Transport applications

Optimal transport has one of the longest application histories in applied mathematics — 240 years from Monge's engineering problem to today's generative model training loops. This page traces where the problem arises, why OT is the right formulation, and how torchmatch's backends map onto each use case.

Supply chain and resource allocation

The first computational use of the transportation problem (the discrete precursor to general OT) was in centrally-planned resource allocation: given factories producing a good and warehouses needing it, at what shipping cost should supply be routed to minimise total freight? Kantorovich developed the LP formulation for exactly this purpose in wartime Soviet planning, and the 1975 Nobel Prize citation mentions "optimal allocation of resources" as the core contribution. Koopmans, sharing the prize, applied the same structure to shipping route optimisation.

The transportation LP — assign continuous mass from supply locations to demand locations at minimum unit transport cost — is the special case where a and b are given histograms and C encodes geographic distance. It remains a daily tool in logistics optimisation and supply chain planning.

Image retrieval and perceptual similarity

The Earth Mover's Distance entered computer vision through image retrieval [Rubner2000]. When comparing two images by their colour histograms, Euclidean distance between bin counts is insensitive to perceptually small shifts: moving probability mass from the "red" bin to the "orange" bin registers as a large L2 distance but a small perceptual distance. OT charges the actual cost of moving mass between bins according to the colour-space ground metric (a distance function defined over the colour space, e.g. Euclidean distance in RGB or Lab), so nearby colours are treated as similar.

The same idea extends to texture descriptors, shape histograms, and any feature whose natural distance is not Euclidean on the raw representation. The EXACT_EMD backend in torchmatch computes the exact earth mover's distance for small histograms where no regularisation is desired.

Colour transfer and style matching

Histogram matching via OT underlies a family of image stylisation techniques. Given a source image and a target whose colour statistics you want to adopt, computing the 1D or 3D OT map between their colour histograms and applying it as a pixel-wise transformation transfers the palette of the target to the source with minimal perceptual distortion. Pitié, Kokaram, and Dahyot [Pitie2007] formalised this as iterated 1D projections (sliced Wasserstein transport — an approximation that computes 1D OT on random projections of the distribution, avoiding the full N-dimensional cost), an approximation that scales to full 3D colour histograms without materialising an n³ cost.

Generative modelling

Wasserstein GAN [Arjovsky2017] reframed generative adversarial training in terms of the Wasserstein-1 distance between the real and generated distributions. The Wasserstein distance is meaningful even when the two distributions have disjoint support (which commonly occurs during early GAN training), whereas Jensen-Shannon divergence and KL divergence both saturate to a constant in this regime, providing no gradient signal. The practical implementation uses the Kantorovich-Rubinstein dual (the equivalent formulation of W1 as a supremum over 1-Lipschitz functions, used to train the discriminator) with gradient-penalised discriminators rather than directly computing the OT plan, but the connection motivates the loss design.

Wasserstein autoencoders (WAE) [Tolstikhin2018] replaced the evidence lower bound of the VAE with a Wasserstein distance between the aggregate posterior and the prior, producing sharper reconstructions on image benchmarks.

Flow matching [Lipman2022]; [Liu2022] frames diffusion-model training as learning a vector field that transports a source distribution (Gaussian noise) to a target distribution (data). The "OT-conditioned flow matching" variant conditions the flow on the OT displacement plan between individual noise samples and data samples, producing straighter trajectories and faster inference.

For all three model classes, transport.matrix.solve with SINKHORN_DIVERGENCE or transport.samples.loss with debias=True provides a differentiable Wasserstein-like training loss that can be substituted for or combined with the standard objectives.

Domain adaptation

A model trained on a labelled source domain often fails on an unlabelled target domain because the marginal feature distributions differ. OT provides a principled way to measure and correct this discrepancy: it computes a transport plan between source and target feature distributions, then uses that plan to move source features toward the target domain before training a classifier.

• OTDA [Courty2017] computes the regularised OT plan between labelled source samples and unlabelled target samples in the feature space of a pretrained network. The plan then transports source features toward the target, producing pseudo-labelled target samples that train a target-domain classifier.
• DeepJDOT [Damodaran2018] integrates OT alignment into an end-to-end deep network: the OT plan between source and target minibatches is recomputed each iteration and used as a re-weighting of the classification loss.
• Distribution matching for dataset distillation: the Sinkhorn divergence between feature distributions of real and distilled datasets serves as the distillation objective, requiring a geometry-aware distance that respects the ground metric of feature space.

The UNBALANCED_SINKHORN backend is particularly useful when source and target have genuinely different class distributions: relaxing the marginal constraints prevents the coupling from being dominated by classes that are abundant in one domain but rare in the other.

Geometric deep learning and 3D vision

Point cloud registration and shape matching require finding correspondences between two unordered point sets. OT provides the soft coupling that minimises transport cost, giving a probabilistic correspondence matrix as the plan. This soft matching initialises or replaces ICP (iterative closest point) in registration pipelines, is differentiable with respect to the point positions, and scales to large point clouds via the samples.loss streaming kernel.

Wasserstein barycenters [Agueh2011] — weighted averages of distributions in Wasserstein space — produce interpolations between shapes that respect the geometry of the ground metric. Interpolating between two 3D shapes in Wasserstein space moves each point smoothly toward its corresponding point in the target, unlike Euclidean averaging which collapses the shape when distributions are disjoint.

Shape completion and generation: point-cloud generative models (PointFlow, ShapeGF, DPM-based point clouds) use Wasserstein or Sinkhorn losses to supervise the generated shape. transport.samples.loss is directly applicable; the streaming Triton kernel avoids the N × M allocation that would dominate memory at generation-scale cloud sizes.

Natural language processing

Word Mover's Distance [Kusner2015] embeds documents as distributions over word embedding vectors (weighted by TF-IDF or uniform) and computes the OT cost between them using pretrained embeddings as the ground metric. The resulting distance is insensitive to synonyms and paraphrases — moving mass from "automobile" to "car" is cheap because the embeddings are nearby — and outperforms bag-of-words similarity on several retrieval benchmarks.

The same idea applies to sentence-level alignment for mining parallel corpora, where OT between sentence embedding distributions identifies likely translations without requiring exact string matches.

Computational biology

Waddington-OT [Schiebinger2019] modelled cellular differentiation as an OT problem: single-cell RNA-seq profiles from adjacent time points define the source and target distributions over gene-expression space, and the OT plan between them gives the most parsimonious account of which early cells give rise to which later cells. This transport interpretation respects the Waddington epigenetic landscape metaphor and produces biologically interpretable developmental trajectories without requiring paired data.

Moscot [Klein2023] scaled this framework to million-cell datasets and extended it to spatial transcriptomics alignment, single-cell multi-omics integration (matching cells across RNA and protein measurements), and lineage tracing. The computational backbone is log-domain Sinkhorn with online batching — structurally identical to LOG_SINKHORN — applied at dataset scales that require careful memory management.

Protein structure comparison: OT between residue-coordinate distributions provides a rotation-invariant distance between two protein chains that accounts for insertions, deletions, and loop flexibility without requiring global structural alignment.