entropic interpolation

The entropic interpolation $[{\mu }_0,{\mu }_1{]}^k:=({\mu }_t^k{)}_{t\in [0,1]}$ is the time-marginal flow of the solution to the Schrödinger problem of order $k$ — that is, with reference measure $R^k$ having generator $L^k=\Delta /(2k)$. It is the Schrödinger bridge analogue of the displacement interpolation in optimal transport.

While the displacement interpolation $[{\mu }_0,{\mu }_1]$ is the geodesic in $({\mathrm{P}}_2(\mathcal{X}),W_2)$ obtained by pushing mass along straight-line optimal transport paths, the entropic interpolation replaces deterministic geodesics with Brownian bridges weighted by the Schrödinger coupling. A key advantage is regularity: the entropic interpolation is always positive and smooth on $(0,1)\times \mathcal{X}$, whereas the displacement interpolation may concentrate on lower-dimensional sets.

The entropic interpolation satisfies an evolution equation governed by the operator $A_t^k=\nabla {\psi }_t^k\cdot \nabla +\Delta /(2k)$, a second-order perturbation of the first-order transport operator $A_t=\nabla {\psi }_t\cdot \nabla$ of the displacement interpolation. The Laplacian term provides smoothing and positivity.

Key Details

• Convergence: ${\lim }_{k\to \infty }[{\mu }_0,{\mu }_1{]}^k=[{\mu }_0,{\mu }_1]$ uniformly on $C([0,1],{\mathrm{P}}_2(\mathcal{X}))$ — the entropic interpolation converges to the displacement interpolation as noise vanishes
• Entropy formula: Along the entropic interpolation, $h_{[{\mu }_0,{\mu }_1{]}^k}^{\prime \prime }(t)=\frac{1}{2}⟨{\Gamma }_2({\phi }_t)+{\Gamma }_2({\psi }_t),{\mu }_t⟩$, where ${\Gamma }_2$ is the iterated carré du champ. This is rigorous (unlike the formal Otto calculus for displacement interpolations) and connects to Ricci curvature via Bochner’s formula: ${\Gamma }_2(\psi )=\Vert \nabla \psi {\Vert }_{\text{HS}}^2+\text{Ric}(\nabla \psi )$
• Lott-Sturm-Villani connection: The entropy convexity along entropic interpolations provides a “very lazy gas” version of the lazy gas thought experiment, and is a starting point for synthetic curvature theory

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