Barycentric Weak Inner-Product Gromov-Wasserstein

• Barycentric wIGW compares source relations with relations among conditional target means, addressing one-to-many settings where coarse states represent refined outcomes.The conditional mean carries the coarse geometry, while conditional variation captures the refinement; ordinary GW is recovered as a special case.Source: §1 · pp. 2, 6
• The discrepancy equals IGW projected onto intermediate laws η satisfying η ⪯cx ν, retaining ν as the target marginal while relaxing only mean-carried geometry.Convex order characterizes attainable conditional means, and martingale gluing realizes an optimal mean map as a coupling with target marginal ν.Source: §1 · pp. 3, 6
• For finite-second-moment measures, coupling, map, and projection formulations admit minimizers, and optimal mean maps can be reconstructed through martingale kernels.The two-stage construction maps X to Z=m(X), then uses a kernel with E[Y | Z]=Z to recover ν.Source: §1 · p. 3
• Ridge regularization yields an A–B min–max formulation whose inner problem is convex barycentric weak optimal transport and whose outer variables optimize moment matrices.The ridge makes the cost coercive and strongly convex in z; the outer exchange is established when ε ≥ 2λmax(Sµ), with convergence using ε > 2λmax(Sµ).Source: §1 · p. 4
• The finite-sample algorithm has an explicit contraction bound controlling inexact inner-oracle error under a quantitative ridge condition, with numerical demonstrations on shape and graph refinements.A martingale refinement can have zero wIGW despite positive ordinary IGW when children preserve the coarse parent means and Gram geometry.Source: §1 · pp. 5–6
• The paper evaluates RNA/ATAC atlas-based cell-type transfer on paired 10x Genomics PBMC multiome data in cell-to-cell and prototype-to-cell settings, excluding target labels from optimization.The prototype-to-cell setting represents the one-to-many case, and sensitivity to the ridge parameter is assessed.Source: §1 · pp. 2, 6

Kantorovich duality then gives

• Kantorovich duality supplies a one-potential formulation for the IGW envelope, and the coupling infimum attains a minimizer.The dual is expressed through a universally measurable transform with the required integrability conditions.Source: Kantorovich duality then gives · p. 7

2.1 Ordinary IGW: OT envelope and computational precedent

• Ordinary IGW admits an outer matrix optimization coupled with an ordinary OT problem, requiring only second moments in its bilinear formulation.The envelope varies A because the target marginal fixes S_ν; this contrasts with quadratic Euclidean GW’s fourth-moment assumption and builds on the variational mechanism in [14].Source: Kantorovich duality then gives · p. 7, §2.1 · p. 7

2.2 Weak optimal transport

• Barycentric weak OT replaces pointwise transport costs by conditional-law costs and reduces, under its stated continuity, convexity, and growth assumptions, to barycentric maps whose pushforward is below the target in convex order.If the conditional cost is uniformly strongly convex, the minimizing barycentric map is unique; convex order is equivalently characterized by a martingale coupling through Strassen’s theorem [8].Source: §2.2 · pp. 8–9
• The weak GW construction applies this conditional mechanism to source pairs, comparing the conditional target laws induced by a coupling with the source relation.The Jensen–Strassen correspondence yields a convex-order projection and, after ridge regularization, the weak OT block used by the numerical method.Source: §2.2 · pp. 8–9

Conditional-law framework

• Weak GW is formulated directly in terms of the conditional laws induced by a coupling.For each coupling π, the construction uses a disintegration π(dx,dy)=μ(dx)π_x(dy).Source: §3 · p. 9
• Two principal lifts either average the original pointwise loss or aggregate each pair of conditional laws before applying the loss.The framework also establishes measurability of the resulting lifts under measurable relation and loss assumptions.Source: §3 · pp. 9–10

Consistency with ordinary GW

• The averaging lift is consistent with ordinary GW because disintegrating both copies of a coupling recovers the original GW objective.This consistency is formalized by Proposition 3.2, which states that ordinary GW is an exact specialization.Source: §3 · p. 10

• Barycentric wIGW evaluates relational loss using the conditional mean map, so target conditional laws with identical means are identified.The construction specializes the weak GW lift to the barycentric inner-product relation.Source: §4 · p. 11
• Conditional Jensen shows that replacing pointwise target relations by conditional expectations weakly reduces the squared relational loss for every fixed coupling.The inequality quantifies the reduction from barycentric averaging before applying the squared loss.Source: §4 · p. 11
• The barycentric construction retains only means, whereas alternative relations can preserve additional distributional information; corresponding projection and finite A–B results remain specific to Dbar.Results for the other listed relations require separate analysis and are left for future work.Source: §4 · p. 11
• Convex order exactly characterizes conditional mean maps generated by couplings with target marginal ν, reducing the primal optimization to admissible maps.The admissible map set is nonempty, convex, and weakly compact in L2(µ; Rdy).Source: §4 · pp. 10–11
• Because m#µ ⪯cx ν is directional, wIGWbar(µ, ν) and wIGWbar(ν, µ) may differ, so the quantity is called a discrepancy rather than a distance.Source: §4 · p. 12

Moment reduction

• The pairwise loss reduces to three finite-dimensional second moments, yielding a tractable moment representation.Source: §5 · p. 12

Zero set

• wIGWbar(µ, ν) = 0 if and only if there exists a Borel conditional-mean map satisfying the zero-set criterion.Existence follows from weak compactness of the feasible map set, lower semicontinuity of the moment objective, and nonnegativity of the squared loss.Source: §5 · p. 12

Isometries with martingale noise

• An isometric embedding T gives zero discrepancy whenever T#µ ⪯cx ν, including target refinements with vanishing conditional-mean noise.The noise may depend on X and be heteroscedastic, non-Gaussian, or anisotropic.Source: §5 · p. 12
• The zero-cost certificate requires E[ξ | X] = 0; global centering E[ξ] = 0 alone is insufficient.Source: §5 · p. 13

Orthogonal invariance

• The discrepancy is invariant under separate orthogonal transformations, but experimental centering is necessary because raw inner products are translation-sensitive.Source: §5 · p. 13

6 Projection onto the convex order cone

• Barycentric weak IGW equals the minimum ordinary IGW discrepancy over intermediate target laws below ν in convex order.The projection identity minimizes IGW over η satisfying η ⪯cx ν.Source: §6 · p. 13
• A mean-preserving spread of the target cannot increase directional wIGW, and barycentric weak wIGW is no larger than ordinary IGW.The first conclusion follows from the expanding feasible projection set; the second chooses ν itself as the intermediate law.Source: §6 · p. 13
• The map, projection, and coupling formulations each admit minimizers, and optimal maps induce optimal projection laws and IGW couplings.The direct method supplies optimizing maps, projection laws, and couplings.Source: §6 · p. 14

Every such kernel defines

• Strassen martingale kernels realize every optimal mean map as an optimal wIGW coupling with target marginal ν.This applies both to maps obtained directly and to conditional means extracted from optimal IGW couplings; the two maps may differ when minimizers are nonunique.Source: Every such kernel defines · p. 14

• The compact-support dual characterizes wIGW using a convex potential for convex order and matrices A and B for cross- and second-moment terms.The potential u enforces convex order, A linearizes the source–map cross moment, and B linearizes the map second moment.Source: §7 · pp. 15–16
• On compact supports, convex order is equivalent to testing continuous convex functions, with Strassen martingale couplings and conditional Jensen providing the key characterization.If m#µ ⪯cx ν, a martingale coupling supported on K × K yields Jensen inequalities; failure of convex order is separated by a globally defined convex function.Source: §7 · p. 15
• Jensen–Strassen identifies the coupling and barycentric-map problems, while Sion’s theorem exchanges compact map minimization with potential maximization to establish dual equality.The required compactness, topology, semicontinuity, and convexity conditions are verified for the minimax applications.Source: §7 · pp. 15–16
• The resulting outer formulation preserves the order inf_A sup_B; B-singular cases may yield nonunique barycentric maps, motivating ridge regularization without compact support.The ridge extension is deferred to Section 8, where it produces a coercive weak OT block.Source: §7 · p. 16

8.1 Unregularized matrix representation

• The unregularized matrix reduction is exact, with Fenchel optimizers A = Mm and B = Sm, but singular B can destroy the coercivity required for weak OT duality.Because B2 includes singular matrices such as B = 0, the fixed-matrix costs cannot satisfy the required coercive lower bound uniformly.Source: §8 · p. 16, §8.1 · p. 17

8.2 Ridge regularization and the weak OT block

• Ridge regularization verifies coercivity and strong convexity in the conditional mean, yielding a unique optimal barycentric mean map but not necessarily a unique coupling.The ridge changes the value by at most its strength times the target second moment, and distinct couplings may share the same conditional mean.Source: §8.2 · pp. 17–18

8.3 The ridge envelope and outer minimax equality

• Under ε ≥ 2λX, the ridge formulation is convex–concave and satisfies the outer A–B minimax equality.The objective is jointly convex in (A, m) and concave in B; strict ε > 2λX is reserved for the convergence modulus in Section 11.Source: §8.3 · p. 20
• For every positive ridge, the weak OT envelope and potential dual follow by exchanging the mean and B variables with A fixed.The additional spectral condition is needed only to exchange the two outer matrix optimizations because separate convexity does not imply joint convexity in (A, m).Source: §8.3 · pp. 19, 21

Equivalent formulations

• Projection identity, compact dual, and ridge envelope present the construction at different levels and are organized by computational role in Table 2.Table 2 collects equivalent formulations of wIGW and its ridge version.Source: §9 · p. 21, Table 2 · p. 21

Computational roles

• The formulation displays a weak OT oracle together with its dual over convex potentials, while its first four rows equal the unregularized wIGW objective.Source: §9 · p. 21

Exact reconstruction

• At each fixed pair of outer matrices, ridge regularization makes the conditional mean map unique, while a martingale kernel supplies the variation needed to realize the target marginal.The weak OT block returns a coupling, whereas the ridge primal is expressed through conditional-mean moments; reconstruction requires these descriptions to agree at an outer optimizer.Source: §10 · pp. 21–22

Compatibility and gluing

• Outer optimality conditions establish compatibility between the weak OT solution and the ridge primal, allowing martingale gluing to recover a coupling.The resulting coupling belongs to Π(µ, ν) and solves the ridge coupling primal (38).Source: §10 · p. 22

Proposition 10.1

• Proposition 10.1 formalizes compatibility, reconstruction, and gluing for µ and ν with finite second moments and ε > 0, using the optimizer A⋆, B⋆ of the nested outer problem.The proposition defines m⋆(x) from the optimal coupling and concludes that the reconstructed coupling solves the ridge coupling primal.Source: §10 · p. 22

11 Finite algorithm and convergence

• The inner algorithm applies normalized KL mirror descent with Sinkhorn projections, using entropy only as mirror geometry rather than adding an entropy penalty to the weak OT objective.The implementation uses an inexact KL projection, marginal repair, and an outer return rule based on the lowest direct ridge primal value; the theorem analyzes iterates rather than the best-so-far tuple.Source: §11 · p. 22, §11.1 · pp. 23–25

11.1 Finite numerical formulation and projected algorithm

• Finite measures reduce each fixed-matrix weak OT block to a convex optimization over a transport polytope because conditional means depend linearly on the coupling matrix.Compatibility diagnostics use moment and marginal or martingale residuals, while data-dependent Frobenius balls bound compatible outer solutions.Source: §11.1 · pp. 22–24

The variational inequality

• Strong convexity in the barycentric vector converts fixed-matrix objective suboptimality into outer operator error, so feasible approximate couplings can support the contraction guarantee.The relevant suboptimality is measured against the unregularized fixed-matrix objective, not a KL proximal or entropy-regularized objective.Source: §11.1 · p. 24, The variational inequality · pp. 27–28

11.2 Projected contraction and inexact oracle control

• The projected outer operator G is globally L-Lipschitz and strongly monotone, with a unique solution; vanishing inner suboptimality yields convergence, while uniformly bounded suboptimality yields an explicit error floor.Its Lipschitz constant satisfies L ≤ 4 + 8(λX + M2(ν))/ε.Source: §11.2 · p. 27, The variational inequality · pp. 27–28

12.1 Synthetic objectives and reported quantities

• The synthetic protocol uses known parent-to-child couplings as algebraically feasible zero-wIGW witnesses, while solver runs start from the product coupling and report finite-budget behavior outside the strong ridge regime.Pcert recovers the clean skeleton through conditional means, but the weak solver does not receive it; ε = 10^-4 approximates the unregularized objective and is outside the regime certified by Theorem 11.3.Source: §12.1 · p. 29

12.2 Shape refinements with martingale noise

• Across shape refinements, the explicit weak certificate remains at arithmetic zero while ordinary IGW at the same coupling increases with conditional spread; numerical solves agree with POT to reported precision.Both envelope runs stopped after three iterations with zero final Frank–Wolfe gap; maximum marginal residuals were zero and 2.95 · 10^-17, and direct objectives agreed with POT within 8.33 · 10^-17.Source: §12.2 · p. 29, Figure 4 · p. 30

12.3 Graph feature refinement

• The graph-feature refinement likewise has a floating-point-zero certificate, while ordinary IGW increases with refinement spread; the A/OT envelope reaches zero final Frank–Wolfe gap and satisfies 10^-12 marginal feasibility.For the reported graph instance, the direct objective agrees with POT to 1.11 · 10^-15 and the maximum marginal residual is 1.39 · 10^-17; near-equivalent barycentric skeletons can yield large orientation-dependent RMS.Source: §12.3 · p. 32, Figure 7 · p. 32

12.4 PBMC multiome atlas based cell type transfer

• 0.041, 0.115, and 0.060 higher mean macro-F1, ARI, and NMI, respectively, wIGW outperforms the IGW envelope in prototype-to-cell transfer; ARI and NMI improve on every split.Against target k-means, wIGW has lower macro-F1 by 0.013 but higher ARI and NMI by 0.083 and 0.040. In cell-to-cell transfer, wIGW exceeds the scaled-identity IGW envelope by 0.030, 0.082, and 0.052 on macro-F1, ARI, and NMI.Source: Table 4 · p. 34, §12.4 · p. 34
• With fixed ε = 10^-4 and budgets, fresh-seed validation preserves higher wIGW ARI and NMI across all five splits in both tasks, but macro-F1 varies and the check is not an independent donor validation.Relative to the IGW envelope, the fresh-seed differences are +0.008, +0.088, and +0.063 for cell-to-cell macro-F1, ARI, and NMI, and −0.054, +0.132, and +0.081 for prototype-to-cell.Source: §12.4 · p. 35
• PBMC results are limited to one donor, six RNA-derived classes, overlapping balanced splits, unmatched solver budgets, and local solutions that are sensitive to coordinate representation and initialization.A sign flip changed mean cell-to-cell wIGW macro-F1 by −0.452 and an orthogonal basis change by −0.351, whereas covariant controls preserved the plan and changed the objective by at most 2.02 · 10^-16.Source: §12.4 · pp. 36–37

13 Conclusion

• The barycentric formulation compares conditional means and uses a martingale kernel to supply target variation, yielding an exact convex-order projection with existing minimizers and optimal martingale gluing.Convex order makes the comparison directional: mean-preserving spreads of the target cannot increase discrepancy.Source: §13 · p. 37
• wIGW exceeds the scaled-identity IGW envelope by 0.115 ARI and 0.060 NMI on all five original PBMC splits.Five additional donor splits preserve the ARI and NMI advantage, but not a uniform macro-F1 advantage; metrics remain stable for ridge values 10^-6 through 10^-2.Source: §13 · p. 37
• Ridge moment duality produces an A–B envelope with a strongly convex weak OT inner problem and an outer projected iteration having an explicit contraction bound under spectral-ridge, step-size, and oracle conditions.For fixed (A, B), the barycentric mean map is unique, although optimal couplings and martingale realizations need not be.Source: §13 · p. 37
• Point-cloud and graph-feature studies isolate mean-preserving refinement as a mechanism that can preserve the relevant geometry while changing the prescribed target representation.The construction is intended for settings where conditional means carry geometry and conditional variation represents target refinement.Source: §13 · p. 37
• Future work includes richer conditional-law relations such as DMCov, DW2, and DMMD, one-to-many quantum comparisons, and operator or barycenter control for infinite-dimensional kernels and nonlinear barycenters.Selecting among nonunique couplings and martingale realizations requires an additional criterion on conditional variability.Source: §13 · p. 37

A Analytic and variational foundations

• The analytic foundations establish weak compactness, weak lower semicontinuity, and a reflexive-space direct method for existence arguments.Weak compactness is treated sequentially, with Kakutani reflexivity supplying compactness and minimizers following from lower semicontinuity.Source: §A.1 · p. 38

A.2.1 Proof of Theorem 2.1

• The proof of Theorem 2.1 expands the IGW objective, completes the square, identifies the unique Fenchel minimizer A = M, and exchanges the coupling and matrix infima.The argument then uses tightness, weak closedness, moment-tail control, and continuity of Mπ to obtain joint minimizers over couplings and matrices.Source: §A.2.1 · pp. 39–40

A.2 Ordinary IGW and barycentric weak OT

• The weak OT dual is reduced to a potential formulation using Kantorovich duality, measurable infimal convolution, and integrability of the resulting Q_Au potential.Replacing φ by Q_Au preserves feasibility and cannot decrease the dual objective, establishing the stated potential class.Source: §A.2.1 · p. 40

A.2.2 Proof of Theorem 2.4

• Theorem 2.4’s weak dual can be restricted to finite convex barycentric potentials without changing its value, while strong convexity makes the optimal barycentric map unique.Existence follows from lower semicontinuity, convexity, quadratic growth, and the noncompact weak optimal-transport theorem.Source: §A.2.2 · pp. 40–42

Barycentric formulations, moments, and projection

• Every feasible coupling induces a barycentric map below ν in convex order, and martingale gluing realizes every such map, so coupling and map infima coincide.The feasible barycentric-map set is convex and weakly compact; its norm boundedness and weak closedness follow from convex-order moment bounds and lower semicontinuity.Source: §B.1.3 · p. 43, §B.1.2 · p. 43

Moments and zero-cost structure

• The inner-product objective admits a moment representation, and its zero set consists of feasible maps preserving the source Gram relation almost everywhere.An isometric linear map has zero cost whenever its pushforward is below ν in convex order; conditionally centered noise preserves this conclusion.Source: §B.2.1 · p. 44, §B.2.2 · p. 44, §B.2.3 · p. 44

B.2.4 Proof of Theorem 6.1

• The barycentric wIGW value equals projection of ν in convex order followed by ordinary IGW, with second moments ensuring integrability and the projected intermediate law remaining in P2.The proof uses conditional Jensen to compare any coupling with its conditional-mean map and convexity of the squared inner-product loss.Source: §B.2.4 · pp. 44–45

B.2.6 Proof of Theorem 6.4

• Under finite second moments, the wIGW map objective attains a minimizer, and martingale gluing converts it into an optimal coupling whose conditional mean is the minimizing map.Every martingale kernel from the projected intermediate law to ν gives an optimal coupling because the objective depends only on the conditional mean.Source: §B.2.6 · p. 46

C Duality and ridge formulations

• For compactly supported targets, convex-order feasibility is encoded by convex test functions, and matrix Fenchel identities plus Sion’s theorem justify exchanging the outer optimization variables in the ridge dual.The minimax argument uses weak L2 compactness for maps, finite-dimensional matrix topology, and convex-concave lower-semicontinuous sections.Source: §C.1.1 · p. 47

C.1.1 Proof of Theorem 7.1; C.1.2 Proof of Lemma 8.1; C.1.4 Proof of Theorem 8.3

• The ridge problem admits a compact potential dual and an A–B min-max representation obtained by restricting Fenchel optimizers to bounded matrix domains and applying Sion’s theorem only to m and B.The matrix domains contain the Fenchel optimizers, while the m-section is weakly lower semicontinuous and the B-section is continuous and concave.Source: §C.1.1 · p. 48, §C.1.2 · p. 48, §C.1.4 · p. 50

C.1.3 Proof of Proposition 8.2

• Under finite second moments and ridge strong convexity, the primal minimizer exists, the map and coupling formulations agree, and the barycentric map is unique.The coupling objective is convex but need not be strictly convex in the coupling itself, because distinct couplings can share conditional means.Source: §C.1.3 · p. 49

C.1.5 Proof of Theorem 8.4

• When ε ≥ 2λ_X, the reduced ridge objective is convex in (A, m) or (A, π), concave in B, and 2-strongly concave in B, enabling the outer minimax exchange.The proof uses the operator bound T*T ⪯ λ_XI and compact convex matrix domains.Source: §C.1.5 · p. 51

D.1.1 Proof of Proposition 10.1

• Exact nested outer optimizers are compatible, and gluing the martingale kernel after the optimal barycentric map reconstructs an optimal coupling with the same ridge wIGW objective.The martingale property follows from E[Y | M] = M, where M is the optimal conditional mean.Source: §D.1.1 · p. 52

D.1.2 Proof of Proposition 11.1

• For finite supports, the inner objective is convex and strongly convex in barycentric vectors, while the inexact mirror step uses KL projection through alternating row and column scalings after flooring the kernel at 10^-300.The objective need not be strongly convex in the transport plan when different plans have identical barycentric rows.Source: §D.1.2 · p. 53

D.1.3 Proof of Proposition 11.2; D.1.4 Proof of Theorem 11.3

• Under compatible moment bounds, the projected outer iteration has a unique solution and contracts for 0 < η < 2α/L^2, with exact-oracle rate d_k ≤ q^k d_0 and bounded-error radius ηē/(1−q).The proof establishes strong monotonicity and a global Lipschitz constant L for the saddle gradient operator.Source: §D.1.3 · p. 54, §D.1.4 · p. 55

D.1.4 Proof of Theorem 11.3

• The convolution-tail argument proves d_k → 0 when the error sequence satisfies ||e_j|| → 0.The finite initial convolution terms vanish with k, while the tail is controlled by η sup_{j≥N} ||e_j||/(1−q).Source: §D.1.4 · p. 55

D.1.5 Proof of Proposition 11.4

• The Proposition 11.4 proof relies on feasibility and unregularized-objective suboptimality, while ridge strong convexity supplies curvature through Hessian 2C_B,ε with smallest eigenvalue 2α_B.A separately entropy-regularized optimizer requires an additional bias bound before this argument applies.Source: §D.1.5 · p. 56

E General geometry with conditional laws

• General weak GW assigns each source point a conditional target law and evaluates a lifted cost on the source relation and two conditional laws.The barycentric wIGW specialization uses inner-product source relations and squared differences, while Wasserstein-distance and covariance-based alternatives capture other conditional-law relations.Source: Figure 10 · p. 56, E General geometry with conditional laws · p. 57
• The generalized covariance, Wasserstein, and kernel constructions involve nested conditional transports or separate measurability and moment analyses, whereas the paper’s structural results cover only finite-dimensional bilinear relations.Corresponding results for D_MCov, D_W2, and D_ip are left for future work.Source: E General geometry with conditional laws · p. 57

F.1 Strassen’s martingale characterization

• Strassen’s theorem characterizes convex order by the existence of a martingale coupling, with finite first moments ensuring well-defined conditional barycenters.The proof combines disintegration, conditional Jensen’s inequality, compactness, separation, and Kantorovich duality.Source: §F.1 · pp. 58–59

F.2 Danskin’s envelope theorem

• Danskin’s theorem gives directional derivatives of optimized objectives as extrema over active optimizers’ gradients and yields differentiability when those gradients coincide.The paper verifies these conditions using ridge-induced uniqueness, common barycentric vectors, and unique maximizing matrices.Source: §F.2 · p. 60

F.3 Sion’s minimax theorem

• Sion’s minimax theorem exchanges minimum and supremum when the minimizing domain is compact convex and the sections are semicontinuous with the required quasi-convexity or quasi-concavity.The theorem is applied in weak, Frobenius, uniform, and finite-dimensional topologies across Theorems 7.1, 8.3, and 8.4.Source: §F.3 · pp. 60–61

G Experimental configurations and evaluation metrics

• Synthetic evaluations recompute objectives and residuals from returned plans or solver states, distinguishing numerical compatibility and marginal checks from additional objectives.The reported quantities include a constructed martingale witness, returned weak-solver state, and ordinary-solver comparison plans.Source: Table 5 · p. 61
• The PBMC dataset contains paired RNA and ATAC measurements from one healthy donor; filtering retains 8212 of 12016 cells, with 7760 entering the six-class eligible pool.Each frozen split uses a disjoint 900-cell atlas and 480-cell evaluation set, with modality transformations fitted on the atlas and applied unchanged to evaluation cells.Source: G Experimental configurations and… · p. 63, Table 9 · p. 63

G.1 PBMC runtime and local solver robustness

• PBMC runtime comparison measures method-call time after preprocessing across five frozen splits, while allowing method-specific budgets rather than matching wall-clock effort.The runtime table documents observed cost and feasibility, not a hardware-normalized efficiency comparison.Source: Table 11 · p. 64
• The robustness audit tests initialization and independent coordinate changes for wIGW and the IGW envelope under the same reported iteration budgets, without using labels or pair identities.The audit includes objective-based selection among three starts plus sign-flip and orthogonal-basis controls.Source: Figure 11 · p. 65, §G.1 · p. 65
• Selected wIGW states show maximum compatibility residuals of 1.42 · 10^-4 for A and 2.07 · 10^-4 for B, with maximum projected residual 5.90 · 10^-4.These are numerical diagnostics, not saddle-point or convergence certificates, because Algorithm 1 retains a best tuple before updating.Source: §G.1 · p. 66

G.2 PBMC evaluation metrics

• PBMC transfer is evaluated using macro-F1, ARI, and NMI, with target labels excluded from transport optimization and Hungarian matching used only to name target k-means clusters for macro-F1.The matching does not alter clustering, ARI, or NMI; ties use fixed conventions.Source: §G.2 · p. 66
• Paired-cell retrieval ranks source RNA cells by transported mass only for the cell-to-cell task, with metrics computed over five frozen splits and random-ranking expectations reported for n = 480.Retrieval is undefined for target k-means and prototype-to-cell settings because they lack cross-modal coupling or a unique paired source cell.Source: Table 13 · p. 67, §G.2 · p. 67