Wasserstein-like Directional Metrics

• Wasserstein-like directional metrics are transport-inspired discrepancies that integrate directional constraints and anisotropic weights into the traditional Wasserstein framework.
• They are applied in varied contexts including spectral analysis on SPD(n), unbalanced transport, graph comparisons, and spherical law decomposition.
• Key methodologies involve quotient structures, asymmetric penalties, and higher-order dynamics, offering flexible tools for modeling complex directional phenomena.

A Wasserstein-like directional metric is a transport-inspired metric or discrepancy in which the geometry depends explicitly on directions, orientations, source terms, or anisotropic structural components rather than only on undirected mass displacement. Recent work uses the expression in several distinct but related senses: spectral anisotropy on $SPD(n)$, prescribed directions for mass creation and annihilation in unbalanced transport, invariance under global isometries, angular–concentration decompositions for spherical laws, and source–target sensitivity for directed graphs. The term therefore denotes a family of constructions rather than a single canonical metric, unified by the reuse of Wasserstein or Benamou–Brenier ideas under additional directional constraints or symmetries (Luo et al., 2020, Mao et al., 19 Jan 2026, Toukam, 20 Mar 2025, You et al., 19 Apr 2025, Patterson, 2019).

1. Spectral directionality on positive-definite matrices

One of the most explicit realizations appears on the manifold $SPD(n)$, where the Wasserstein metric is constructed as a quotient metric induced from $GL(n)$. In that setting, the Sylvester operator $\Gamma_A$ is defined by

$A\,\Gamma_A[X]+\Gamma_A[X]\,A=X,$

and the metric is

$$g_W|_A(X,Y)=\mathrm{tr}\!\big(\Gamma_A[Y]\,A\,\Gamma_A[X]\big)=\frac12\,\mathrm{tr}\!\big(\Gamma_A[Y]\,X\big).$$

The directional character is spectral: in an eigenbasis of $A$, $(\Gamma_A[X])_{ij}$ is weighted by $(\lambda_i+\lambda_j)^{-1}$, so directions associated with large eigenvalues are scaled down, while directions involving small eigenvalues are amplified. The paper explicitly states that curvatures and radii depend only on eigenvalues after orthogonal reduction, so the geometry is controlled by spectral data rather than just matrix entries (Luo et al., 2020).

The same work derives explicit local and global geometry. Between any $A_1,A_2\in SPD(n)$, the minimal Wasserstein geodesic is

$SPD(n)$0

with logarithm

$SPD(n)$1

and exponential map

$SPD(n)$2

The manifold is globally geodesic convex, has non-negative sectional curvature, has no conjugate pair and no cut locus, and its curvature is controlled by the smallest eigenvalue through bounds such as

$SPD(n)$3

A plausible implication is that this is a prototype of a directional Wasserstein geometry in which degeneracy is detected spectrally and anisotropy is intrinsic rather than imposed externally (Luo et al., 2020).

Closely related is the Bures–Wasserstein metric on $SPD(n)$4, which the literature interprets through optimal alignment of square roots. Its classical closed form is

$SPD(n)$5

and geometrically

$SPD(n)$6

Here the optimizer $SPD(n)$7 is the best alignment or transport direction between square-root orbits. The same note shows that this metric admits an exact convex semidefinite reformulation, which yields convex programs for pairwise distances, barycenters, distances between convex subsets, and BW ball constraints. In this matrix setting, “directional” refers less to asymmetry than to alignment along orthogonal directions selected by the optimizer (Mohan, 2023).

2. Directional mass variation in unbalanced transport

A second major meaning of directionality arises in unbalanced transport, where mass variation is permitted but constrained to occur in prescribed ways. On finite reversible Markov chains, one paper introduces a Benamou–Brenier-type metric on nonnegative measures with nonconservative continuity equation

$SPD(n)$8

where $SPD(n)$9 is a fixed strictly positive reference direction. The action is

$GL(n)$0

so the metric weights source and transport costs separately. The resulting space $GL(n)$1 is geodesic, but its geodesics exhibit a non-locality property: for $GL(n)$2, optimal curves satisfy $GL(n)$3 for almost every $GL(n)$4, and in fact $GL(n)$5 for almost every $GL(n)$6. The source amplitude can be rearranged into a decreasing function without increasing the action, so mass creation tends to occur early and then decay later, and an optimal source cannot vanish on a time set of positive measure. The dual formula is expressed through Hamilton–Jacobi subsolutions. In this framework, directionality is literal: all creation and annihilation must occur along one fixed positive vector $GL(n)$7 (Mao et al., 19 Jan 2026).

A more general $GL(n)$8-type theory allows directionality through asymmetric penalties. In that framework, the generalized discrepancy is given in dual form by local pointwise constraints on 1-Lipschitz potentials and in infimal-convolution form by

$GL(n)$9

The paper emphasizes that asymmetry can be encoded by different penalty functions $\Gamma_A$0 and $\Gamma_A$1, by an asymmetric constraint set $\Gamma_A$2, or by local discrepancies placed before, between, or after transport. A canonical example is generalized total variation with different costs for mass increase and mass decrease, so that growth and shrinkage can be penalized differently. The framework is therefore not always a metric in the strict symmetric sense; in general it is a convex discrepancy or semimetric-like object (Schmitzer et al., 2017).

Another extension separates radial mass motion from transport of normalized shape. For positive finite measures,

$\Gamma_A$3

The dynamic formulation reads

$\Gamma_A$4

and the tangent space splits into a transport component $\Gamma_A$5 and a mass component $\Gamma_A$6. Geodesics linearly interpolate the mass coordinate while the normalized profile follows a reparameterized $\Gamma_A$7-geodesic; barycenters likewise decompose into weighted average mass and a Wasserstein barycenter of normalized measures. This suggests a second sense of directionality: movement occurs in coupled transport and mass directions rather than along transport alone (Leblanc et al., 2023).

3. Invariance, quotient structure, and the status of symmetry

A different line of work treats some global motions as costless. The Procrustes Wasserstein metric is defined by a modified Benamou–Brenier approach in which the velocity is augmented by a global rigid motion that does not change direction or speed of particle trajectories. After centering, translations disappear and the construction reduces to orthogonal transformations: $\Gamma_A$8 The orthogonal curve is generated by a skew-symmetric field through $\Gamma_A$9, and the static problem becomes an optimal Procrustes alignment followed by ordinary $A\,\Gamma_A[X]+\Gamma_A[X]\,A=X,$0. For Gaussian measures, this collapses to

$A\,\Gamma_A[X]+\Gamma_A[X]\,A=X,$1

where $A\,\Gamma_A[X]+\Gamma_A[X]\,A=X,$2 are the ordered eigenvalue vectors of the covariance matrices. Here “directional” means that orientation mismatch is quotiented out, leaving only intrinsic spectral mismatch (Toukam, 20 Mar 2025).

The $A\,\Gamma_A[X]+\Gamma_A[X]\,A=X,$3-Gromov–Wasserstein framework provides an important corrective to a common misconception. It compares $A\,\Gamma_A[X]+\Gamma_A[X]\,A=X,$4-valued kernels through optimal couplings,

$A\,\Gamma_A[X]+\Gamma_A[X]\,A=X,$5

and subsumes classical GW, Wasserstein distance, fused GW, spectral GW, and other variants by varying the target space $A\,\Gamma_A[X]+\Gamma_A[X]\,A=X,$6. The paper proves that $A\,\Gamma_A[X]+\Gamma_A[X]\,A=X,$7 is a genuine metric on weak isomorphism classes and that separability, completeness, and geodesicity are inherited from $A\,\Gamma_A[X]+\Gamma_A[X]\,A=X,$8. However, it also states explicitly that the $A\,\Gamma_A[X]+\Gamma_A[X]\,A=X,$9-GW distance itself is symmetric, not directional. Directed graphs or asymmetric objects may be encoded in the kernel $$g_W|_A(X,Y)=\mathrm{tr}\!\big(\Gamma_A[Y]\,A\,\Gamma_A[X]\big)=\frac12\,\mathrm{tr}\!\big(\Gamma_A[Y]\,X\big).$$0, but the distance formula remains symmetric. This suggests that directional data and directional metric are distinct notions: asymmetry may live in the objects while the comparison functional remains symmetric (Bauer et al., 2024).

4. Directional structure for spherical laws, graphs, and matrix surrogates

For directional statistics on the sphere, one paper introduces a geometry-aware distance on non-degenerate von Mises–Fisher distributions that decomposes the discrepancy into a geodesic term between mean directions and a variance-like term comparing concentrations. The derivation uses a high-concentration Gaussian approximation in the tangent space,

$$g_W|_A(X,Y)=\mathrm{tr}\!\big(\Gamma_A[Y]\,A\,\Gamma_A[X]\big)=\frac12\,\mathrm{tr}\!\big(\Gamma_A[Y]\,X\big).$$1

followed by parallel transport and a Bures–Wasserstein comparison of the resulting tangent-space covariance objects. The parameter space $$g_W|_A(X,Y)=\mathrm{tr}\!\big(\Gamma_A[Y]\,A\,\Gamma_A[X]\big)=\frac12\,\mathrm{tr}\!\big(\Gamma_A[Y]\,X\big).$$2 thereby acquires a product-manifold structure: the angular coordinate uses the spherical geodesic distance, while the concentration coordinate becomes Euclidean after the reparameterization $$g_W|_A(X,Y)=\mathrm{tr}\!\big(\Gamma_A[Y]\,A\,\Gamma_A[X]\big)=\frac12\,\mathrm{tr}\!\big(\Gamma_A[Y]\,X\big).$$3. The paper proves that the resulting distance is well-defined, continuous, symmetric, and topologically consistent, and uses it for barycenters and vMF mixture reduction. In this setting, directionality is literal angular structure, separated from concentration (You et al., 19 Apr 2025).

For graphs and other structured data, a Wasserstein-style metric can be directional because source and target maps are compared separately. Directed graphs are treated as $$g_W|_A(X,Y)=\mathrm{tr}\!\big(\Gamma_A[Y]\,A\,\Gamma_A[X]\big)=\frac12\,\mathrm{tr}\!\big(\Gamma_A[Y]\,X\big).$$4-sets with structure maps $$g_W|_A(X,Y)=\mathrm{tr}\!\big(\Gamma_A[Y]\,A\,\Gamma_A[X]\big)=\frac12\,\mathrm{tr}\!\big(\Gamma_A[Y]\,X\big).$$5, and the relaxed comparison replaces deterministic graph morphisms by Markov kernels satisfying commutativity constraints in $$g_W|_A(X,Y)=\mathrm{tr}\!\big(\Gamma_A[Y]\,A\,\Gamma_A[X]\big)=\frac12\,\mathrm{tr}\!\big(\Gamma_A[Y]\,X\big).$$6. The resulting metric on $$g_W|_A(X,Y)=\mathrm{tr}\!\big(\Gamma_A[Y]\,A\,\Gamma_A[X]\big)=\frac12\,\mathrm{tr}\!\big(\Gamma_A[Y]\,X\big).$$7-sets is

$$g_W|_A(X,Y)=\mathrm{tr}\!\big(\Gamma_A[Y]\,A\,\Gamma_A[X]\big)=\frac12\,\mathrm{tr}\!\big(\Gamma_A[Y]\,X\big).$$8

For directed graphs, source and target enter as separate terms, so orientation is not collapsed away; the paper also states that the Wasserstein-style version is generally not symmetric and becomes a linear program for finite structures. Here directionality is structural and categorical rather than spectral or mass-based (Patterson, 2019).

A more heuristic example is the Quasi Manhattan Wasserstein Distance for matrices. QMWD does not compute a genuine 2D transport plan; instead it combines three 1D Wasserstein distances computed after row-major vectorization, $$g_W|_A(X,Y)=\mathrm{tr}\!\big(\Gamma_A[Y]\,A\,\Gamma_A[X]\big)=\frac12\,\mathrm{tr}\!\big(\Gamma_A[Y]\,X\big).$$9 rotation, and transposition, followed by dimension-dependent scaling, a modulo correction, and a maximum aggregation. The stated motivation is to approximate Manhattan Wasserstein behavior with lower time and space complexity. The paper explicitly notes that the transformation is ad hoc, that no proof of the triangle inequality is given, and that “metric” should therefore be read informally. This is an example of a computational surrogate with axis-aware flavor rather than a rigorously developed transport geometry (Lim, 2023).

5. Tangent directions, energy Hessians, and directional calculus on Wasserstein space

Not every directional construction is a new distance. In Wasserstein regression, the central object is the transport-induced tangent vector rather than a modified metric. For univariate distributions,

$A$0

and, at an atomless reference $A$1,

$A$2

The tangent space $A$3 is a Hilbert space, the exponential map is an isometric homeomorphism on the image of the log map, and regression operators act on $A$4-transformed predictors rather than on raw distributions. Parallel transport

$A$5

then allows comparison of operators defined at different tangent spaces. The directional content is local linearization by optimal transport maps (Chen et al., 2020).

A more explicit metric-level modification is the transport Hessian metric, defined as the Wasserstein Hessian of a functional $A$6: $A$7 Its dynamic distance is obtained by minimizing an action under the continuity equation $A$8. For quadratic $A$9, the construction recovers the standard Benamou–Brenier formula for $(\Gamma_A[X])_{ij}$0; for general $(\Gamma_A[X])_{ij}$1, the quadratic form is weighted by second variational derivatives of $(\Gamma_A[X])_{ij}$2 and therefore becomes anisotropic in the transport directions $(\Gamma_A[X])_{ij}$3. The paper presents this as a Wasserstein-induced, energy-dependent Riemannian metric with links to Newton flow, shallow water equations, heat flow via Fisher information, and Stein metrics with a mean-field kernel (Li, 2020).

The metric-viscosity literature on Wasserstein space sharpens another distinction. There, the standard metric remains $(\Gamma_A[X])_{ij}$4, and no new directional metric is introduced. Instead, direction-like behavior is recovered from distance-like functions, local slopes,

$(\Gamma_A[X])_{ij}$5

negative gradient curves, and co-rays. On complete unbounded length spaces, metric viscosity solutions of $(\Gamma_A[X])_{ij}$6 are exactly distance-like functions, and strong metric viscosity solutions are those for which every point admits a negative gradient curve. In this setting, the directional structure is induced by calibrated geodesics rather than by replacing the ambient Wasserstein metric (Jiang et al., 2023).

6. Higher-order and lifted Riemannian generalizations

Some recent constructions alter the differential order of the geometry itself. In a gradient-flow formulation motivated by econophysics, the manifold

$(\Gamma_A[X])_{ij}$7

is endowed with a metric whose tangent vectors satisfy not only zero total mass but also zero first moment. More generally, tangent vectors are orthogonal to harmonic functions, and in one dimension this means orthogonal to affine functions. The metric tensor is

$(\Gamma_A[X])_{ij}$8

with Onsager operator

$(\Gamma_A[X])_{ij}$9

The paper describes this as a fourth-order, nonlocal, Wasserstein-like metric whose distance is the action of the dual norm of a second-order homogeneous Sobolev factor space. The Gini coefficient becomes a Lyapunov functional and a formal gradient-flow driver in this geometry. Here directionality refers to admissible tangent directions constrained by conserved moments (Cohen, 2024).

A further lift extends unbalanced transport from densities to the space $(\lambda_i+\lambda_j)^{-1}$0 of Riemannian metrics. The Wasserstein–Ebin metric is built from the evolution equation

$(\lambda_i+\lambda_j)^{-1}$1

where $(\lambda_i+\lambda_j)^{-1}$2 is a transport vector field and $(\lambda_i+\lambda_j)^{-1}$3 is a source term. The transport part is penalized by an $(\lambda_i+\lambda_j)^{-1}$4-metric on vector fields, while the source part is penalized by the Ebin metric. The main theorem states that the volume map $(\lambda_i+\lambda_j)^{-1}$5 is a Riemannian submersion from the Wasserstein–Ebin metric to the Wasserstein–Fisher–Rao metric on smooth densities. The paper also constructs a Riemannian submersion from the automorphism group of the tangent bundle onto $(\lambda_i+\lambda_j)^{-1}$6, generalizing Otto’s geometric description of Wasserstein geometry. A static KL-type formulation is proposed, but the link between static and dynamic formulations is left open. This suggests that directional Wasserstein-like geometry can be lifted from scalar densities to anisotropic tensor fields while preserving the transport-plus-source paradigm (Bauer et al., 26 May 2026).

Across these constructions, the phrase “Wasserstein-like directional metric” names a recurring design principle rather than a single object: retain coupling-based, geodesic, or Benamou–Brenier structure, then inject directionality through spectral weights, fixed source directions, asymmetric mass penalties, quotient symmetries, tangent-space decompositions, higher-order constraints, or anisotropic state variables. The literature also draws a consistent boundary around the term: some geometries are genuinely directional or asymmetric, while others are only defined on directional data, or recover directional behavior from the standard Wasserstein metric without altering the metric itself (Bauer et al., 2024, Jiang et al., 2023).