deterministic correspondences and efficient computation.
• The framework underpins applications in graph partitioning, dictionary learning, dimensionality reduction, and stochastic block model inference with solid theoretical guarantees.

Semi-relaxed Gromov-Wasserstein (srGW) is an asymmetric relaxation of the Gromov-Wasserstein (GW) comparison problem in which the source marginal is fixed while the target marginal is left free and is therefore learned from the optimization itself. In the discrete setting, this replaces the balanced transport polytope used by classical GW with a one-sided constraint set, and turns graph, metric-space, and latent-structure matching problems into continuous optimization over probabilistic couplings that can absorb unequal masses, ignore irrelevant target support, and induce sparse or near-deterministic correspondences. Across recent work, srGW has been used for graph partitioning and dictionary learning, manifold-valued multidimensional scaling, joint clustering and dimensionality reduction, stochastic block model and graphon estimation, and robustness to partial matching and outliers (Vincent-Cuaz et al., 2021).

1. Definition and relation to balanced GW

Classical discrete GW compares two structures by optimizing a quadratic distortion functional over couplings with both marginals fixed. For graphs represented by structure matrices $C$ and $\bar C$ with node weights $h$ and $\bar h$, the balanced formulation is

$$GW_q^q(C,h,\bar C,\bar h)=\min_{T\in\mathcal U(h,\bar h)} \mathcal L_q(C,\bar C,T),$$

with

$$\mathcal U(h,\bar h)=\{T\in\mathbb R_+^{n\times m}\mid T\mathbf 1_m=h,\;T^\top \mathbf 1_n=\bar h\},$$

and quadratic GW loss

$$\mathcal L_q(C,\bar C,T)=\sum_{i,j\in[n]^2}\sum_{k,l\in[m]^2}|C_{ij}-\bar C_{kl}|^q\,T_{ik}T_{jl}.$$

This is the mass-conserving formulation used by standard GW (Vincent-Cuaz et al., 2021).

The srGW divergence relaxes only the second marginal. In the graph formulation introduced for graph learning,

$$srGW_q^q(C,h,\bar C)=\min_{\bar h\in\Sigma_m} GW_q^q(C,h,\bar C,\bar h),$$

and equivalently

$$srGW_q^q(C,h,\bar C)=\min_{T\in\mathcal U(h,m)} \mathcal L_q(C,\bar C,T), \qquad \mathcal U(h,m)=\{T\in\mathbb R_+^{n\times m}\mid T\mathbf 1_m=h\}.$$

Under any feasible $T$, the induced target marginal is

$\bar C$0

so the target weights are not prescribed a priori but learned by the coupling itself (Vincent-Cuaz et al., 2021).

The same asymmetric principle appears in metric-measure formulations. For metric measure spaces $\bar C$1 and $\bar C$2, a semi-coupling is a probability measure $\bar C$3 on $\bar C$4 such that

$\bar C$5

The srGW $\bar C$6-distance is then

$\bar C$7

with

$\bar C$8

For $\bar C$9,

$h$0

This formulation makes explicit that only the source measure is constrained, whereas no marginal constraint is imposed on $h$1 (Clark et al., 2024).

A common misconception is to conflate srGW with any computational relaxation of GW. That identification is not correct. In particular, the formation-shape control paper employs the exact balanced discrete GW objective with both marginals fixed and uniform,

$h$2

and then solves it through a semidefinite programming relaxation; the paper explicitly does not use srGW (Nakashima et al., 27 Mar 2025).

2. Canonical formulations and model variants

In graph problems with adjacency or structure matrices $h$3 and $h$4, the squared srGW objective used for network learning is

$h$5

Here the left/source marginal is fixed, each source node keeps mass $h$6, and the right/target marginal is free. This is the formulation used to learn latent block proportions from a single observed graph (Dufour et al., 1 Jun 2026).

In stochastic block model inference, the same idea is written as

$h$7

with

$h$8

The implicit target masses are $h$9, so the cluster proportions are inferred rather than fixed (Queric et al., 27 May 2026).

For dimensionality reduction and clustering, srGW is used as an optimization over a learned target graph or embedding. The semi-relaxed transport polytope is

$\bar h$0

and the core objective is

$\bar h$1

The corresponding embedding problem is

$\bar h$2

and the barycentric version is

$\bar h$3

Because the target marginal is relaxed, some prototypes may receive zero mass (Assel et al., 2023).

A fused extension incorporates node features. In the graph-learning formulation,

$\bar h$4

and in the dimensionality-reduction formulation,

$\bar h$5

These fused variants combine structural discrepancy and feature matching in a single semi-relaxed OT objective (Vincent-Cuaz et al., 2021).

3. Theoretical properties and geometric interpretation

The defining interpretation of srGW is that it searches for a reweighted version of the target structure that best matches the source. In the graph setting, if the optimal target marginal $\bar h$6 is sparse, only a subgraph of the target is effectively used. This yields a soft subgraph matching or soft partial matching interpretation (Vincent-Cuaz et al., 2021).

The basic zero property is correspondingly asymmetric. If $\bar h$7, $\bar h$8, and $\bar h$9 has full support, then

$$GW_q^q(C,h,\bar C,\bar h)=\min_{T\in\mathcal U(h,\bar h)} \mathcal L_q(C,\bar C,T),$$0

if and only if there exists $$GW_q^q(C,h,\bar C,\bar h)=\min_{T\in\mathcal U(h,\bar h)} \mathcal L_q(C,\bar C,T),$$1 with $$GW_q^q(C,h,\bar C,\bar h)=\min_{T\in\mathcal U(h,\bar h)} \mathcal L_q(C,\bar C,T),$$2 and a bijection

$$GW_q^q(C,h,\bar C,\bar h)=\min_{T\in\mathcal U(h,\bar h)} \mathcal L_q(C,\bar C,T),$$3

such that

$$GW_q^q(C,h,\bar C,\bar h)=\min_{T\in\mathcal U(h,\bar h)} \mathcal L_q(C,\bar C,T),$$4

and

$$GW_q^q(C,h,\bar C,\bar h)=\min_{T\in\mathcal U(h,\bar h)} \mathcal L_q(C,\bar C,T),$$5

Thus srGW vanishes exactly when the source graph is isomorphic to a reweighted subgraph of the target graph (Vincent-Cuaz et al., 2021).

For finite-source metric spaces, srGW admits a Monge characterization. If $$GW_q^q(C,h,\bar C,\bar h)=\min_{T\in\mathcal U(h,\bar h)} \mathcal L_q(C,\bar C,T),$$6 is finite with fully supported $$GW_q^q(C,h,\bar C,\bar h)=\min_{T\in\mathcal U(h,\bar h)} \mathcal L_q(C,\bar C,T),$$7, and $$GW_q^q(C,h,\bar C,\bar h)=\min_{T\in\mathcal U(h,\bar h)} \mathcal L_q(C,\bar C,T),$$8 is a proper metric space with a cocompact isometric group action, then for every $$GW_q^q(C,h,\bar C,\bar h)=\min_{T\in\mathcal U(h,\bar h)} \mathcal L_q(C,\bar C,T),$$9 there exists $$\mathcal U(h,\bar h)=\{T\in\mathbb R_+^{n\times m}\mid T\mathbf 1_m=h,\;T^\top \mathbf 1_n=\bar h\},$$0 such that

$$\mathcal U(h,\bar h)=\{T\in\mathbb R_+^{n\times m}\mid T\mathbf 1_m=h,\;T^\top \mathbf 1_n=\bar h\},$$1

Moreover, for $$\mathcal U(h,\bar h)=\{T\in\mathbb R_+^{n\times m}\mid T\mathbf 1_m=h,\;T^\top \mathbf 1_n=\bar h\},$$2, any optimal semi-coupling with the same distortion must be induced by a function. This reduces the finite-source srGW optimization to a Monge problem over maps $$\mathcal U(h,\bar h)=\{T\in\mathbb R_+^{n\times m}\mid T\mathbf 1_m=h,\;T^\top \mathbf 1_n=\bar h\},$$3 (Clark et al., 2024).

That Monge theorem yields a direct connection to multidimensional scaling. If $$\mathcal U(h,\bar h)=\{T\in\mathbb R_+^{n\times m}\mid T\mathbf 1_m=h,\;T^\top \mathbf 1_n=\bar h\},$$4 is a finite metric space with uniform measure and $$\mathcal U(h,\bar h)=\{T\in\mathbb R_+^{n\times m}\mid T\mathbf 1_m=h,\;T^\top \mathbf 1_n=\bar h\},$$5 realizes $$\mathcal U(h,\bar h)=\{T\in\mathbb R_+^{n\times m}\mid T\mathbf 1_m=h,\;T^\top \mathbf 1_n=\bar h\},$$6, then the point cloud $$\mathcal U(h,\bar h)=\{T\in\mathbb R_+^{n\times m}\mid T\mathbf 1_m=h,\;T^\top \mathbf 1_n=\bar h\},$$7 solves the classical MDS problem

$$\mathcal U(h,\bar h)=\{T\in\mathbb R_+^{n\times m}\mid T\mathbf 1_m=h,\;T^\top \mathbf 1_n=\bar h\},$$8

This establishes that metric MDS is a special case of srGW with $$\mathcal U(h,\bar h)=\{T\in\mathbb R_+^{n\times m}\mid T\mathbf 1_m=h,\;T^\top \mathbf 1_n=\bar h\},$$9 and Euclidean target space (Clark et al., 2024).

At the $$\mathcal L_q(C,\bar C,T)=\sum_{i,j\in[n]^2}\sum_{k,l\in[m]^2}|C_{ij}-\bar C_{kl}|^q\,T_{ik}T_{jl}.$$0 level, srGW is linked to a modified Gromov-Hausdorff geometry. The symmetrized semi-relaxed Gromov-Hausdorff distance equals Memoli’s modified Gromov-Hausdorff distance, and under full support,

$$\mathcal L_q(C,\bar C,T)=\sum_{i,j\in[n]^2}\sum_{k,l\in[m]^2}|C_{ij}-\bar C_{kl}|^q\,T_{ik}T_{jl}.$$1

By contrast, for $$\mathcal L_q(C,\bar C,T)=\sum_{i,j\in[n]^2}\sum_{k,l\in[m]^2}|C_{ij}-\bar C_{kl}|^q\,T_{ik}T_{jl}.$$2, the symmetrized srGW is not a pseudometric in general (Clark et al., 2024).

The robust partial-GW analysis makes this failure precise. For the limiting case $$\mathcal L_q(C,\bar C,T)=\sum_{i,j\in[n]^2}\sum_{k,l\in[m]^2}|C_{ij}-\bar C_{kl}|^q\,T_{ik}T_{jl}.$$3, one marginal is fixed exactly and the other is unconstrained, and the paper explicitly identifies

$$\mathcal L_q(C,\bar C,T)=\sum_{i,j\in[n]^2}\sum_{k,l\in[m]^2}|C_{ij}-\bar C_{kl}|^q\,T_{ik}T_{jl}.$$4

with the semi-relaxed Gromov-Wasserstein divergence previously introduced by Vincent-Cuaz et al. The same paper shows that relaxed GW generally fails both non-degeneracy and the triangle inequality, even though optimal relaxed couplings exist and the relaxed objectives converge to classical GW as the relaxation vanishes (Chhoa et al., 2024).

4. Optimization structure and computational methods

A major reason srGW is used in practice is that its one-sided marginal structure simplifies the inner linearized transport steps. In the graph formulation of the conditional gradient solver, given $$\mathcal L_q(C,\bar C,T)=\sum_{i,j\in[n]^2}\sum_{k,l\in[m]^2}|C_{ij}-\bar C_{kl}|^q\,T_{ik}T_{jl}.$$5,

$$\mathcal L_q(C,\bar C,T)=\sum_{i,j\in[n]^2}\sum_{k,l\in[m]^2}|C_{ij}-\bar C_{kl}|^q\,T_{ik}T_{jl}.$$6

and if $$\mathcal L_q(C,\bar C,T)=\sum_{i,j\in[n]^2}\sum_{k,l\in[m]^2}|C_{ij}-\bar C_{kl}|^q\,T_{ik}T_{jl}.$$7 are symmetric,

$$\mathcal L_q(C,\bar C,T)=\sum_{i,j\in[n]^2}\sum_{k,l\in[m]^2}|C_{ij}-\bar C_{kl}|^q\,T_{ik}T_{jl}.$$8

The linearized subproblem is

$$\mathcal L_q(C,\bar C,T)=\sum_{i,j\in[n]^2}\sum_{k,l\in[m]^2}|C_{ij}-\bar C_{kl}|^q\,T_{ik}T_{jl}.$$9

which decomposes row-by-row into simplex problems. Each row is therefore obtained by selecting a minimum-cost target node and placing all row mass there, an $$srGW_q^q(C,h,\bar C)=\min_{\bar h\in\Sigma_m} GW_q^q(C,h,\bar C,\bar h),$$0 operation with easy GPU parallelization (Vincent-Cuaz et al., 2021).

The entropic mirror-descent version is even simpler than balanced entropic GW because only one marginal must be enforced. The update is

$$srGW_q^q(C,h,\bar C)=\min_{\bar h\in\Sigma_m} GW_q^q(C,h,\bar C,\bar h),$$1

with

$$srGW_q^q(C,h,\bar C)=\min_{\bar h\in\Sigma_m} GW_q^q(C,h,\bar C,\bar h),$$2

followed by the one-sided scaling

$$srGW_q^q(C,h,\bar C)=\min_{\bar h\in\Sigma_m} GW_q^q(C,h,\bar C,\bar h),$$3

Unlike balanced entropic GW, this does not require Sinkhorn scaling on both marginals (Vincent-Cuaz et al., 2021).

In generalized dimensionality reduction, the outer optimization is handled by block coordinate descent. The procedure alternates between solving srGW for fixed $$srGW_q^q(C,h,\bar C)=\min_{\bar h\in\Sigma_m} GW_q^q(C,h,\bar C,\bar h),$$4 using the conditional gradient solver of Vincent-Cuaz et al., and updating $$srGW_q^q(C,h,\bar C)=\min_{\bar h\in\Sigma_m} GW_q^q(C,h,\bar C,\bar h),$$5 by gradient descent with adaptive learning rate (Adam). The paper states the per-update complexity as

$$srGW_q^q(C,h,\bar C)=\min_{\bar h\in\Sigma_m} GW_q^q(C,h,\bar C,\bar h),$$6

For the manifold-valued MDS framework, the computational pipeline is: discretize the target space $$srGW_q^q(C,h,\bar C)=\min_{\bar h\in\Sigma_m} GW_q^q(C,h,\bar C,\bar h),$$7 by a finite subset $$srGW_q^q(C,h,\bar C)=\min_{\bar h\in\Sigma_m} GW_q^q(C,h,\bar C,\bar h),$$8; solve an srGW problem from $$srGW_q^q(C,h,\bar C)=\min_{\bar h\in\Sigma_m} GW_q^q(C,h,\bar C,\bar h),$$9 to $$srGW_q^q(C,h,\bar C)=\min_{T\in\mathcal U(h,m)} \mathcal L_q(C,\bar C,T), \qquad \mathcal U(h,m)=\{T\in\mathbb R_+^{n\times m}\mid T\mathbf 1_m=h\}.$$0; interpret the optimal semi-coupling as a map $$srGW_q^q(C,h,\bar C)=\min_{T\in\mathcal U(h,m)} \mathcal L_q(C,\bar C,T), \qquad \mathcal U(h,m)=\{T\in\mathbb R_+^{n\times m}\mid T\mathbf 1_m=h\}.$$1 via the Monge theorem; and use $$srGW_q^q(C,h,\bar C)=\min_{T\in\mathcal U(h,m)} \mathcal L_q(C,\bar C,T), \qquad \mathcal U(h,m)=\{T\in\mathbb R_+^{n\times m}\mid T\mathbf 1_m=h\}.$$2 as initialization for gradient descent on the continuous distortion objective

$$srGW_q^q(C,h,\bar C)=\min_{T\in\mathcal U(h,m)} \mathcal L_q(C,\bar C,T), \qquad \mathcal U(h,m)=\{T\in\mathbb R_+^{n\times m}\mid T\mathbf 1_m=h\}.$$3

This hybrid method is named SRGW+GD (Clark et al., 2024).

For network learning, optimization proceeds by block-coordinate minimization. With $$srGW_q^q(C,h,\bar C)=\min_{T\in\mathcal U(h,m)} \mathcal L_q(C,\bar C,T), \qquad \mathcal U(h,m)=\{T\in\mathbb R_+^{n\times m}\mid T\mathbf 1_m=h\}.$$4 fixed, the block matrix update has closed form,

$$srGW_q^q(C,h,\bar C)=\min_{T\in\mathcal U(h,m)} \mathcal L_q(C,\bar C,T), \qquad \mathcal U(h,m)=\{T\in\mathbb R_+^{n\times m}\mid T\mathbf 1_m=h\}.$$5

and with $$srGW_q^q(C,h,\bar C)=\min_{T\in\mathcal U(h,m)} \mathcal L_q(C,\bar C,T), \qquad \mathcal U(h,m)=\{T\in\mathbb R_+^{n\times m}\mid T\mathbf 1_m=h\}.$$6 fixed, the semi-relaxed coupling $$srGW_q^q(C,h,\bar C)=\min_{T\in\mathcal U(h,m)} \mathcal L_q(C,\bar C,T), \qquad \mathcal U(h,m)=\{T\in\mathbb R_+^{n\times m}\mid T\mathbf 1_m=h\}.$$7 is updated by Frank-Wolfe or conditional gradient. The paper states that Frank-Wolfe converges to stationary points for nonconvex problems with primal-dual gap rate $$srGW_q^q(C,h,\bar C)=\min_{T\in\mathcal U(h,m)} \mathcal L_q(C,\bar C,T), \qquad \mathcal U(h,m)=\{T\in\mathbb R_+^{n\times m}\mid T\mathbf 1_m=h\}.$$8, and that computation is GPU-parallelized through POT (Dufour et al., 1 Jun 2026).

5. Statistical and structural results

A recurring theme in recent srGW work is that the semi-relaxation yields soft assignments computationally but often recovers hard or nearly hard latent structure.

For the srGW barycenter problem used to interpolate between clustering and dimensionality reduction, the key theoretical result is a hard-clustering recovery statement. With $$srGW_q^q(C,h,\bar C)=\min_{T\in\mathcal U(h,m)} \mathcal L_q(C,\bar C,T), \qquad \mathcal U(h,m)=\{T\in\mathbb R_+^{n\times m}\mid T\mathbf 1_m=h\}.$$9, if

$T$0

is convex on the transport polytope, then the srGW barycenter problem admits scaled membership matrices as optimal transport plans. Because the objective becomes concave over $T$1 under that condition, minima occur at extreme points, which are membership matrices. The paper states that this condition is satisfied for many classical dimensionality-reduction settings, for example when $T$2 is PSD or NSD, and that the result extends to squared Euclidean distance matrices (Assel et al., 2023).

In network learning, the relaxed solution is shown to be typically deterministic. For fixed $T$3, if $T$4 is a local minimizer of

$T$5

then

$T$6

If the argmin is unique, node $T$7 must already be assigned deterministically. With the rounding map

$T$8

the paper proves that for any local minimizer $T$9, there exists a deterministic labeling $\bar C$00 such that

$\bar C$01

The resulting $\bar C$02 soft-to-hard gap is independent of $\bar C$03 (Dufour et al., 1 Jun 2026).

The same paper derives finite-sample and asymptotic statistical guarantees. For an SBM with true parameters $\bar C$04, with high probability,

$\bar C$05

with probability at least

$\bar C$06

For $\bar C$07-Hölder graphons and

$\bar C$08

the estimator satisfies

$\bar C$09

with probability at least

$\bar C$10

These rates are stated to be minimax-optimal (Dufour et al., 1 Jun 2026).

In the stochastic block model inference framework based on srGW, the statistical interpretation is different but parallel. Maximum likelihood variational inference can be reformulated as an entropically regularized srGW problem: $\bar C$11 where the first term is exactly the srGW objective with $\bar C$12. The paper then studies the unregularized srGW estimator and proves consistency of both latent assignments and connectivity recovery under a Bernoulli SBM and assumptions A1–A3. In particular, when $\bar C$13 is known, the expected srGW loss is minimized by

$\bar C$14

for a permutation matrix $\bar C$15, and the estimator satisfies

$\bar C$16

For unknown connectivity,

$\bar C$17

The paper also proves the likelihood comparison

$\bar C$18

which links unregularized srGW directly to profile likelihood (Queric et al., 27 May 2026).

6. Applications, neighboring frameworks, and scope

The earliest graph-focused srGW paper uses the divergence for graph dictionary learning, partitioning, clustering, and completion. In dictionary learning, a single graph atom $\bar C$19 is learned through

$\bar C$20

and each graph is represented by the induced target marginal

$\bar C$21

This yields sparse weight vectors on a common template graph rather than linear combinations of multiple atoms (Vincent-Cuaz et al., 2021).

The same semi-relaxed mechanism underlies recent generalized dimension-reduction methods. SRGW+GD applies srGW to initialize manifold-valued MDS into Euclidean spaces, circles, and spheres, and is used in the paper to visualize ensembles of political redistricting plans by embedding them into $\bar C$22. The circle-valued visualization is constructed by solving srGW on a 1000-point discretization of the circle and then refining via gradient descent on the circle-valued MDS objective (Clark et al., 2024).

A separate line of work uses srGW to interpolate between clustering and dimensionality reduction. When the embedding sample size equals the input sample size, $\bar C$23, the method recovers classical dimensionality-reduction behavior; when the embedding dimensionality is unconstrained, the transport plan acts as a clustering or prototype-assignment device, and because the target marginal is relaxed, some prototypes can receive zero mass. This suggests an operational continuum between graph-based clustering and low-dimensional embedding rather than a categorical distinction between them (Assel et al., 2023).

In statistical network analysis, srGW is used in two related but distinct ways. One approach formulates block-model and graphon estimation directly as a semi-relaxed GW barycenter problem over couplings and block matrices, with near-deterministic solutions and minimax-optimal rates (Dufour et al., 1 Jun 2026). Another shows that variational EM for stochastic block models is an entropically regularized srGW projection, then removes the entropy term and adds the sparsity-promoting regularizer

$\bar C$24

to promote exact empty clusters and enable intrinsic model selection from a single optimization run (Queric et al., 27 May 2026).

Finally, srGW should be distinguished from broader unbalanced or lifted GW formulations. The multi-marginal GW paper does not define srGW as a separate named case, but its unbalanced and fixed-support dummy-factor constructions are described as a close analogue of asymmetric marginal relaxation. Conversely, semidefinite relaxations of GW preserve both marginals exactly and relax only the quadratic rank-one relation; they are therefore not semi-relaxed models, even if they are computational relaxations of GW (Beier et al., 2022).

A plausible implication is that srGW now occupies a distinct methodological position within the GW family: it is neither merely a heuristic weakening of balanced GW nor interchangeable with unbalanced, partial, or semidefinite formulations. Its defining feature is one-sided marginal freedom, and the literature uses that asymmetry to learn latent proportions, select substructures, compress support, and bridge transport-based geometry with statistical estimation.