Classical transport requires equal total mass and preserves every particle. Unbalanced transport permits mass to be created, removed, or attenuated, charging separately for spatial motion and variation of mass. The resulting geometry combines Wasserstein and Fisher--Rao structures.
Relaxing the marginal constraints
Definition. Let \(\mu,\nu\) be finite nonnegative measures, not necessarily of equal mass. For divergences \(D_1,D_2\) on positive measures, define \[ \operatorname{UOT}(\mu,\nu) = \inf_{\pi\in\mathcal M_+(X\times Y)} \left\{ \int c\,\mathrm{d}\pi +\tau_1D_1(\pi_X\,\|\,\mu) +\tau_2D_2(\pi_Y\,\|\,\nu) \right\}. \] The marginals of \(\pi\) need not equal \(\mu,\nu\); the divergences penalize their mismatch.
Transported mass is represented by \(\pi\). Source mass not used by \(\pi_X\) may be destroyed, and missing target mass relative to \(\pi_Y\) may be created. The parameters \(\tau_1,\tau_2\) determine whether it is cheaper to move mass or modify it.
KL-penalized unbalanced transport
Dual formulation. For \(D_i=\operatorname{KL}\), \[ \operatorname{UOT}(\mu,\nu) = \sup_{\varphi+\psi\le c} \left[ -\tau_1\int\{e^{-\varphi/\tau_1}-1\}\,\mathrm{d}\mu -\tau_2\int\{e^{-\psi/\tau_2}-1\}\,\mathrm{d}\nu \right]. \]
Proof. The entropy conjugacy formula is \[ \tau\operatorname{KL}(\alpha\,\|\,\mu) = \sup_f\left\{\int f\,\mathrm{d}\alpha -\tau\int(e^{f/\tau}-1)\,\mathrm{d}\mu\right\}. \] Apply it to \(\pi_X,\pi_Y\), then minimize over \(\pi\ge0\). Finiteness requires \(c+f+g\ge0\). Setting \(\varphi=-f,\psi=-g\) gives the constraint and dual objective.
Unlike balanced dual potentials, \(\varphi,\psi\) cannot be shifted by opposite constants without changing the objective: the total transported mass is itself an optimized quantity.
Entropic scaling for unequal mass
Adding coupling entropy with parameter \(\varepsilon\) again gives a Gibbs plan \(P=\operatorname{diag}(u)K\operatorname{diag}(v)\). For discrete KL marginal penalties, generalized Sinkhorn updates take the form \[ u\leftarrow\left(a\oslash Kv\right)^{\tau_1/(\tau_1+\varepsilon)}, \qquad v\leftarrow\left(b\oslash K^{\mathsf T}u\right)^{\tau_2/(\tau_2+\varepsilon)}. \] Exponents below one perform only a partial marginal correction. As \(\tau_i\to\infty\), the exponents approach one and balanced Sinkhorn is recovered.
The continuity equation with reaction
Definition. A dynamic unbalanced path consists of a measure curve \(\mu_t\), a velocity \(v_t\), and a reaction rate \(\alpha_t\) satisfying \[ \partial_t\mu_t+\nabla\cdot(v_t\mu_t)=\alpha_t\mu_t. \] In one common normalization, its action is \[ \mathcal A_\delta(\mu,v,\alpha) = \int_0^1\int \{|v_t|^2+\delta^2|\alpha_t|^2\} \,\mathrm{d}\mu_t\,\mathrm{d}t. \]
The left side moves existing mass; the right side changes its amplitude. The parameter \(\delta\) sets the relative price of reaction. The induced distance is a form of the Hellinger--Kantorovich or Wasserstein--Fisher--Rao metric, with constants depending on normalization.
Pure growth and Fisher--Rao geometry
Scalar mass distance. If all mass stays at one point and only its amount changes from \(m_0\) to \(m_1\), the minimal action is \[ 4\delta^2(\sqrt{m_1}-\sqrt{m_0})^2. \]
Proof. The reaction equation is \(m'=\alpha m\). Put \(r=\sqrt m\), so \(\alpha=2r'/r\). Then \[ \delta^2\int_0^1\alpha_t^2m_t\,\mathrm{d}t =4\delta^2\int_0^1|r_t'|^2\,\mathrm{d}t. \] The minimum over curves with fixed endpoints is attained by linear \(r_t\), giving the formula.
Cone geometry and Dirac measures
Dirac formula. In the normalization above, \[ \operatorname{HK}_\delta^2(a\delta_x,b\delta_y) = 4\delta^2\left[ a+b-2\sqrt{ab}\, \cos_+\!\left(\frac{d(x,y)}{2\delta}\right) \right], \] where \(\cos_+(r)=\cos(\min\{r,\pi/2\})\).
The square root of mass is a radial coordinate on a metric cone over \(X\). Nearby Dirac masses are partly transported; beyond the cutoff angle it is cheaper to annihilate at \(x\) and recreate at \(y\). This gives a precise geometric meaning to the transport--reaction tradeoff.
Dynamic Hellinger--Kantorovich formula
Dynamic representation. The squared Hellinger--Kantorovich distance is the infimum of \(\mathcal A_\delta\) over all reaction--continuity triples connecting the endpoint measures. Minimizers exist under standard compactness assumptions, and the distance metrizes weak convergence together with convergence of total mass on bounded spaces.
Proof mechanism. Lift each measure to the cone by representing mass as squared radius. Ordinary quadratic transport on the cone simultaneously changes position and radius. Projecting cone geodesics back to \(X\) yields the reaction--continuity equation and its action. Conversely, every finite-action triple admits a cone lift. The Benamou--Brenier formula on the cone gives equality.
Unbalanced barycenters
An unbalanced barycenter minimizes \[ \nu\longmapsto\sum_i\lambda_i\operatorname{HK}_\delta^2(\nu,\mu_i) \] or an analogous static UOT objective. It can average measures with different total mass and can suppress features present in only a few inputs rather than transporting mass over a long distance to accommodate them.
Gradient flows with mass variation
The formal tangent norm now contains transport and reaction: \[ \|(\nabla\phi,\alpha)\|_\mu^2 =\int(|\nabla\phi|^2+\delta^2\alpha^2)\,\mathrm{d}\mu. \] Gradient flows combine spatial drift with a reaction term. For a first variation \(F=\delta\mathcal E/\delta\rho\), one obtains, up to normalization, \[ \partial_t\rho =\nabla\cdot(\rho\nabla F)-\delta^{-2}\rho F. \] The first term transports mass down the energy gradient; the second creates or destroys it according to the sign of \(F\).
Balanced and pure-reaction limits
As reaction becomes infinitely expensive, the dynamic constraint approaches mass conservation and the geometry tends toward Wasserstein distance on equal-mass measures. If spatial motion becomes infinitely expensive, only Fisher--Rao change remains. Unbalanced transport is therefore not a small variant of classical transport but a geometry interpolating two distinct mechanisms.