Optimal Transport and Schrödinger Bridge
1. Optimal transport: comparing distributions geometrically
Many statistical divergences compare probability distributions pointwise. For example, KL divergence compares densities by
\[\mathrm{KL}(P \Vert Q) = \int \log\frac{dP}{dQ} dP.\]This is powerful, but it does not directly encode the geometry of the sample space. If two distributions are slightly shifted in space, KL can behave poorly when their supports barely overlap.
Optimal transport instead compares distributions through the cost of moving probability mass.
For discrete probability measures
\[\mu = \sum_{i=1}^n a_i \delta_{x_i}, \nu = \sum_{j=1}^m b_j \delta_{y_j},\]a transport plan is a nonnegative matrix $\pi$ satisfying
\[\sum_{j=1}^m \pi_{ij} = a_i, \sum_{i=1}^n \pi_{ij} = b_j.\]The set of all such couplings is
\[\Pi(a,b) = \{ \pi \in \mathbb{R}_+^{n \times m} : \pi \mathbf{1}_m = a, \pi^\top \mathbf{1}_n = b \}.\]Given a cost matrix $C_{ij} = c(x_i,y_j)$, the Kantorovich optimal transport problem is
\[\mathrm{OT}_c(\mu,\nu) = \min_{\pi \in \Pi(a,b)} \sum_{i=1}^n \sum_{j=1}^m C_{ij}\pi_{ij}.\]When $c(x,y)=|x-y|^p$, this gives the $p$-Wasserstein distance:
\[W_p^p(\mu,\nu) = \min_{\pi \in \Pi(\mu,\nu)} \int \|x-y\|^p d\pi(x,y).\]So OT is not merely a divergence. It is a geometry-aware comparison of measures.
2. OT as a variational principle
A variational principle characterizes an object as the minimizer or maximizer of a functional.
In OT, the unknown object is not a parameter vector. It is a coupling $\pi$ between two distributions. The variational problem is
\[\pi^* = \arg\min_{\pi \in \Pi(\mu,\nu)} \int c(x,y) d\pi(x,y).\]This means OT chooses the joint distribution $\pi$ with fixed marginals $\mu$ and $\nu$ that minimizes expected transportation cost.
The dual formulation makes the variational structure even clearer:
\[\mathrm{OT}_c(\mu,\nu) = \sup_{\varphi,\psi} \left\{ \int \varphi(x)d\mu(x) + \int \psi(y)d\nu(y) : \varphi(x)+\psi(y) \le c(x,y) \right\}.\]The primal searches over transport plans.
The dual searches over potential functions.
This duality is one reason OT appears naturally in optimization, PDEs, economics, probability, and machine learning.
3. Dynamic OT: moving a distribution through time
For quadratic cost, OT also has a dynamic formulation.
Instead of directly choosing a coupling between $\rho_0$ and $\rho_1$, we choose a time-indexed path of densities $\rho_t$ and a velocity field $v_t$ that transports mass from the initial density to the final density.
The Benamou-Brenier formulation is
\[W_2^2(\rho_0,\rho_1) = \inf_{\rho_t,v_t} \int_0^1 \int \|v_t(x)\|^2 \rho_t(x) dx dt\]subject to the continuity equation
\[\partial_t \rho_t(x) + \nabla \cdot [\rho_t(x)v_t(x)] = 0,\]with boundary constraints
\[\rho_{t=0}=\rho_0, \qquad \rho_{t=1}=\rho_1.\]This says:
Among all deterministic flows that move $\rho_0$ into $\rho_1$, choose the one with minimum kinetic energy.
This is the fluid-dynamic variational principle behind quadratic optimal transport.
4. Entropic OT: adding randomness and computation
Classical OT can be computationally expensive and statistically unstable in high dimensions. A modern relaxation adds entropy:
\[\mathrm{OT}_{\varepsilon}(\mu,\nu) = \min_{\pi \in \Pi(\mu,\nu)} \int c(x,y)d\pi(x,y) + \varepsilon \mathrm{KL}(\pi \Vert \mu \otimes \nu).\]Here $\varepsilon > 0$ controls the strength of regularization.
This changes the geometry:
- When $\varepsilon$ is small, entropic OT behaves more like classical OT.
- When $\varepsilon$ is large, the optimal coupling becomes more diffuse.
- Computationally, the solution can be obtained by Sinkhorn scaling.
In the discrete case, define
\[K_{ij} = \exp\left(-\frac{C_{ij}}{\varepsilon}\right).\]The optimal coupling has the form
\[\pi^* = \mathrm{diag}(u) K \mathrm{diag}(v),\]where $u$ and $v$ are scaling vectors chosen so that $\pi^*$ has marginals $a$ and $b$.
This is why entropic OT became central in modern computational OT: it converts a hard transport problem into a scalable matrix-scaling problem.
5. Schrödinger bridge: from transport plans to stochastic paths
Optimal transport asks: what is the cheapest way to move one probability distribution into another?
Schrödinger bridge asks: what is the most likely stochastic evolution that transforms one distribution into another while staying close to a reference dynamics?
The two are deeply connected:
Optimal transport is a deterministic variational principle over couplings or probability flows.
Schrödinger bridge is an entropy-regularized stochastic variational principle over path measures.
Schrödinger bridge starts from a different question.
Suppose $R$ is a reference stochastic process on path space. For example, $R$ may be Brownian motion or a Markov diffusion. We observe that the process should start with distribution $\mu$ and end with distribution $\nu$.
The Schrödinger bridge problem asks:
Among all path measures with endpoint marginals $\mu$ and $\nu$, which one is closest to the reference process $R$?
Mathematically, let $\Omega$ be path space and let $P_0$ and $P_1$ denote the time-0 and time-1 marginals of a path measure $P$. The Schrödinger bridge is
\[P^* = \arg\min_{P} \mathrm{KL}(P \Vert R)\]subject to
\[P_0=\mu, \qquad P_1=\nu.\]This is a variational principle over probability measures on paths.
Compared with OT:
- OT searches over couplings $\pi(x,y)$.
- Dynamic OT searches over deterministic flows $(\rho_t,v_t)$.
- Schrödinger bridge searches over stochastic path laws $P$.
So Schrödinger bridge can be viewed as a stochastic, entropy-regularized analogue of optimal transport.
6. Static view: Schrödinger bridge as entropic OT
If we only look at the endpoints $(X_0,X_1)$ of the reference process $R$, we get a reference endpoint law $R_{01}$.
The static Schrödinger problem becomes
\[\pi^* = \arg\min_{\pi \in \Pi(\mu,\nu)} \mathrm{KL}(\pi \Vert R_{01}).\]Now suppose the reference endpoint law has a Gibbs form
\[dR_{01}(x,y) \propto \exp\left(-\frac{c(x,y)}{\varepsilon}\right) d\mu(x)d\nu(y).\]Then minimizing $\mathrm{KL}(\pi \Vert R_{01})$ is equivalent, up to constants, to solving
\[\min_{\pi \in \Pi(\mu,\nu)} \int c(x,y)d\pi(x,y) + \varepsilon \mathrm{KL}(\pi \Vert \mu \otimes \nu).\]This is exactly entropic optimal transport.
So entropic OT is not just a numerical trick. It is also the static endpoint version of a stochastic path-space inference problem.
7. Dynamic view: Schrödinger bridge as stochastic control
When the reference process is Brownian motion or a diffusion, the Schrödinger bridge has a stochastic control formulation.
For example, consider the controlled diffusion
\[dX_t = u_t(X_t)dt + \sqrt{\varepsilon}dW_t,\]with endpoint constraints
\[X_0 \sim \rho_0, \qquad X_1 \sim \rho_1.\]The density $\rho_t$ evolves according to the Fokker-Planck equation
\[\partial_t \rho_t + \nabla \cdot [\rho_t u_t] = \frac{\varepsilon}{2}\Delta \rho_t.\]The bridge can be written as the variational problem
\[\inf_{\rho_t,u_t} \int_0^1 \int \frac{1}{2}\|u_t(x)\|^2 \rho_t(x) dx dt\]subject to
\[\partial_t \rho_t + \nabla \cdot [\rho_t u_t] = \frac{\varepsilon}{2}\Delta \rho_t,\]and
\[\rho_{t=0}=\rho_0, \qquad \rho_{t=1}=\rho_1.\]Compare this with dynamic OT:
\[\partial_t \rho_t + \nabla \cdot [\rho_t v_t] = 0.\]The difference is the diffusion term
\[\frac{\varepsilon}{2}\Delta \rho_t.\]Thus:
OT is deterministic mass transport.
Schrödinger bridge is stochastic mass transport under diffusion.
As $\varepsilon \to 0$, the diffusion term vanishes, and the Schrödinger bridge approaches optimal transport.
8. The Fisher-information correction
There is another useful way to see the relation.
Using a current velocity representation, the Schrödinger bridge objective can be written as a kinetic-energy term plus a Fisher-information correction:
\[\int_0^1 \int \frac{1}{2}\|v_t(x)\|^2 \rho_t(x) dx dt + \frac{\varepsilon^2}{8} \int_0^1 I(\rho_t)dt,\]where
\[I(\rho_t) = \int \|\nabla \log \rho_t(x)\|^2 \rho_t(x)dx.\]The first term is the same type of kinetic energy used in dynamic OT.
The second term penalizes sharp or irregular density paths. It appears because Schrödinger bridge is not just moving mass deterministically; it is moving mass through a noisy stochastic process.
This makes the variational relation very clear:
Schrödinger bridge equals optimal-transport-like kinetic energy plus an entropy-driven regularity term.
When $\varepsilon$ becomes small, the Fisher-information correction disappears, recovering the OT variational principle.
9. Schrödinger system and Sinkhorn scaling
The optimal path measure often has a multiplicative tilting form:
\[dP^* = f(X_0)g(X_1)dR.\]This means the optimal bridge is obtained by reweighting the reference process using two endpoint potentials.
In the static discrete case, this becomes
\[\pi^*_{ij} = u_i K_{ij} v_j,\]where
\[K_{ij} = \exp\left(-\frac{C_{ij}}{\varepsilon}\right).\]The vectors $u$ and $v$ are chosen so that
\[\sum_j \pi^*_{ij}=a_i, \qquad \sum_i \pi^*_{ij}=b_j.\]This is exactly the Sinkhorn scaling structure.
Therefore, the same mathematical object appears from three viewpoints:
- Entropic OT: regularized coupling minimization.
- Schrödinger bridge: KL projection on path space.
- Sinkhorn algorithm: matrix scaling to enforce marginals.
10. Why this matters in modern AI and statistics
Optimal transport and Schrödinger bridge are important because they give a geometry-aware language for probability distributions.
In modern AI, we often need to compare, interpolate, transform, or generate distributions:
- generated samples versus real samples;
- source domain versus target domain;
- prior distribution versus posterior distribution;
- noisy dynamics versus constrained endpoint behavior;
- empirical measures versus compressed or learned measures.
OT gives a deterministic geometry of distributions.
Schrödinger bridge adds stochastic dynamics and entropy.
Together, they provide a bridge between:
- probability,
- optimization,
- stochastic control,
- PDEs,
- generative modeling,
- simulation,
- and statistical inference.
The important conceptual point is:
OT is a variational principle over how mass should move.
Schrödinger bridge is a variational principle over how probability laws should evolve under uncertainty.
11. One-sentence summary
Optimal transport finds the least-cost deterministic rearrangement between probability measures, while Schrödinger bridge finds the most likely stochastic path-space evolution between them by minimizing relative entropy to a reference process.
References
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