Intrinsic Wasserstein Distance

• Intrinsic Wasserstein Distance is a metric concept that generalizes classical optimal transport by integrating mass variation with geometric structure.
• It employs a generalized Benamou–Brenier formulation by adding source terms to handle mass creation and annihilation in transport problems.
• The duality through the flat metric links mass differences with Lipschitz constraints, enabling robust analysis in metric measure spaces and PDEs.

The intrinsic Wasserstein distance is a concept that encompasses a family of constructions and theoretical frameworks wherein the metric structure or geometry induced by Wasserstein (optimal transport) distances is made intrinsic to the space of probability measures or tailored to the underlying data geometry. This notion appears across several research domains, including extensions to measures with varying mass, variants that incorporate underlying geometric or manifold structure, dual formulations, and dynamical/variational representations. Intrinsic Wasserstein distances play a central role in analysis on metric measure spaces, high-dimensional statistics, partial differential equations, generative modeling, and in optimal transport–based comparison of structured objects.

1. Structural Limitations of Classical Wasserstein Distances and the Need for Intrinsic Constructions

Classical Wasserstein distances $W_p$ (for $p \geq 1$), rooted in the Monge–Kantorovich optimal transport problem, are defined for probability measures of equal total mass. The primal formulation is

$$W_p(\mu, \nu) = \left( \inf_{T \in \Gamma(\mu, \nu)} \int |x - y|^p dT(x, y)\right)^{1/p},$$

where $\Gamma(\mu, \nu)$ denotes the set of couplings with $\mu$ and $\nu$ as marginals. This restriction to equal mass is an intrinsic limitation: in applications such as transport equations with sources, mass may not be conserved; hence, $W_p$ is undefined or infinite for measures of unequal mass (Piccoli et al., 2013). This motivates intrinsic generalizations that retain the geometric and variational flavor of optimal transport but allow for broader applicability.

2. Generalized Wasserstein Distances: Intrinsic Extension via Mass Variation

Piccoli and Rossi introduced the generalized Wasserstein distance $W_p^{a,b}$ to circumvent the equal-mass limitation (Piccoli et al., 2013). The construction “mixes” two types of costs:

• Transport cost (as in classical Wasserstein, weighted by $b$)
• Mass variation cost (total variation or $L^1$ distance, weighted by $a$)

Formally, for finite positive measures $\mu$ and $\nu$,

$$T_{a,b}(\mu, \nu) = \inf \left\{a(|\mu - \mu_1| + |\nu - \nu_1|)^p + b[W_p(\mu_1, \nu_1)]^p\right\},$$

where the infimum is taken over all decompositions $\mu = \mu_1 + \mu_2$, $\nu = \nu_1 + \nu_2$, with $\mu_1, \nu_1$ being the parts to be transported. Then,

$$W_p^{a,b}(\mu, \nu) = \left(T_{a,b}(\mu, \nu)\right)^{1/p}.$$

This distance interpolates between transport and total variation: for large $a$, the cost of mass variation dominates, recovering an $L^1$-type metric; for small $a$ (relative to $b$), transport (i.e., geometric movement) dominates. This extension renders the metric “intrinsic” in the sense that all finite measures, regardless of total mass, can be assigned a finite distance.

3. Generalized Dynamical Formulation: The Intrinsic Benamou–Brenier Framework

The classical Benamou–Brenier formula relates the squared $W_2$ distance between measures $\mu_0$ and $\mu_1$ to the infimum of an action functional over solutions to the continuity equation: $$W_2^2(\mu_0, \mu_1) = \inf_{(\mu_t, v_t)}\int_0^1 \int |v_t(x)|^2 d\mu_t(x) dt, \quad \text{subject to } \partial_t \mu_t + \nabla\cdot(v_t\mu_t) = 0.$$ In the generalized setting, to allow for mass sources/sinks, Piccoli and Rossi (Piccoli et al., 2013) extend this formula by introducing a source term: $\partial_t \mu_t + \nabla\cdot(v_t\mu_t) = h_t.$ An associated action functional becomes

$$\mathcal{B}_{a,b}[\mu, v, h] = \frac{a^2}{2}\int_0^1\left(\int |h_t| dx\right)^2dt + \frac{1}{2}\int_0^1\int |v_t(x)|^2 d\mu_t(x) dt.$$

The generalized Benamou–Brenier formula states

$$\inf\{\mathcal{B}_{a,b}[\mu, v, h]: (\mu, v, h)\ \text{solve the transport–source problem}\} = T_{a,b}(\mu_0, \mu_1),$$

which links the metric directly to a variational problem that simultaneously handles mass transport and net creation/annihilation. This provides an “intrinsic” dynamical metric that naturally extends beyond mass-preserving flows.

4. Duality and the Flat Metric: Intrinsic Kantorovich–Rubinstein Theorem

For $p=1$ and $a=b=1$, the generalized Wasserstein distance $W_1^{1,1}$ admits a dual characterization: $$W_1^{1,1}(\mu, \nu) = \sup\left\{\int f d(\mu - \nu): \|f\|_\infty \leq 1, \operatorname{Lip}(f)\leq 1\right\},$$ i.e., the supremum is over all functions $f$ that are both Lipschitz with constant at most one and uniformly bounded by one [(Piccoli et al., 2013), Theorem 13]. This is precisely the so-called “flat metric” or bounded Lipschitz distance. In contrast to the classical $W_1$, which requires equal mass and only the Lipschitz constraint, the flat metric accounts also for pure mass differences, reflecting a fully intrinsic metric between finite signed measures: $W_1^{1,1}(\mu, \nu) = d(\mu, \nu).$ This dual formulation explicitly ties geometric (transport) and measure-theoretic (mass change) aspects, generalizing the Kantorovich–Rubinstein theorem to the non-conservative setting.

5. Mathematical Formalism and Parameter Effects

The core mathematical constructs for the intrinsic Wasserstein strategy are summarized as follows:

Generalized Cost	Definition
$T_{a,b}(\mu,\nu)$	$$\inf \{ a|\mu - \mu'| + a|\nu - \nu'| + b W_p(\mu', \nu') : \mu' \leq \mu, \nu' \leq \nu \}$$
Generalized Distance	$W_p^{a,b}(\mu,\nu) = [T_{a,b}(\mu,\nu)]^{1/p}$
Benamou–Brenier (gen.)	$$T_{a,b}(\mu_0,\mu_1) = \inf\{\mathcal{B}_{a,b}[\mu, v, h]:\cdots\}$$
Duality (p=1, a=b=1)	$$W_1^{1,1}(\mu, \nu) = \sup \left\{ \int f d(\mu-\nu) : \|f\|_\infty \leq 1, \text{Lip}(f)\leq 1\right\}$$

The parameters $a$ and $b$ provide a mechanism for controlling the relative penalization. In the extreme $a\to\infty$, the metric is dominated by pure mass differences and approaches the $L^1$ distance. For small $a/b$, $W_p^{a,b}$ approaches the classical transport metric. This parametric flexibility supports modeling scenarios with varying dominance of creation/annihilation versus rearrangement of mass.

6. Applications and Theoretical Implications

By extending Wasserstein distances intrinsically in this manner, several new analytical opportunities emerge:

• Partial Differential Equations with Sources: The generalized Benamou–Brenier framework enables the variational paper of PDEs where mass is not conserved, such as transport equations with source or sink terms.
• Measure Comparison: $W_p^{a,b}$ quantifies distance between arbitrary finite measures, not just probability measures of equal mass, which is pivotal for modeling, inference, and statistics in inhomogeneous or dissipative systems.
• Gradient Flow and Numerical Schemes: The inclusion of source terms in the continuity equation and dual formulations facilitates the design and analysis of new numerical schemes for gradient flows and variational approximations in optimal transport problems with mass variation.
• Theoretical Generality: The identification of the dual (flat metric) representation and extension of the Benamou–Brenier formula deepen the structural analogy between the analysis on optimal transport spaces and more classical Riemannian or metric geometry, but in a context that is fully intrinsic to the geometry of arbitrary measures.

7. Connections to Broader Intrinsic Optimal Transport Theory

Intrinsic modifications of the Wasserstein metric reflect a broader trend in adapting optimal transport to underlying geometric or application-specific structure. For example, in the context of measure-valued equations on manifolds, intrinsic constructions appear via the use of geodesic distances in the base space (manifold), replacement of the ground cost with one induced by spectral or localized structure, or adaptation to variable-mass settings (such as unbalanced transport or the inclusion of boundary sources). The generalized Wasserstein distances serve as foundational models for such extensions.

A plausible implication is that, by providing a robust variational framework and dual characterization, intrinsic Wasserstein distances can serve as the basis for future developments in comparative geometry of measure spaces, efficient computation on non-conservative or heterogeneous datasets, and further connections to information geometry and entropic transport formulations.

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In summary, intrinsic Wasserstein distances generalize and adapt the classical optimal transport metric structure by incorporating mass variation, duality with bounded Lipschitz functions, and variational dynamical formulations. These developments enable the application of optimal transport methods to a broader array of problems in analysis, partial differential equations, and applied mathematics where the geometry of finite measures—not just probability measures of equal mass—is central (Piccoli et al., 2013).