Hierarchical Optimal Transport: Concepts & Applications

• Hierarchical Optimal Transport is a framework that compares structured collections of probability measures by lifting OT problems to higher-level abstractions.
• It leverages lower-level OT computations, such as Wasserstein distances and entropy regularization, to efficiently address complex alignment challenges.
• HOT is applied in diverse domains like domain adaptation, document analysis, and histopathology, offering both theoretical guarantees and practical scalability.

Searching arXiv for papers on Hierarchical Optimal Transport to ground the article in current literature. Hierarchical Optimal Transport (HOT) denotes a family of optimal transport constructions in which the transported objects are themselves organized into higher-level structures—such as classes, clusters, topics, slides, tiles, views, modalities, layers, or node groups—and the transport problem is lifted from points to structured collections of probability measures. In common formulations, source and target are represented as measures over measures, the ground cost of the upper-level problem is itself a Wasserstein or OT distance between lower-level distributions, and the resulting hierarchy is used to preserve structure, define new divergences, or reduce the effective solution space of otherwise intractable alignment problems (Hamri et al., 2021, Hamri et al., 2022, Yurochkin et al., 2019, Yeaton et al., 2022).

1. Mathematical formulation

Classical discrete optimal transport compares two empirical measures,

$$\mu=\sum_{i=1}^n a_i\delta_{x_i},\qquad \nu=\sum_{j=1}^m b_j\delta_{y_j},$$

by solving

$$(\mathcal D_{\mathcal{MK}})\quad \min_{\gamma\in U(a,b)} \langle \gamma,C\rangle_F,$$

where $C$ is the ground cost matrix and $U(a,b)$ is the set of couplings with prescribed marginals. HOT replaces this flat point-to-point comparison with transport between structured components, so that the outer problem acts on distributions whose support consists of lower-level probability measures rather than individual samples (Hamri et al., 2021).

A standard HOT construction considers two sets of measures,

$$\theta=(\mu_1,\ldots,\mu_h),\qquad \vartheta=(\nu_1,\ldots,\nu_l),$$

and their empirical distributions

$$\phi=\sum_{i=1}^h \alpha_i\delta_{\mu_i},\qquad \varphi=\sum_{j=1}^l \beta_j\delta_{\nu_j}.$$

The corresponding HOT problem is

$$(\mathcal{HOT})\quad \min_{\Gamma\in U(\alpha,\beta)} \langle \Gamma,\mathcal W\rangle_F, \qquad \mathcal W_{i,j}=W_p^p(\mu_i,\nu_j),$$

so the upper-level ground cost is a Wasserstein distance between lower-level measures. For computational efficiency, entropy regularization is often used: $$\min_{\Gamma\in U(\alpha,\beta)} \langle \Gamma,\mathcal W\rangle_F-\varepsilon \mathcal H(\Gamma).$$ This formulation appears explicitly in HOT-based domain adaptation, where source classes and target clusters are matched by transporting probability mass between class-conditioned and cluster-conditioned empirical distributions (Hamri et al., 2021).

A closely related theoretical object is the Hierarchical Wasserstein distance,

$$\mathcal{HW}_{p}(\phi,\varphi) = \left( \inf_{\eta\in\Pi(\phi,\varphi)} \int_{\mathcal P_p(\mathcal X)^2} \mathcal W_p(\rho,\varrho)^p\,d\eta(\rho,\varrho) \right)^{1/p},$$

defined on probability measures over $\mathcal P_p(\mathcal X)$. In the domain-adaptation theory built around this construction, $$(\mathcal D_{\mathcal{MK}})\quad \min_{\gamma\in U(a,b)} \langle \gamma,C\rangle_F,$$0 is a true metric on the hierarchical Wasserstein space $$(\mathcal D_{\mathcal{MK}})\quad \min_{\gamma\in U(a,b)} \langle \gamma,C\rangle_F,$$1, and it satisfies a hierarchical monotonicity property,

$$(\mathcal D_{\mathcal{MK}})\quad \min_{\gamma\in U(a,b)} \langle \gamma,C\rangle_F,$$2

where $$(\mathcal D_{\mathcal{MK}})\quad \min_{\gamma\in U(a,b)} \langle \gamma,C\rangle_F,$$3 are the corresponding flattened measures (Hamri et al., 2022).

2. Structural patterns across HOT models

Across the literature, HOT is instantiated by choosing a hierarchy that reflects the data-generating structure and then defining an outer OT whose cost matrix is induced by lower-level OT or Wasserstein computations. The same principle appears in domain adaptation, document comparison, histopathology, multi-view learning, graph matching, and representational alignment, but the meaning of “hierarchy” depends on the application (Yurochkin et al., 2019, Yeaton et al., 2022, Luo et al., 2020, Cheng et al., 2023, Shah et al., 2 Oct 2025).

Setting	Hierarchy	Upper-level cost
Domain adaptation	classes / target clusters	Wasserstein between class and cluster measures
Document representation	documents / topics / words	WMD between topics
Histopathology	datasets / slides / tiles	slide OT built from tile OT
Multi-view learning	views / prototype clusters	sliced Wasserstein-based costs
Multi-network alignment	networks / aligned clusters / nodes	per-cluster MFGW

In document representation, documents are modeled as distributions over topics and topics as distributions over words. The hierarchical optimal topic transport distance is

$$(\mathcal D_{\mathcal{MK}})\quad \min_{\gamma\in U(a,b)} \langle \gamma,C\rangle_F,$$4

where the ground cost between topics is the word mover’s distance between their word distributions. Under uniqueness of the topic decomposition and distinctness of topics, this construction defines a metric on document space, and when $$(\mathcal D_{\mathcal{MK}})\quad \min_{\gamma\in U(a,b)} \langle \gamma,C\rangle_F,$$5 with singleton word topics it reduces to WMD (Yurochkin et al., 2019).

In histopathology, the hierarchy is dataset $$(\mathcal D_{\mathcal{MK}})\quad \min_{\gamma\in U(a,b)} \langle \gamma,C\rangle_F,$$6 slides $$(\mathcal D_{\mathcal{MK}})\quad \min_{\gamma\in U(a,b)} \langle \gamma,C\rangle_F,$$7 tiles. Tile features are encoded by a pretrained extractor $$(\mathcal D_{\mathcal{MK}})\quad \min_{\gamma\in U(a,b)} \langle \gamma,C\rangle_F,$$8, tile-level costs are

$$(\mathcal D_{\mathcal{MK}})\quad \min_{\gamma\in U(a,b)} \langle \gamma,C\rangle_F,$$9

slide-to-slide distances are computed by entropy-regularized OT over tile embeddings, and dataset-to-dataset distances are then computed by OT over slides using the slide-level OT matrix as the upper-level ground cost. The resulting HHOT construction preserves the slide–tile structure imposed by tiling rather than flattening all tiles into a single empirical measure (Yeaton et al., 2022).

In multi-view learning, HOT relaxes alignment assumptions by penalizing sliced Wasserstein distances between latent-view distributions and then using those distances as the ground distance for an entropic OT problem across views or between views and prototype clusters. This produces either implicit clustering through pairwise couplings or explicit clustering through reference latent codes $C$0 (Luo et al., 2020).

3. Algorithms and computational strategies

A recurrent theme in HOT is that the hierarchy is not only semantic but computational. Entropic regularization and Sinkhorn-type solvers are pervasive because the outer transport problem is often expensive even before the lower-level OT computations are included (Hamri et al., 2021, Yeaton et al., 2022, Luo et al., 2020).

Several papers replace exact lower-level OT with approximations. DeepHOT, which combines domain-level OT with image-level OT for unsupervised domain adaptation, approximates the image-level OT by sliced Wasserstein distance over random one-dimensional projections of patch features and uses mini-batch unbalanced OT at the domain level. Its image-level approximation has complexity $C$1, while the mini-batch domain-level UOT is handled by a generalized Sinkhorn algorithm with $C$2 per batch (Xu et al., 2022). The multi-view HOT model likewise uses empirical sliced Wasserstein distance,

$C$3

as a practical surrogate for explicit sample alignment, then updates the OT coupling $C$4 by Sinkhorn scaling (Luo et al., 2020).

Other HOT models exploit hierarchy to decompose large problems. In histopathology, HHOT uses a two-stage Sinkhorn procedure: tile-level OT fills a slide-level cost matrix, then slide-level OT compares datasets. The paper states that the hierarchical approach is much faster than naive flat OT between all tiles and gives a total complexity dominated by

$C$5

under the stated accuracy parameters (Yeaton et al., 2022). In multi-network alignment, HOT first aligns clusters through an FGW barycenter and then performs node-level alignment only within matched clusters; a proximal point method with KL regularization and generalized Sinkhorn iterations is used for the multi-marginal FGW subproblems, with guaranteed convergence to a local optimum (Zeng et al., 2023).

A distinct computational line is Hierarchical Refinement, which recursively solves low-rank OT subproblems and refines co-clusters until singleton matches are obtained. Under the stated assumptions, it uses linear space $C$6, has log-linear runtime $C$7, and culminates in a bijective coupling, thereby converting low-rank OT factors into a multiscale construction of a Monge map (Halmos et al., 4 Mar 2025).

4. Theoretical guarantees

HOT has generated several forms of theory: metric properties, generalization bounds, and existence–uniqueness results for generalized hierarchical transport models. These results differ by application, but they share the idea that multilevel structure changes both the geometry of the comparison and the admissible transport plans (Yurochkin et al., 2019, Hamri et al., 2022, Xu, 16 Dec 2025).

In domain adaptation, the hierarchical Wasserstein formalism yields explicit target-risk bounds. One bound given in the literature is

$C$8

where the hierarchical Wasserstein term measures class–cluster alignment rather than only the divergence of flattened source and target measures. An expanded form further decomposes the bound into matched class–cluster Wasserstein terms, an off-diagonal coupling penalty, and the combined error of the ideal joint hypothesis (Hamri et al., 2022). This makes the trade-off between source risk, structural alignment, and joint error more explicit than in flat OT-based domain-adaptation theory.

In document representation, the HOT distance is a metric provided that each document can be written uniquely as a linear combination of the given topics, each topic is unique, and the topic-level base distance is itself a metric. The same paper also gives an upper bound relating WMD and HOT through the quality of the topic approximation, which quantifies the loss induced by replacing the original word distribution by its topic-mixture representation (Yurochkin et al., 2019).

A broader abstract generalization is Hierarchical Jump multi-marginal transport (HJMOT), where mass may skip intermediate spaces through augmented isolated points. On Polish spaces, lower semicontinuity and coercivity yield existence of Kantorovich minimizers, and strong coercivity together with sequential differentiability, sequential twistedness, and a local control condition yield a unique Borel Monge solution. The framework extends to smooth complete boundaryless Riemannian manifolds by replacing ordinary differentiability with differentiability along geodesics and replacing twist by geodesic twistedness (Xu, 16 Dec 2025).

5. Major application domains

HOT has been used wherever data possess a natural multilevel organization and flat OT risks mixing semantically distinct structures. In unsupervised domain adaptation, HOT-DA groups labeled source samples by class, learns target clusters through Wasserstein-Spectral clustering, solves the entropy-regularized HOT problem between the resulting class and cluster measures, and then transports source samples to target clusters via barycentric mapping. The method is explicitly motivated by the claim that structure-agnostic OT may blur class boundaries, whereas HOT yields compact, often block-diagonal transport plans (Hamri et al., 2021).

In few-shot learning, hierarchical OT is used for adaptive distribution calibration. A high-level OT aligns base classes and novel samples, a low-level OT computes the adaptive cost between each base class and each novel sample by considering all samples inside the base class, and the resulting transport plan serves as an adaptive weight matrix for transferring base-class statistics to calibrate novel-class distributions (Guo et al., 2022). In DeepHOT for unsupervised domain adaptation, the outer OT aligns source and target images while an inner image-level OT aligns local patch distributions, so the domain-level ground metric becomes a structural distance that includes local category-discriminative information (Xu et al., 2022).

In vision and graph problems, HOT appears in both pairwise and multi-object matching. The UHOT graph-matching framework computes node-level FGW alignments for all modality pairs and then fuses them through an unbalanced modality-level OT plan, producing a final node correspondence by weighted averaging of the lower-level transport plans (Cheng et al., 2023). For multi-network alignment, HOT uses an FGW barycenter to decompose several graphs into aligned clusters and then applies hierarchical multi-marginal FGW within each cluster, reducing the effective search space from a full $C$9 tensor to smaller cluster-restricted problems (Zeng et al., 2023). In sparse stereo and unsupervised object matching, HOT-POT computes local partial OT costs between objects and then solves a global partial OT at the object level, using epipolar or 3D ray distances as the lower-level geometry-aware cost (Clerc et al., 18 Jan 2026).

HOT also appears in scientific data comparison, graph abstraction, and representation analysis. HOTT compares documents at topic level rather than word level and reports better interpretability and scalability with comparable performance to current methods at a fraction of the cost (Yurochkin et al., 2019). HHOT defines distances between histopathology datasets without training, preserves the slide–tile hierarchy, and predicts difficulty of transferability in a tumor-vs.-normal transfer setting (Yeaton et al., 2022). OTCoarsening uses optimal transport as an unsupervised loss for learning a hierarchy of coarsened graphs, where $U(a,b)$0 and the coarsening matrix $U(a,b)$1 is learned end to end (Ma et al., 2019). More recently, HOT has been used to align model layers and brain regions by jointly optimizing neuron-level and layer-level transport plans, yielding a single alignment score and soft mappings that distribute mass across multiple layers when network depths differ (Shah et al., 2 Oct 2025).

6. Scope, variants, and recurring misconceptions

The literature does not treat HOT as a single canonical objective. Some works use balanced transport between measures of measures; others use unbalanced OT, partial OT, multi-marginal FGW, or hierarchical decompositions built from low-rank or blockwise subproblems. Concrete examples include entropy-regularized HOT in domain adaptation, unbalanced modality-level OT in graph matching, partial HOT in stereo/object matching, multi-marginal HOT in network alignment, and refinement-based HOT for scalable Monge-map recovery (Hamri et al., 2021, Cheng et al., 2023, Clerc et al., 18 Jan 2026, Zeng et al., 2023, Halmos et al., 4 Mar 2025).

A second misconception is that the hierarchy must be a literal tree of labels. In practice, the hierarchy may be semantic or computational: topics over words, slides over tiles, views over latent prototype clusters, layers over neurons, or multistage path spaces in which intermediate spaces may even be skipped by hierarchical jumps (Yurochkin et al., 2019, Yeaton et al., 2022, Luo et al., 2020, Xu, 16 Dec 2025). A plausible implication is that HOT is best understood as a modeling principle rather than a single solver: identify a multilevel organization, define lower-level transport or Wasserstein costs that respect that organization, and then solve an outer transport problem whose coupling encodes structure-aware alignment.

The same diversity explains why claims about HOT’s advantages must be read in context. Some papers emphasize preservation of class integrity and structural discriminativeness, others emphasize interpretability, scalability, or theoretical tractability, and still others use HOT primarily as a way to encode multi-level correspondence constraints. What is consistent across these variants is that the hierarchy is not decorative: it changes the admissible transport geometry, the meaning of the coupling, and often the downstream object being estimated—whether that object is a classifier, a barycentric map, a graph alignment tensor, a bijective correspondence, or a global representational alignment score (Hamri et al., 2022, Xu et al., 2022, Zeng et al., 2023, Shah et al., 2 Oct 2025).