Semi-discrete Optimal Transport

• Semi-discrete Optimal Transport is a formulation where a continuous source measure is optimally mapped to a discrete target using piecewise-constant maps and Laguerre partitions.
• The approach leverages a finite-dimensional dual formulation with weighted Voronoi diagrams and concave objectives to ensure mass balance and computational efficiency.
• Regularization techniques and diverse numerical solvers, including Newton and stochastic methods, enhance solution precision across applications in machine learning, PDEs, and geometry.

Searching arXiv for recent and foundational papers on semi-discrete optimal transport to ground the article in current literature. Semi-discrete optimal transport is the optimal transport problem in which one marginal is continuous and the other is discrete. In its standard form, a continuous source measure $\mu$ on a domain $X$ is transported to an atomic target $\nu=\sum_{i=1}^m a_i\delta_{y_i}$ by minimizing transport cost over couplings or, when a Monge solution exists, over piecewise-constant maps that send each source point to one target atom. This setting admits a finite-dimensional dual, a partition interpretation through Laguerre cells, and a computational-geometry realization through weighted Voronoi or power diagrams, which together make it a central interface between optimal transport theory, numerical optimization, and applications in machine learning, geometry, and PDEs (Wolansky, 2019, Khamlich et al., 31 Jul 2025).

1. Formal setup and geometric structure

The semi-discrete problem is typically posed with a continuous source $\mu$ and a discrete target

$$\nu=\sum_{i=1}^{n} w_i\,\delta_{x_i}, \qquad w_i>0,\qquad \sum_i w_i=1.$$

For quadratic cost $c(z,x)=\tfrac12\|z-x\|^2$, one seeks a measurable map $T$ with $T_\#\mu=\nu$ minimizing

$\int c(z,T(z))\,d\mu(z).$

In the semi-discrete setting with quadratic cost and absolutely continuous $\mu$, the optimal coupling is induced by a deterministic transport map, and that map is piecewise constant: each $X$0 is sent to a single atom $X$1 (Kong et al., 16 Oct 2025).

The associated geometry is encoded by Laguerre cells. Given dual potentials $X$2, the $X$3-th cell is

$X$4

These cells form a partition up to boundaries, and for quadratic cost they are weighted Voronoi or power cells. The dual objective is concave: $X$5 with gradient

$X$6

First-order optimality therefore enforces the mass constraints $X$7, and the optimal map is $X$8 on $X$9 (Kong et al., 16 Oct 2025).

This dual description is equivalent to the older partition formulation in which one minimizes $\nu=\sum_{i=1}^m a_i\delta_{y_i}$0 over measurable partitions $\nu=\sum_{i=1}^m a_i\delta_{y_i}$1 with prescribed masses $\nu=\sum_{i=1}^m a_i\delta_{y_i}$2. In the semi-discrete case, the Monge map is exactly the piecewise-constant map induced by such a partition, while the Kantorovich relaxation can be written as a family of nonnegative measures $\nu=\sum_{i=1}^m a_i\delta_{y_i}$3 summing to $\nu=\sum_{i=1}^m a_i\delta_{y_i}$4 and carrying masses $\nu=\sum_{i=1}^m a_i\delta_{y_i}$5 (Wolansky, 2019).

Two structural facts recur throughout the literature. First, the dual potentials are identifiable only up to an additive constant, so one fixes a gauge such as $\nu=\sum_{i=1}^m a_i\delta_{y_i}$6 or $\nu=\sum_{i=1}^m a_i\delta_{y_i}$7. Second, under standard regularity assumptions—absolute continuity of $\nu=\sum_{i=1}^m a_i\delta_{y_i}$8, generic position of the atoms, and suitable twist or convexity hypotheses—the maximizer exists and is unique modulo constants, and the associated cell partition is unique $\nu=\sum_{i=1}^m a_i\delta_{y_i}$9-almost everywhere (2405.14459).

2. Regularization, soft partitions, and asymptotics

A standard modification is entropic regularization. In the semi-discrete setting, adding $\mu$0 yields a reduced dual over discrete potentials $\mu$1,

$\mu$2

or equivalently a concave maximization form $\mu$3. The gradient components are

$\mu$4

where $\mu$5 is a soft assignment. For $\mu$6, the reduced dual is smooth and strictly convex in minimization form, and the hard Laguerre partition is replaced by a soft log-sum-exp partition $\mu$7. As $\mu$8, the soft assignment collapses toward the Laguerre diagram (Khamlich et al., 31 Jul 2025).

This regularization changes both geometry and bias. Larger $\mu$9 improves conditioning and smooths cell interfaces, but it perturbs the exact semi-discrete solution. In the semi-discrete setting this perturbation has an unusual asymptotic signature: the first-order term in the expansion of the regularized cost vanishes, while the second-order term does not. More precisely, for inverse regularization parameter $$\nu=\sum_{i=1}^{n} w_i\,\delta_{x_i}, \qquad w_i>0,\qquad \sum_i w_i=1.$$0, the cost gap satisfies

$$\nu=\sum_{i=1}^{n} w_i\,\delta_{x_i}, \qquad w_i>0,\qquad \sum_i w_i=1.$$1

where $$\nu=\sum_{i=1}^{n} w_i\,\delta_{x_i}, \qquad w_i>0,\qquad \sum_i w_i=1.$$2 is the $$\nu=\sum_{i=1}^{n} w_i\,\delta_{x_i}, \qquad w_i>0,\qquad \sum_i w_i=1.$$3-dimensional integral of $$\nu=\sum_{i=1}^{n} w_i\,\delta_{x_i}, \qquad w_i>0,\qquad \sum_i w_i=1.$$4 along the interface between cells $$\nu=\sum_{i=1}^{n} w_i\,\delta_{x_i}, \qquad w_i>0,\qquad \sum_i w_i=1.$$5 and $$\nu=\sum_{i=1}^{n} w_i\,\delta_{x_i}, \qquad w_i>0,\qquad \sum_i w_i=1.$$6. This distinguishes the semi-discrete regime from the discrete-discrete case, where the suboptimality is exponentially small, and from the continuous-continuous case, where the cost-only suboptimality is typically $$\nu=\sum_{i=1}^{n} w_i\,\delta_{x_i}, \qquad w_i>0,\qquad \sum_i w_i=1.$$7 (Altschuler et al., 2021).

Regularization also appears in nonstandard ways. In one-dimensional semi-discrete partial optimal transport with quadratic cost, the unregularized dual has reduced regularity. A geometric regularization is introduced by thickening the density along an auxiliary dimension; the resulting regularized dual becomes $$\nu=\sum_{i=1}^{n} w_i\,\delta_{x_i}, \qquad w_i>0,\qquad \sum_i w_i=1.$$8, its maximizer is unique, and the maximizers converge to those of the original problem at quadratic rate in the thickness parameter (Cances et al., 10 Sep 2025).

A common misconception is that regularization in semi-discrete OT is merely a numerical convenience. The asymptotic analysis shows instead that regularization interacts in a highly structured way with the discontinuity set of the unregularized map: the leading constant depends explicitly on the density of $$\nu=\sum_{i=1}^{n} w_i\,\delta_{x_i}, \qquad w_i>0,\qquad \sum_i w_i=1.$$9 at cell interfaces, not on bulk behavior alone (Altschuler et al., 2021).

3. Solver families and numerical architectures

Semi-discrete OT is computationally attractive because the dual lives in $c(z,x)=\tfrac12\|z-x\|^2$0, but this does not make all instances easy. The classical numerical route is to optimize dual weights and repeatedly compute cell masses. For quadratic cost, the Hessian can be written in terms of interface integrals, and for costs given by positive combinations of $c(z,x)=\tfrac12\|z-x\|^2$1-norms with $c(z,x)=\tfrac12\|z-x\|^2$2, Laguerre cells are star-shaped with respect to the target points. That star-shapedness permits a robust Newton implementation based on boundary tracing and analytic Hessian assembly, with local quadratic convergence on the subspace orthogonal to constants (Dieci et al., 2023).

For entropy-regularized semi-discrete OT on complex domains, finite-element discretization turns the continuous integral into quadrature over $c(z,x)=\tfrac12\|z-x\|^2$3 source points, after which the naive cost of a dual evaluation is $c(z,x)=\tfrac12\|z-x\|^2$4. A recent large-scale framework reduces this bottleneck by combining distance-based truncation, fast spatial queries using R-trees, source and target multilevel hierarchies, and $c(z,x)=\tfrac12\|z-x\|^2$5-scaling. The reported effects are substantial: target-only hierarchy with softmax refinement reduced finest-level iterations to $c(z,x)=\tfrac12\|z-x\|^2$6–$c(z,x)=\tfrac12\|z-x\|^2$7 and produced up to $c(z,x)=\tfrac12\|z-x\|^2$8 speedup, while strong scaling reached $c(z,x)=\tfrac12\|z-x\|^2$9 on $T$0 nodes (Khamlich et al., 31 Jul 2025).

A different line of work characterizes the regularized semi-discrete solution by an ODE in the regularization parameter. In that formulation, the minimizers $T$1 solve

$T$2

with explicit initial condition at $T$3. This yields an RK3 continuation method that is competitive with Newton for squared Euclidean cost and performs better for powered Euclidean costs, especially when the target points lie outside the support of the source (Nenna et al., 3 Apr 2025).

Stochastic first-order methods are now equally important. In one-sample semi-discrete OT, one can sample only from $T$4 and store the discrete target $T$5, producing unbiased stochastic gradients of the semi-dual. Projected SGD, mini-batching, Polyak–Ruppert averaging, and decreasing entropic regularization all fit naturally in this setting, and their per-iteration cost is linear in the number of target atoms times dimension (2405.14459, Genans et al., 31 Oct 2025, Genans et al., 29 Oct 2025).

In practice, these solver families are complementary. Newton and geometric power-diagram methods remain natural when exact cell geometry is accessible; FE and truncation-based methods dominate on complex geometries and densities; stochastic solvers are preferable when $T$6 is only sampled, dimensions are high, or semi-discrete OT is embedded inside a larger learning loop.

4. Statistical theory and optimization rates

A distinctive feature of semi-discrete OT is that the discrete target can be known exactly while only the continuous source is sampled. This one-sample regime is statistically easier than the two-sample regime. For semi-discrete map estimation, the minimax lower bound in the one-sample setting is $T$7 for the OT map and $T$8 for the discrete Kantorovich potential, compared with the classical $T$9 map rate when both measures are sampled (2405.14459).

Projected SGD with adaptive entropic regularization exploits this gap. In the DRAG algorithm, one performs stochastic gradient updates on the entropic semi-dual with schedules $T_\#\mu=\nu$0 and $T_\#\mu=\nu$1, together with projection onto a bounded gauge-fixed set and Polyak–Ruppert averaging. The analysis shows that decreasing regularization is not merely heuristic: it accelerates convergence relative to fixed-$T_\#\mu=\nu$2 schemes and achieves unbiased $T_\#\mu=\nu$3 sample and iteration complexity for the OT cost and potential estimation, together with a $T_\#\mu=\nu$4 rate for the OT map (Genans et al., 31 Oct 2025).

A complementary 2025 analysis studies averaged projected SGD directly on the unregularized semi-dual. It identifies a projection set that contains a minimizer even when the source measure is not compactly supported, and proves computational and statistical convergence for MTW costs, including $T_\#\mu=\nu$5 with $T_\#\mu=\nu$6. In that framework, the OT map estimator attains the minimax $T_\#\mu=\nu$7 rate, while the averaged potential estimator reaches $T_\#\mu=\nu$8 under additional Hölder regularity (Genans et al., 29 Oct 2025).

These results refine earlier rate statements for projected SGD with adaptive regularization. In that earlier theory, non-averaged iterates track the regularized minimizer at rate $T_\#\mu=\nu$9 in mean squared norm, while Polyak–Ruppert averaging yields $\int c(z,T(z))\,d\mu(z).$0 for the potential and $\int c(z,T(z))\,d\mu(z).$1 for the map, so that letting $\int c(z,T(z))\,d\mu(z).$2 nearly recovers the minimax lower bounds (2405.14459).

A common misconception is that semi-discrete OT inherits only generic nonsmooth stochastic-optimization rates. The recent theory shows that the finite-dimensional semi-dual, local strong convexity on the gauge-fixed subspace, and the geometry of Laguerre cells together permit sharp statistical statements all the way to minimax map estimation (Genans et al., 29 Oct 2025).

5. Hardness, degeneracies, and other limitations

The finite-dimensional dual does not remove worst-case computational hardness. Semi-discrete OT with cost $\int c(z,T(z))\,d\mu(z).$3 is $\int c(z,T(z))\,d\mu(z).$4-hard even when one measure is Lebesgue on the unit hypercube and the other is supported on only two points. This hardness result gives a theoretical basis for the long-standing perception that exact semi-discrete Wasserstein computation can be difficult, and it motivates approximation, smoothing, and stochastic optimization (Taskesen et al., 2021).

Geometric degeneracies create a second class of limitations. When the support of the source measure is disconnected, the Hessian can lose a useful spectral gap unless the target masses satisfy a subset-sum separation condition. A two-stage algorithm addresses this setting by combining regularized gradient descent with damped Newton, establishing global linear convergence, local superlinear convergence, and convergence of Laguerre cells in both symmetric-difference and Hausdorff senses (Bansil, 2019).

Topology can also be a hidden failure mode. In triangular-mesh implementations of geometric semi-discrete OT, weighted Delaunay flips are often used to maintain convexity and nonempty cells, but those flips alter mesh connectivity. QC-OT shows that this is not harmless in applications requiring non-flip physical deformations: it relaxes Delaunay and convexity checks, then uses quasiconformal correction to guarantee orientation preservation and topology preservation of the transport map (Lv et al., 2 Jul 2025).

Reduced regularity is another source of difficulty. In unidimensional semi-discrete partial optimal transport, the dual functional is less regular than in higher dimensions, which is why thickening regularization is introduced before Newton-type methods are applied (Cances et al., 10 Sep 2025). This suggests that semi-discrete OT is not a monolithic class: even within quadratic cost, dimension and mass constraints can qualitatively alter the optimization landscape.

The literature therefore supports an objective correction to a frequent overstatement: semi-discrete OT is structurally simpler than continuous-continuous OT, but it is not uniformly easy. Hardness results, disconnected-support pathologies, mesh-topology issues, and reduced regularity all persist in important regimes (Taskesen et al., 2021).

6. Applications and domain-specific extensions

Semi-discrete OT now appears in a wide spectrum of application domains. In flow-based generative modeling, AlignFlow computes a semi-discrete OT map from a continuous noise law to a discrete empirical dataset, then uses the resulting deterministic Noise–Data Alignment as a plug-and-play replacement for independent sampling in Flow Matching, Shortcut Models, MeanFlow, and related architectures. The reported overhead is negligible—less than $\int c(z,T(z))\,d\mu(z).$5 of training time for the SDOT stage—and the method improves FID and convergence on CIFAR-10 and ImageNet256 (Kong et al., 16 Oct 2025).

A related generative construction, DPM-OT, computes a semi-discrete OT map between Gaussian noise and early diffusion latents. The resulting “expressway” from prior to data latent space allows generation in around $\int c(z,T(z))\,d\mu(z).$6–$\int c(z,T(z))\,d\mu(z).$7 function evaluations while mitigating mode mixture through the discontinuous geometry of semi-discrete Brenier maps (Li et al., 2023).

In computational geometry and scientific computing, entropy-regularized semi-discrete OT has been used for Wasserstein barycenters, shape registration, and blue noise sampling on manifolds. The FE-based framework with truncation, multilevel, and $\int c(z,T(z))\,d\mu(z).$8-scaling produces high-resolution barycenters of complex $\int c(z,T(z))\,d\mu(z).$9D vascular geometries, smooth deformation fields in registration, and manifold-adapted sampling on $\mu$0 (Khamlich et al., 31 Jul 2025). A symmetrized semi-discrete construction further introduces coupled power diagrams with identical geometry, enabling interpolation by linear interpolation of mesh vertices and joint sampling of measures supported on polygonal or polyhedral domains (Herrou et al., 2022).

In geophysical PDEs, semi-discrete OT provides a constructive route to the semi-geostrophic equations. Discrete solutions in geostrophic coordinates are characterized by an ODE on seed locations coupled through Laguerre tessellations, yielding global-in-time weak solutions for compactly supported initial data and a practical numerical method based on alternating semi-discrete OT solves and ODE integration (Bourne et al., 2020).

Several recent extensions move beyond classical expected-cost OT. In “Semidiscrete optimal transport with unknown costs,” the cost functions are linear but unknown and can only be learned by noisy single-function observations; the proposed semi-myopic algorithm combines stochastic approximation with online regression and attains $\mu$1 rates for dual and regression error, with $\mu$2 decision error (Zhu et al., 2023). In “Quantile optimization in semidiscrete optimal transport,” the objective is a quantile of the transport cost rather than its mean; the solution requires randomized tie-breaking that preserves the target marginal and produces a geometric structure different from classical power diagrams (Zhu et al., 11 Feb 2026). In one-dimensional partial transport, the semi-discrete formulation is motivated by risk management, crowd motion, and sliced partial transport for point-cloud registration (Cances et al., 10 Sep 2025).

These developments suggest a broad shift in how semi-discrete OT is used. It is no longer only a geometric solver for transporting a density to atoms. It has become a modular template: exact partition geometry when needed, smooth or stochastic approximations when scale demands it, and application-specific variants—partial, quantile, unknown-cost, topology-preserving—when the classical Monge–Kantorovich objective is not the right model.