Generalized Optimal Partial Transport (GOPT)

• Generalized Optimal Partial Transport (GOPT) is a framework that relaxes full mass matching by employing spatially varying penalties for unmatched mass.
• It builds on classical and partial transport theories, integrating variable destruction/creation parameters to improve flexibility in uncertain or incomplete data scenarios.
• GOPT enables enhanced scalability and accuracy in applications such as machine learning, computer vision, and fluid simulation by precisely controlling mass transfer.

Generalized Optimal Partial Transport (GOPT) denotes a class of optimal transport models that relax full mass preservation while allowing the cost of leaving mass unmatched to depend on location or support point. In the formulation introduced under the name GOPT, the constant destruction/creation parameter of classical optimal partial transport is replaced by measurable, bounded functions $\lambda_1,\lambda_2$, so that unmatched mass is penalized in a spatially varying way rather than by a single scalar (Bai, 2024). Closely related developments extend partial transport through prescribed transported mass constraints, pointwise rejection costs, structural matching terms such as Gromov–Wasserstein and fused Gromov–Wasserstein objectives, and generalized multi-marginal or fluctuating-population formulations (Chapel et al., 2020, Tripathi et al., 19 May 2026, Kitagawa et al., 2014, Marino et al., 2022). Across these variants, the common objective is to compare measures when total masses differ, outliers or missing components are present, or only a subset of the support should participate in the coupling.

1. Core formulations

Classical optimal transport imposes exact marginal constraints and therefore requires all mass to be matched. In discrete form, with marginals $\bp,\bq$ and cost matrix $\bC$, the standard problem is

$\min_{\bT \in \Pi(\bp,\bq)} \langle \bC, \bT \rangle_F,$

where $\Pi(\bp,\bq)$ enforces $\bT \mathbf{1} = \bp$ and $\bT^\top \mathbf{1} = \bq$ (Chapel et al., 2020). Classical optimal partial transport relaxes this by allowing only part of the mass to be moved. A standard partial coupling set is

$\Pi^u(\bp, \bq) = \bigg\{ \bT \in \mathbb{R}_+^{n \times m} : \bT \mathbf{1} \leq \bp,\, \bT^\top \mathbf{1} \leq \bq,\, \mathbf{1}^\top \bT \mathbf{1} = s \bigg\},$

with $0 < s \leq \min(\|\bp\|_1, \|\bq\|_1)$, and the corresponding partial Wasserstein objective is

$PW_p^p(\bp,\bq) = \min_{\bT \in \Pi^u(\bp,\bq)} \langle \bC, \bT \rangle_F$

(Chapel et al., 2020).

The GOPT formulation introduced in 2024 starts from classical optimal partial transport with a total-variation penalty: $\bp,\bq$0 or equivalently

$\bp,\bq$1

where $\bp,\bq$2. GOPT replaces the constant $\bp,\bq$3 by spatially varying functions $\bp,\bq$4 and defines

$\bp,\bq$5

with choices such as $\bp,\bq$6 or $\bp,\bq$7, and similarly on the target side. Classical OPT is recovered when $\bp,\bq$8 (Bai, 2024).

A related discrete generalization is intent-controlled partial optimal transport (IC-POT), which makes rejection explicit through side-specific slack variables and pointwise unmatched costs $\bp,\bq$9 and $\bC$0: $\bC$1 Its reduced sub-coupling form is

$\bC$2

which shows that constant rebates recover classical partial transport (Tripathi et al., 19 May 2026).

2. Duality, penalties, and equivalent reformulations

The analytical structure of generalized partial transport is closely tied to duality. For entropic GOPT, the dual developed in 2024 is

$\bC$3

where $\bC$4 and the feasible set depends on whether the penalty uses $\bC$5 or $\bC$6. Setting $\bC$7 constant recovers the classical OPT dual, while $\bC$8 yields the unregularized problem if the minimizer is absolutely continuous (Bai, 2024).

When both penalties are $\bC$9, GOPT admits an explicit linear reformulation: $\min_{\bT \in \Pi(\bp,\bq)} \langle \bC, \bT \rangle_F,$0 This leads to an augmented-space reduction with augmented marginals $\min_{\bT \in \Pi(\bp,\bq)} \langle \bC, \bT \rangle_F,$1, $\min_{\bT \in \Pi(\bp,\bq)} \langle \bC, \bT \rangle_F,$2, and an augmented cost with zeros on dummy locations, enabling the use of classical OT linear programming solvers (Bai, 2024).

IC-POT has a dual with explicit local caps: $\min_{\bT \in \Pi(\bp,\bq)} \langle \bC, \bT \rangle_F,$3 In this form, $\min_{\bT \in \Pi(\bp,\bq)} \langle \bC, \bT \rangle_F,$4 and $\min_{\bT \in \Pi(\bp,\bq)} \langle \bC, \bT \rangle_F,$5 act as local acceptance thresholds. Complementary slackness gives

$\min_{\bT \in \Pi(\bp,\bq)} \langle \bC, \bT \rangle_F,$6

and the admissible-support condition

$\min_{\bT \in \Pi(\bp,\bq)} \langle \bC, \bT \rangle_F,$7

holds in every optimal solution. IC-POT is also equivalent to balanced Kantorovich OT on an augmented support with

$\min_{\bT \in \Pi(\bp,\bq)} \langle \bC, \bT \rangle_F,$8

although proper equivalence fails under entropic regularization (Tripathi et al., 19 May 2026).

These model-specific duals fit a broader functional-analytic perspective. In an abstract Banach lattice setting, constrained optimal transport can be written as a primal supremum over a convex subset of the positive unit sphere of the dual, with a corresponding dual on the bi-dual, and the equality $\min_{\bT \in \Pi(\bp,\bq)} \langle \bC, \bT \rangle_F,$9 together with dual attainment holds under mild assumptions. This framework is presented as directly accommodating constrained OT, martingale OT, and generalized or partial OT through modifications of the annihilator subspace that encodes the constraints (Ekren et al., 2016). A different but related duality line is the weak optimal total variation transport theory, which establishes a Kantorovich duality for total-variation-penalized transport and recovers a version of the Caffarelli–McCann duality for partial optimal transport (Chung et al., 2019).

3. Exact algorithms, Sinkhorn-type methods, and large-scale solvers

For discrete partial Wasserstein transport, an exact reduction via virtual points replaces entropic approximation by an augmented balanced OT problem. Given transported mass $\Pi(\bp,\bq)$0, one adds a new bin $\Pi(\bp,\bq)$1 to $\Pi(\bp,\bq)$2 with mass $\Pi(\bp,\bq)$3, a new bin $\Pi(\bp,\bq)$4 to $\Pi(\bp,\bq)$5 with mass $\Pi(\bp,\bq)$6, and the extended cost

$\Pi(\bp,\bq)$7

with $\Pi(\bp,\bq)$8. The top-left block of the optimal transport plan on the extended problem is the exact partial plan, and

$\Pi(\bp,\bq)$9

The same work develops a Frank–Wolfe method for partial Gromov–Wasserstein in which each linearized subproblem is solved as a partial Wasserstein problem via the dummy-point construction, with convergence to a stationary point stated in the supplementary material (Chapel et al., 2020).

For entropic GOPT, the generalized Sinkhorn procedure updates dual scalings through proximal-divide operators. With $\bT \mathbf{1} = \bp$0 penalties, one update has the form

$\bT \mathbf{1} = \bp$1

while with $\bT \mathbf{1} = \bp$2 penalties,

$\bT \mathbf{1} = \bp$3

and similarly for $\bT \mathbf{1} = \bp$4. This yields Algorithm 1 (opt-Sinkhorn). For non-entropic GOPT with $\bT \mathbf{1} = \bp$5 penalties, Algorithm 2 (opt-lp) solves the augmented OT linear program and projects the result back to the original partial problem (Bai, 2024).

The numerical status of Sinkhorn for partial transport is more delicate than for balanced OT. A 2023 study identifies the incompatibility of standard OT rounding with the POT sum constraint $\bT \mathbf{1} = \bp$6 and shows that the resulting violation can be lower bounded by

$\bT \mathbf{1} = \bp$7

It proposes a new feasible rounding procedure, Round-POT, and derives a feasible Sinkhorn complexity of $\bT \mathbf{1} = \bp$8, together with two first-order methods: Adaptive Primal-Dual Accelerated Gradient Descent with complexity $\bT \mathbf{1} = \bp$9 and Dual Extrapolation with complexity $\bT^\top \mathbf{1} = \bq$0 (Nguyen et al., 2023).

A separate large-scale optimization line addresses generalized transport problems through an inexact primal-dual method derived from the time discretization of a dynamical system. The inner nonsmooth subproblem is solved by semismooth Newton iterations, and the resulting linear systems are transformed into graph Laplacian systems handled by algebraic multigrid. The method is stated to cover a large class of generalized optimal transport problems, including partial transport, and to achieve global linear or superlinear convergence for function residual and feasibility violation via a Lyapunov analysis (Hu et al., 2022).

For high-dimensional settings, slicing provides another scalable route. Sliced optimal partial transport defines

$\bT^\top \mathbf{1} = \bq$1

with Monte Carlo approximation over random directions. The same work provides a fast one-dimensional primal-dual algorithm, proves that $\bT^\top \mathbf{1} = \bq$2 is a metric when the cost is a $\bT^\top \mathbf{1} = \bq$3-th power of a metric and $\bT^\top \mathbf{1} = \bq$4, and uses slicing to reduce the computational burden of partial transport in high dimension (Bai et al., 2022).

4. Geometric structure, regularity, and free boundaries

A defining analytical feature of partial transport is the emergence of active regions and free boundaries. For bounded domains $\bT^\top \mathbf{1} = \bq$5, with $\bT^\top \mathbf{1} = \bq$6 a non-convex polygonal domain and $\bT^\top \mathbf{1} = \bq$7, there is an active region $\bT^\top \mathbf{1} = \bq$8 of mass $\bT^\top \mathbf{1} = \bq$9 that is transported to $\Pi^u(\bp, \bq) = \bigg\{ \bT \in \mathbb{R}_+^{n \times m} : \bT \mathbf{1} \leq \bp,\, \bT^\top \mathbf{1} \leq \bq,\, \mathbf{1}^\top \bT \mathbf{1} = s \bigg\},$0, and the free boundaries are

$\Pi^u(\bp, \bq) = \bigg\{ \bT \in \mathbb{R}_+^{n \times m} : \bT \mathbf{1} \leq \bp,\, \bT^\top \mathbf{1} \leq \bq,\, \mathbf{1}^\top \bT \mathbf{1} = s \bigg\},$1

When $\Pi^u(\bp, \bq) = \bigg\{ \bT \in \mathbb{R}_+^{n \times m} : \bT \mathbf{1} \leq \bp,\, \bT^\top \mathbf{1} \leq \bq,\, \mathbf{1}^\top \bT \mathbf{1} = s \bigg\},$2 and $\Pi^u(\bp, \bq) = \bigg\{ \bT \in \mathbb{R}_+^{n \times m} : \bT \mathbf{1} \leq \bp,\, \bT^\top \mathbf{1} \leq \bq,\, \mathbf{1}^\top \bT \mathbf{1} = s \bigg\},$3 are separated by a straight line, the free boundary $\Pi^u(\bp, \bq) = \bigg\{ \bT \in \mathbb{R}_+^{n \times m} : \bT \mathbf{1} \leq \bp,\, \bT^\top \mathbf{1} \leq \bq,\, \mathbf{1}^\top \bT \mathbf{1} = s \bigg\},$4 is smooth except for a finite number of singular points. The singular set consists of points mapped to vertices of $\Pi^u(\bp, \bq) = \bigg\{ \bT \in \mathbb{R}_+^{n \times m} : \bT \mathbf{1} \leq \bp,\, \bT^\top \mathbf{1} \leq \bq,\, \mathbf{1}^\top \bT \mathbf{1} = s \bigg\},$5 and points whose normal subdifferential contacts multiple open edges; away from these finite exceptional sets, the free boundary is locally smooth (Chen et al., 2023).

The same paper gives a corresponding result for full transport to non-convex polygonal targets: the singular set of the convex potential is locally a $\Pi^u(\bp, \bq) = \bigg\{ \bT \in \mathbb{R}_+^{n \times m} : \bT \mathbf{1} \leq \bp,\, \bT^\top \mathbf{1} \leq \bq,\, \mathbf{1}^\top \bT \mathbf{1} = s \bigg\},$6-dimensional smooth curve, except for a finite set of points. Its analysis uses localization of sublevel sets, the interior ball property, Monge–Ampère regularity theory, convex duality, and a bootstrapping argument from $\Pi^u(\bp, \bq) = \bigg\{ \bT \in \mathbb{R}_+^{n \times m} : \bT \mathbf{1} \leq \bp,\, \bT^\top \mathbf{1} \leq \bq,\, \mathbf{1}^\top \bT \mathbf{1} = s \bigg\},$7 to $\Pi^u(\bp, \bq) = \bigg\{ \bT \in \mathbb{R}_+^{n \times m} : \bT \mathbf{1} \leq \bp,\, \bT^\top \mathbf{1} \leq \bq,\, \mathbf{1}^\top \bT \mathbf{1} = s \bigg\},$8 (Chen et al., 2023). Within the broader GOPT context, this establishes that partial-transport free boundaries can remain sharply structured even under non-convex geometry.

Regularity questions also appear in barycentric and weak formulations. Weak optimal total variation transport yields generalized Wasserstein distances of Piccoli–Rossi type, including

$\Pi^u(\bp, \bq) = \bigg\{ \bT \in \mathbb{R}_+^{n \times m} : \bT \mathbf{1} \leq \bp,\, \bT^\top \mathbf{1} \leq \bq,\, \mathbf{1}^\top \bT \mathbf{1} = s \bigg\},$9

and supports a barycenter theory with existence for measures with compact support and consistency under weak convergence (Chung et al., 2019). A plausible implication is that free-boundary geometry and generalized barycentric structure are complementary aspects of the same non-conservative transport regime: one describes the active set in physical space, the other the induced geometry on spaces of measures.

5. Structural extensions and neighboring generalizations

One important extension replaces extrinsic pointwise costs by intrinsic structural comparisons. Partial Gromov–Wasserstein transport compares intra-domain geometries and is defined by

$0 < s \leq \min(\|\bp\|_1, \|\bq\|_1)$0

with

$0 < s \leq \min(\|\bp\|_1, \|\bq\|_1)$1

and fused Gromov–Wasserstein adds a trade-off between attribute and structural terms. In topology tracking, merge trees are modeled as measure networks $0 < s \leq \min(\|\bp\|_1, \|\bq\|_1)$2, partial fused Gromov–Wasserstein is used to compute a partial coupling $0 < s \leq \min(\|\bp\|_1, \|\bq\|_1)$3 between adjacent time steps, and the resulting coupling is interpreted as a probabilistic tracking graph. The same work proves the stability estimate

$0 < s \leq \min(\|\bp\|_1, \|\bq\|_1)$4

for merge trees encoded via the LCA matrix (Li et al., 2023).

Multi-marginal partial transport generalizes the two-marginal problem to measures $0 < s \leq \min(\|\bp\|_1, \|\bq\|_1)$5 and a transported mass $0 < s \leq \min(\|\bp\|_1, \|\bq\|_1)$6, with admissible plans in $0 < s \leq \min(\|\bp\|_1, \|\bq\|_1)$7 and quadratic cost

$0 < s \leq \min(\|\bp\|_1, \|\bq\|_1)$8

Under the sharp condition $0 < s \leq \min(\|\bp\|_1, \|\bq\|_1)$9, where $PW_p^p(\bp,\bq) = \min_{\bT \in \Pi^u(\bp,\bq)} \langle \bC, \bT \rangle_F$0, the optimal plan is unique. The work also establishes equivalence with a Monge–Kantorovich partial barycenter problem and shows that monotonicity of active marginals may fail for $PW_p^p(\bp,\bq) = \min_{\bT \in \Pi^u(\bp,\bq)} \langle \bC, \bT \rangle_F$1, unlike in the two-marginal case (Kitagawa et al., 2014).

In a different direction, grand-canonical optimal transport allows the number of marginals to fluctuate. The admissible object is a family $PW_p^p(\bp,\bq) = \min_{\bT \in \Pi^u(\bp,\bq)} \langle \bC, \bT \rangle_F$2 with

$PW_p^p(\bp,\bq) = \min_{\bT \in \Pi^u(\bp,\bq)} \langle \bC, \bT \rangle_F$3

and average density

$PW_p^p(\bp,\bq) = \min_{\bT \in \Pi^u(\bp,\bq)} \langle \bC, \bT \rangle_F$4

The associated cost is

$PW_p^p(\bp,\bq) = \min_{\bT \in \Pi^u(\bp,\bq)} \langle \bC, \bT \rangle_F$5

and the formulation is presented as a generalization of classical multi-marginal OT that also relates to partial and unbalanced transport (Marino et al., 2022). This suggests that GOPT sits within a larger family of transport models in which either the amount of transported mass, the admissible support, or even the number of participating marginals becomes variable.

A further terminological extension appears in econometrics, where “Generalized Optimal Transport” denotes an infinite-dimensional optimization framework over Borel measures constrained by arbitrary identified joint subdistributions and structural conditions such as independence or conditional independence. The model is encoded through moment equalities of characteristic kernels and reduced, via duality and approximation theory, to finite-dimensional convex programs. In the special case of empirical optimal transport with Lipschitz cost, the resulting estimator is stated to be $PW_p^p(\bp,\bq) = \min_{\bT \in \Pi^u(\bp,\bq)} \langle \bC, \bT \rangle_F$6-uniformly valid from one side (Voronin, 30 Jul 2025). This use of the term is distinct from function-weighted partial transport, but it reinforces the broader pattern that “generalized” transport often means replacing fixed marginal constraints by richer collections of admissibility conditions.

6. Applications

Machine learning has been a major application area for generalized partial transport. In positive-unlabeled learning, partial Wasserstein transport is used to move a mass equivalent to the class prior $PW_p^p(\bp,\bq) = \min_{\bT \in \Pi^u(\bp,\bq)} \langle \bC, \bT \rangle_F$7 from unlabeled to positive samples, with negatives assigned to a dummy point; an optional group-lasso penalty on the transport plan enforces cleaner separation. On UCI, MNIST, and domain adaptation datasets, partial-W and partial-GW are reported to match or outperform state-of-the-art PU methods. In standard SCAR settings, partial-W usually performs best, while under selection bias (SAR) or feature distribution shift, partial-GW remains accurate because it exploits intra-domain structure (Chapel et al., 2020).

IC-POT extends this application pattern to settings where rejection must depend on side information. The reported examples include positive-unlabeled learning, open-partial domain adaptation, and multi-modal satellite ocean measurements. In the PU and OPDA cases, pointwise rejection rules encode local selection probability, posterior confidence, or neighborhood-based confidence rather than a blanket global rule. In the geophysical case, physical priors and sensor reliability maps define $PW_p^p(\bp,\bq) = \min_{\bT \in \Pi^u(\bp,\bq)} \langle \bC, \bT \rangle_F$8 so that only physically meaningful or available data are compared (Tripathi et al., 19 May 2026).

Topology and scientific visualization provide a structurally different application. Using partial fused Gromov–Wasserstein couplings between merge trees, topological features in time-varying scalar fields are assigned probabilistically across time, allowing splits and merges to appear as multiple nonzero couplings. The reported datasets include 2D Heated Cylinder, 2D Unsteady Cylinder Flow, 2D von Kármán Vortex Street, 2D Ionization Front, and 3D Hurricane Isabel. Compared with Global Feature Tracking and Lifted Wasserstein Matcher, pFGW yields more continuous, fewer oversegmented, and more robust matchings, especially under subsampling (Li et al., 2023).

Computer vision and geometry processing use partial transport for point cloud registration and color transfer. Sliced optimal partial transport is applied to noisy point cloud registration and is reported to deliver computational and accuracy benefits through one-dimensional slicing (Bai et al., 2022). Partial optimal transport has also been used in color transfer, point-cloud registration, mini-batch OT, and domain adaptation, which is precisely why feasible approximations and correct rounding become important in large-scale settings (Nguyen et al., 2023).

Simulation and computational geometry form another application line. In partial optimal transport-based fluid simulation, generalized Laguerre cells are defined as intersections between Laguerre cells and spheres. An analytic construction computes these restricted cells directly, replacing convex cell clipping and enabling more precise evaluation of cell volume and facet area. The method is stated to support free-surface fluid simulation and deformation mechanics in schemes such as Gallouët–Mérigot and Power Particles, while also providing a rendering framework based on the computed volumetric structure (Plateau--Holleville et al., 9 Jan 2026).

Taken together, these applications show that GOPT is used whenever transport must remain selective: only a prescribed fraction of mass should move, rejection should depend on spatial or statistical reliability, geometry should be matched without a common ambient metric, or free boundaries and unmatched regions carry intrinsic information rather than being nuisances.