--- title: Generalized Optimal Partial Transport (GOPT) url: https://www.emergentmind.com/topics/generalized-optimal-partial-transport-gopt type: topic --- # Generalized Optimal Partial Transport (GOPT) 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 [2407.06481]. 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 [2002.08276][2605.20030][1401.7255][2201.06859]. 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 $\bT^\top \mathbf{1} = \bq$ [2002.08276]. 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 \[ PW_p^p(\bp,\bq) = \min_{\bT \in \Pi^u(\bp,\bq)} \langle \bC, \bT \rangle_F \] [2002.08276]. The GOPT formulation introduced in 2024 starts from classical optimal partial transport with a total-variation penalty: \[ OPT(\mu, \nu; \lambda) = \inf_{\gamma \in \mathcal{M}_+(\Omega^2)} \langle c, \gamma \rangle + \lambda \left[TV(\gamma_1 \parallel \mu) + TV(\gamma_2 \parallel \nu)\right] \] or equivalently \[ OPT(\mu, \nu; \lambda) = \inf_{\gamma \in \Gamma_\leq(\mu,\nu)} \langle c, \gamma \rangle + \lambda\Big(\|\mu - \gamma_1\| + \|\nu - \gamma_2\|\Big), \] where $\Gamma_\leq(\mu,\nu)=\{\gamma \in \mathcal{M}_+(\Omega^2): \gamma_1 \leq \mu, \gamma_2 \leq \nu\}$. GOPT replaces the constant $\lambda$ by spatially varying functions $\lambda_1,\lambda_2:\Omega\to\mathbb{R}_+$ and defines \[ GOPT(\mu, \nu; \lambda_1, \lambda_2) := \inf_{\gamma \in \mathcal{M}_+(\Omega^2)} \, \langle c, \gamma \rangle + \mathcal{R}_1(\gamma_1 \parallel \mu) + \mathcal{R}_2(\gamma_2 \parallel \nu), \] with choices such as $TV(\lambda_1 \gamma_1 \parallel \lambda_1 \mu)$ or $\overline{PTV}(\lambda_1 \gamma_1 \parallel \lambda_1 \mu)$, and similarly on the target side. Classical OPT is recovered when $\lambda_1=\lambda_2=\lambda$ [2407.06481]. 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 $c_s \in \mathbb{R}_+^n$ and $c_t \in \mathbb{R}_+^m$: \[ \begin{aligned} \min_{P, u, v} \quad & \langle C, P \rangle + \langle c_s, u \rangle + \langle c_t, v \rangle \\ \text{s.t.} \quad & P\mathbf{1}_m + u = \mu \\ & P^\top \mathbf{1}_n + v = \nu \\ & P_{ij} \geq 0,\ u_i \geq 0,\ v_j \geq 0. \end{aligned} \] Its reduced sub-coupling form is \[ \min_{P \in \Gamma_{\le}(\mu, \nu)} \sum_{i=1}^n \sum_{j=1}^m \bigl(C_{ij} - c_s(i) - c_t(j)\bigr) P_{ij}, \] which shows that constant rebates recover classical partial transport [2605.20030]. ## 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 \[ \sup_{\phi, \psi} \quad -\epsilon \left\langle e^{\frac{\phi \oplus \psi}{\epsilon} -1} K, dxdy \right\rangle + \langle \min(\phi, \lambda_1),\mu\rangle + \langle \min(\psi, \lambda_2), \nu\rangle, \] where $K=e^{-c/\epsilon}$ and the feasible set depends on whether the penalty uses $TV$ or $PTV$. Setting $\lambda_1,\lambda_2$ constant recovers the classical OPT dual, while $\epsilon\to 0$ yields the unregularized problem if the minimizer is absolutely continuous [2407.06481]. When both penalties are $PTV$, GOPT admits an explicit linear reformulation: \[ GOPT(\mu, \nu) = \inf_{\gamma \in \Gamma_\leq(\mu, \nu)} \langle c - (\lambda_1 \oplus \lambda_2), \gamma \rangle + \langle \lambda_1, \mu \rangle + \langle \lambda_2, \nu \rangle. \] This leads to an augmented-space reduction with augmented marginals $\hat{\mu} = \mu + |\nu|\delta_{\hat{\infty}}$, $\hat{\nu} = \nu + |\mu|\delta_{\hat{\infty}}$, and an augmented cost with zeros on dummy locations, enabling the use of classical OT linear programming solvers [2407.06481]. IC-POT has a dual with explicit local caps: \[ \begin{aligned} \max_{f, g} \quad & \sum_{i=1}^n f_i \mu_i + \sum_{j=1}^m g_j \nu_j \\ \text{s.t.} \quad & f_i + g_j \leq C_{ij} \quad \forall i, j \\ & f_i \leq c_s(i) \quad \forall i \\ & g_j \leq c_t(j) \quad \forall j. \end{aligned} \] In this form, $f_i \leq c_s(i)$ and $g_j \leq c_t(j)$ act as local acceptance thresholds. Complementary slackness gives \[ P_{ij} > 0 \implies f_i + g_j = C_{ij}, \qquad u_i > 0 \implies f_i = c_s(i),\quad v_j > 0 \implies g_j = c_t(j), \] and the admissible-support condition \[ C_{ij} > c_s(i) + c_t(j) \implies P_{ij}=0 \] holds in every optimal solution. IC-POT is also equivalent to balanced Kantorovich OT on an augmented support with \[ \bar{\mu} = (\mu, \|\nu\|_1), \quad \bar{\nu} = (\nu, \|\mu\|_1),\quad \bar{C} = \begin{pmatrix} C & c_s \\ c_t^\top & 0 \end{pmatrix}, \] although proper equivalence fails under entropic regularization [2605.20030]. 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 $P(a)=D(a)$ 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 [1610.02940]. 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 [1909.05517]. ## 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 $s$, one adds a new bin $n+1$ to $\bp$ with mass $p_{n+1} = \|\bq\|_1 - s$, a new bin $m+1$ to $\bq$ with mass $q_{m+1} = \|\bp\|_1 - s$, and the extended cost \[ \overline{\bC} = \begin{bmatrix} \bC & \xi \mathbf{1} \\ \xi \mathbf{1}^\top & 2\xi + A \end{bmatrix}, \] with $A > \max C_{ij}$. The top-left block of the optimal transport plan on the extended problem is the exact partial plan, and \[ W_p^p(\overline{\bp}, \overline{\bq}) - PW_p^p(\bp,\bq) = \xi(\|\bp\|_1+\|\bq\|_1-2s). \] 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 [2002.08276]. For entropic GOPT, the generalized Sinkhorn procedure updates dual scalings through proximal-divide operators. With $TV$ penalties, one update has the form \[ u \gets \text{clip}\left(\frac{d\mu/dx}{Kv}, [e^{-\lambda_1/\epsilon}, e^{\lambda_1/\epsilon}]\right), \] while with $PTV$ penalties, \[ u \gets \min\left(\frac{d\mu/dx}{Kv}, e^{\lambda_1/\epsilon}\right), \] and similarly for $v$. This yields Algorithm 1 (opt-Sinkhorn). For non-entropic GOPT with $PTV$ penalties, Algorithm 2 (opt-lp) solves the augmented OT linear program and projects the result back to the original partial problem [2407.06481]. 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 $\mathbf{1}^\top X \mathbf{1}=s$ and shows that the resulting violation can be lower bounded by \[ V := \mathbf{1}^\top \bar{X} \mathbf{1} - s \geq \exp\left(-\frac{12A \log n}{\varepsilon} - O(\log n)\right). \] It proposes a new feasible rounding procedure, Round-POT, and derives a feasible Sinkhorn complexity of $\widetilde O(n^2/\varepsilon^4)$, together with two first-order methods: Adaptive Primal-Dual Accelerated Gradient Descent with complexity $\widetilde O(n^{2.5}/\varepsilon)$ and Dual Extrapolation with complexity $\widetilde O(n^2/\varepsilon)$ [2312.13970]. 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 [2207.14082]. For high-dimensional settings, slicing provides another scalable route. Sliced optimal partial transport defines \[ SOPT_\lambda(\mu, \nu) = \int_{\mathbb{S}^{d-1}} \text{OPT}_{\lambda(\theta)}(\langle \theta, \cdot \rangle_{\#} \mu, \langle \theta, \cdot \rangle_{\#} \nu) \, d\sigma(\theta), \] with Monte Carlo approximation over random directions. The same work provides a fast one-dimensional primal-dual algorithm, proves that $(SOPT_\lambda(\mu,\nu))^{1/p}$ is a metric when the cost is a $p$-th power of a metric and $\lambda(\theta)>0$, and uses slicing to reduce the computational burden of partial transport in high dimension [2212.08049]. ## 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 $\Omega,\Omega^*\subset\mathbb{R}^2$, with $\Omega^*$ a non-convex polygonal domain and $0