Topological Optimal Transport (TpOT)

• Topological Optimal Transport (TpOT) is a framework that couples optimal transport with topological and geometric constraints to study existence, uniqueness, and regularity of transport maps.
• It employs differential geometry, such as the cross-difference and pseudo-Riemannian metrics, alongside persistence diagrams to capture intrinsic topological features.
• TpOT unifies diverse formulations—including cost-geometry, persistence-based methods, and structural matching—to advance applications from mesh processing to anomaly detection.

Searching arXiv for recent and foundational papers on Topological Optimal Transport and closely related usages of the term. Topological Optimal Transport (TpOT) denotes a family of research programs in which optimal transport is coupled to topological or geometric structure. In one foundational formulation, it studies how the differential topology and geometry encoded by a transportation cost $c(x,y)$ on $M^+\times M^-$ determine the existence, uniqueness, regularity, and geometric/topological structure of optimal transport (McCann, 2012). In another, it is the optimal partial transport formalism specialized to persistence diagrams, turning matching-based metrics into an optimal transport problem on measures with a built-in mass reservoir at the diagonal (Divol et al., 2019). More recent work defines TpOT as a framework for matching geometric point clouds while explicitly coupling their persistent homology features through geometric, diagrammatic, and hypergraph transport terms (Zhang et al., 2024). A plausible unifying description is that topology enters the transport problem through the cost, the admissible couplings, the transported descriptors, or explicit topology-preserving constraints.

1. Terminological scope and major formulations

The expression “Topological Optimal Transport” is not attached to a single universally fixed formalism. The literature uses it for several related constructions, each of which links transport to topology in a different way. In the differential-geometric tradition, topology refers to the topology and geometry induced by the transport cost itself. In topological data analysis, it refers to persistence diagrams and related descriptors viewed as transportable measures. In structure-matching and application papers, it often refers to transport objectives that preserve or exploit topological summaries, graph structure, cycle incidence, or orientation-preserving constraints (McCann, 2012, Divol et al., 2019, Zhang et al., 2024).

Strand	Core object	Representative formulation
Cost-geometry TpOT	Cost $c(x,y)$ on $M^+\times M^-$	Cross-difference, pseudo-Riemannian metric, MTW geometry
Persistence-diagram TpOT	Radon measures on the upper half-plane	OT with boundary and diagonal reservoir
Coupled topological matching	Point clouds, PH cycles, merge trees, Morse graphs	GW, Wasserstein, co-optimal transport, partial FGW
Topology-preserving applications	Meshes, images, anomaly maps, cortical maps	QC correction, persistence regularization, topological smoothing

This multiplicity is substantive rather than merely terminological. The cost-geometric program is concerned with Monge maps, Kantorovich plans, and the geometry of supports. The persistence-diagram program treats the diagonal as a free reservoir and studies the metric topology of the diagram space. The point-cloud and descriptor programs integrate geometry, persistent homology, and higher-order relations into a single objective. Application papers then specialize one or more of these ideas to retinal enhancement, anomaly segmentation, cortical retinotopy, mesh deformation, and structured generation.

2. Differential topology and geometry of the transport cost

In the foundational differential-geometric formulation, the central objects are the Monge and Kantorovich problems. Given Borel probability measures $\mu$ on $M^+$ and $\nu$ on $M^-$ and a cost $c:M^+\times M^-\to\mathbb{R}$, the Monge problem minimizes

$\int_{M^+} c(x,T(x))\,d\mu(x)$

over measurable maps $M^+\times M^-$0 with $M^+\times M^-$1, while the Kantorovich problem minimizes

$M^+\times M^-$2

over couplings $M^+\times M^-$3. The dual formulation,

$M^+\times M^-$4

introduces $M^+\times M^-$5-convex potentials and contact sets on which optimal plans concentrate (McCann, 2012).

The geometric mechanism is the cross-difference

$M^+\times M^-$6

For $M^+\times M^-$7-cyclically monotone sets, $M^+\times M^-$8 encodes the impossibility of lowering cost by swapping partners. When $M^+\times M^-$9, the Hessian of the cross-difference at $c(x,y)$0 defines a symmetric bilinear form

$c(x,y)$1

with block form involving the mixed Hessian $c(x,y)$2. This yields a pseudo-Riemannian metric on $c(x,y)$3, with signature

$c(x,y)$4

where $c(x,y)$5 is the rank of the cost at the point. The involution $c(x,y)$6 then produces a $c(x,y)$7-form $c(x,y)$8, which is symplectic when $c(x,y)$9 has full rank. In this setting, $M^+\times M^-$0-monotone sets are $M^+\times M^-$1-spacelike, and optimal graphs are $M^+\times M^-$2-Lagrangian (McCann, 2012).

Several structural conditions govern existence, uniqueness, and regularity. The twist condition is equivalently the injectivity of the momentum map $M^+\times M^-$3 for fixed $M^+\times M^-$4, or the condition that the cross-difference has no critical points except the base point. Twist forces the $M^+\times M^-$5-subdifferential to be single-valued $M^+\times M^-$6-a.e., so optimal couplings collapse to Monge maps. Non-degeneracy of $M^+\times M^-$7 gives sharp dimension bounds on supports, local graph structure, and spacelike geometry. The Ma–Trudinger–Wang tensor,

$M^+\times M^-$8

reappears as sectional curvature of the pseudo-Riemannian manifold $M^+\times M^-$9 in planes spanned by horizontal and vertical directions. Under the geometric MTW framework $\mu$0–$\mu$1, strong MTW curvature and geodesic convexity control Hölder continuity and smoothness of optimal maps. Brenier’s theorem for quadratic cost, and smoothness results associated with Caffarelli, Urbas, and Delanoë, are presented as canonical instances (McCann, 2012).

The same strand also emphasizes support geometry. If the cost has rank $\mu$2 at $\mu$3, then the tangent space of any $\mu$4-monotone support satisfies

$\mu$5

Pass and McCann–Pass–Warren show that such supports lie locally in spacelike Lipschitz submanifolds. This converts cyclical monotonicity into a geometric regularity statement about the dimension and rectifiability of optimal plans.

A related but distinct extension replaces the ordinary Wasserstein topology by a fibered transport topology with fixed marginal on a base space. In the Euclidean model $\mu$6, the fibered distance

$\mu$7

constrains transport to fixed fibers, produces a Polish space, and supports an Otto-type gradient-flow theory for heterogeneous PDEs and Cucker–Smale-type alignment models (Peszek et al., 2022). Here the term “topology” refers to the metric topology induced by the constrained transport distance rather than to topological data analysis.

3. Persistence diagrams as measures and OT with boundary

A second major formulation of TpOT arises in topological data analysis by identifying persistence diagrams with measures on the open upper half-plane

$\mu$8

with diagonal

$\mu$9

A persistence diagram is treated as a discrete Radon measure

$M^+$0

and the framework extends this to arbitrary Radon measures $M^+$1 on $M^+$2. The $M^+$3-persistence functional

$M^+$4

defines the space

$M^+$5

which contains both finite diagrams and non-discrete limits such as persistence surfaces, expected diagrams, and weak limits of random diagrams (Divol et al., 2019).

The crucial move is to recast diagram distances as optimal partial transport with boundary. The admissible plans are Radon measures on $M^+$6 whose constraints apply only to the $M^+$7-marginals, so the diagonal functions as a free reservoir. For $M^+$8,

$M^+$9

with cost on $\nu$0 given by $\nu$1 and cost to the diagonal given by $\nu$2. On diagrams, this recovers the classical persistence-diagram metrics exactly: $\nu$3 on $\nu$4, and likewise in the bottleneck case (Divol et al., 2019).

This measure-theoretic reformulation yields a strong ambient topology. $\nu$5 is Polish, and the diagram subspace $\nu$6 is closed in it. Convergence has a Wasserstein-type characterization:

$\nu$7

Relative compactness is characterized by tightness of the weighted measures $\nu$8 together with uniform bounds on $\nu$9. For finite-mass measures, the problem reduces to an ordinary Wasserstein problem on a quotient space $M^-$0 endowed with

$M^-$1

so finite barycenters are standard Wasserstein barycenters after collapsing the diagonal to a single point (Divol et al., 2019).

The same framework gives a precise theory of Fréchet means and continuous representations. For $M^-$2 and $M^-$3, the set of $M^-$4-Fréchet means is non-empty, compact, and convex; if the law is supported on diagrams, there exist diagram-valued barycenters. Continuous linear maps $M^-$5 are exactly those of the form

$M^-$6

with bounded continuous $M^-$7. This characterizes persistence surfaces, silhouettes, landscapes, and weighted Betti curves that are continuous for the OT topology. The framework also supports random persistence diagrams: under a thermodynamic regime for Čech or Rips constructions, empirical diagrams converge almost surely in $M^-$8 to a non-discrete limit measure, and expected diagrams satisfy stability inequalities driven by Wasserstein distances between input distributions (Divol et al., 2019).

4. Coupled transport on structured topological objects

A third line of work uses TpOT to align not only topological summaries but also the ambient geometry and the representatives of homology classes. In the Measure Topological Network (MTN) formalism, an object is

$M^-$9

where $c:M^+\times M^-\to\mathbb{R}$0 is a gauged measure space, $c:M^+\times M^-\to\mathbb{R}$1 indexes homology classes, $c:M^+\times M^-\to\mathbb{R}$2 maps them to birth–death coordinates, and $c:M^+\times M^-\to\mathbb{R}$3 encodes cycle incidence. The TpOT distance couples three discrepancies simultaneously: a Gromov–Wasserstein geometric term, a Wasserstein term on measure persistence diagrams, and a co-optimal transport term on hypernetwork kernels,

$c:M^+\times M^-\to\mathbb{R}$4

On equivalence classes of weakly isomorphic MTNs, this is a geodesic metric, and for $c:M^+\times M^-\to\mathbb{R}$5 the quotient space has non-negative Alexandrov curvature and unique geodesics of explicit convex-interpolation form (Zhang et al., 2024).

This construction yields a direct transport-based methodology for geometric cycle matching. Persistent homology classes are matched through $c:M^+\times M^-\to\mathbb{R}$6, point clouds through $c:M^+\times M^-\to\mathbb{R}$7, and the co-optimal term aligns point-level and cycle-level transport. In the discrete $c:M^+\times M^-\to\mathbb{R}$8 setting, the objective becomes a weighted sum of a GW distortion term, an augmented persistence-diagram cost with diagonal matching, and a hypergraph-incidence term. Practical solvers use entropic regularization with Sinkhorn scaling or alternating minimization with fused GW subproblems. The reported empirical behavior is that TpOT can preserve cycle correspondences that pure GW or pure diagram Wasserstein leave ambiguous, especially when many bars are similar but cycle geometry is distinct (Zhang et al., 2024).

Merge trees supply another structured TpOT model. A merge tree is encoded as a measure network $c:M^+\times M^-\to\mathbb{R}$9, where $\int_{M^+} c(x,T(x))\,d\mu(x)$0 are critical points, $\int_{M^+} c(x,T(x))\,d\mu(x)$1 is a probability measure on nodes, and $\int_{M^+} c(x,T(x))\,d\mu(x)$2 is either shortest-path distance or lowest-common-ancestor ultrametric. Partial transport is built by relaxing the coupling constraints to

$\int_{M^+} c(x,T(x))\,d\mu(x)$3

and the comparison cost is a partial fused Gromov–Wasserstein objective combining attribute distances and structure distortions. In this framework, the coupling is explicitly interpreted as a probabilistic correspondence between topological features at adjacent time steps, which induces topology-tracking graphs and provides robustness to appearing, disappearing, and noisy features (Li et al., 2023).

Morse graphs admit a closely related treatment as attributed measure networks with degree-based node masses, shortest geodesic path lengths as intrinsic structure, and spatial coordinates as attributes. Wasserstein, GW, FGW, and their partial versions are then used to compare Morse complexes. The optimal coupling acts as a soft registration of nodes, and hard correspondences can be extracted by columnwise maximization or thresholding. The reported guideline is that FGW is generally the most effective default for correspondence tasks, GW is advantageous when intrinsic structure is discriminative and geometry is misleading, and partial variants improve robustness to noisy or unmatched features (Li et al., 2023).

Dynamic extensions move from a single scalar distance to time-resolved diagnostics. One such framework reconstructs point clouds along a TpOT geodesic by interpolating geometry, recomputing persistent homology and cycle incidence at each intermediate state, and then evaluating macroscopic and mesoscopic indicators. The topological distortion curve is identified with the time-indexed topology term, persistence entropy measures lifespan redistribution, and a dual-perspective hypergraph entropy detects asynchronous rewiring of cycle incidence. A point-level topological field is then obtained by propagating entropy changes from cycles to vertices (Wang et al., 15 Mar 2026). This shifts TpOT from static comparison to dynamic topological analysis of phase transitions and longitudinal data.

5. Topology-preserving and application-specific frameworks

Several application papers use TpOT, TPOT, or TopoOT for objectives in which transport is constrained to preserve topology, orientation, or persistent structure. On triangular meshes, QC-OT introduces a topology structure-preserving optimal transport map that relaxes Delaunay triangulation in semi-discrete OT and then applies quasiconformal correction. The stated goal is to preserve mesh connectivity and avoid non-flip violations by enforcing orientation-preserving quasiconformal maps with Beltrami coefficient satisfying $\int_{M^+} c(x,T(x))\,d\mu(x)$4. A temporal extension, tt-OT, iterates relaxed Brenier optimization and QC correction to obtain spatial-temporal topology-preserving transport on dynamic meshes (Lv et al., 2 Jul 2025).

In cortical retinotopy, optimal transport is combined with topological smoothing to quantify planar cortical magnification. The cortical surface is flattened to a disk by semi-discrete OT that preserves local cortical surface areas through Laguerre-cell area constraints, and retinotopic maps are then corrected by Beltrami-coefficient-based smoothing so that $\int_{M^+} c(x,T(x))\,d\mu(x)$5 on each triangle. The paper describes this as an instance of TpOT because OT enforces measure-preserving parameterization while the smoothing stage maintains topological correctness of the retinotopic map (Xiong et al., 6 Dec 2025).

In retinal fundus image enhancement, “TPOT” denotes a topology-preserving training paradigm rather than a transport metric on spaces of descriptors. A generator is trained with a WGAN-type objective and a diagram-based regularizer computed from the persistence diagrams of vessel probability maps produced by a fixed segmentation network. The topological penalty is

$\int_{M^+} c(x,T(x))\,d\mu(x)$6

applied patchwise to discourage spurious or missing vessel structures. The paper explicitly notes that it follows Hu et al. for a differentiable persistence-based correspondence and does not instantiate the full $\int_{M^+} c(x,T(x))\,d\mu(x)$7-Wasserstein or bottleneck machinery with diagonal matching (Dong et al., 2024).

In anomaly segmentation under distribution shift, “TopoOT” integrates multi-filtration persistence diagrams with entropically regularized OT. Successive diagrams across thresholds and between sublevel and superlevel filtrations are aligned by “Optimal Transport Chaining,” and the resulting transport plans define geodesic-like stability scores for persistent features. These stability-aware pseudo-labels supervise a lightweight test-time adaptation head through OT-consistency and contrastive objectives (Zia et al., 28 Jan 2026). Here topology enters through persistent homology, while OT provides a stability criterion across scales rather than a single end-to-end metric between datasets.

A different usage appears in structured recipe generation. There, “Topological Optimal Transport” represents ingredient lists as point clouds in embedding space and compares predicted and gold ingredient sections with a Sinkhorn divergence. The paper explicitly states that “topological” denotes the geometry or shape of the embedding point cloud rather than algebraic topology. The loss therefore uses transport between sets of soft token embeddings,

$\int_{M^+} c(x,T(x))\,d\mu(x)$8

to improve ingredient- and action-level metrics in sequence generation (Ottoborgo et al., 5 Jan 2026).

These application-specific frameworks share a family resemblance rather than a single formal core. Some use persistent homology directly, some enforce orientation-preserving maps on meshes, some attach topological smoothing after transport, and some use “topological” in a geometric set-shape sense. The acronym therefore ranges from mathematically intrinsic transport geometries to domain-specific regularizers.

6. Conceptual issues, misconceptions, and open problems

A common misconception is that TpOT denotes one settled theory. The literature instead contains several distinct but intersecting constructions. In the cost-geometric line, the object of study is the transport cost and the induced pseudo-Riemannian or symplectic structure (McCann, 2012). In the persistence-diagram line, the central object is the space of Radon measures on the upper half-plane with the diagonal as a mass reservoir (Divol et al., 2019). In structured matching, the focus is a coupled objective involving point geometry, topological descriptors, and higher-order relations (Zhang et al., 2024). Application papers may then use the same acronym for topology-preserving mesh OT, diagram-based learning losses, or OT-guided topological smoothing (Lv et al., 2 Jul 2025, Dong et al., 2024).

Another misconception is that “topological” always means algebraic topology. That is true for persistence-diagram, merge-tree, Morse-complex, and cycle-matching formulations, but not for every use. In the recipe-generation paper, the term refers to the geometry of ingredient point clouds in embedding space rather than Betti numbers or persistence. In the fibered-transport paper, it refers to the metric topology of a constrained Wasserstein space (Ottoborgo et al., 5 Jan 2026, Peszek et al., 2022).

Open problems differ by strand. In the differential-geometric theory, regularity can fail without the MTW curvature condition or geodesic convexity; discontinuous optimal maps remain poorly understood outside special cases, regularity near cut loci is still open for squared Riemannian distance, and subtwist-type uniqueness criteria beyond spheres are unknown (McCann, 2012). In persistence-diagram TpOT, current directions include gradient flows in diagram spaces, entropic regularization, and further exploitation of OT tools for learning with distances and barycenters (Divol et al., 2019). In MTN-based TpOT, full stability bounds under joint perturbations, completeness properties, and dependence on cycle representatives remain unresolved, while fourth-order tensor contractions create scalability bottlenecks (Zhang et al., 2024). Merge-tree tracking via partial FGW inherits the non-convexity of the objective and offers no global optimality guarantee beyond convergence to stationary points (Li et al., 2023).

A final conceptual issue concerns topological irregularity in the transport constraint itself. For reflexive, transitive, $\int_{M^+} c(x,T(x))\,d\mu(x)$9 relations $M^+\times M^-$00, the zero–one transport cost $M^+\times M^-$01 need not be lower semicontinuous, yet the dual problem remains well behaved if the cost is a non-increasing limit of lower semicontinuous functions. In this setting, a Strassen-type theorem “nearly holds”: dual equality and $M^+\times M^-$02-approximate couplings survive, but exact primal minimizers may fail to exist (Jaffe et al., 25 Sep 2025). This underscores a broad lesson shared with other TpOT strands: topological regularity of the constraint set, the cost, or the transported descriptor often decides whether one obtains exact couplings, good dual formulations, or only approximations.

Taken together, these developments present TpOT not as a single doctrine but as a technically diverse research area. Its central ambition is consistent across usages: to make topology, geometry, and transport mutually constraining, so that existence, uniqueness, support structure, stability, and practical matching are governed by more than mass displacement alone.