Entropic Interpolation Overview

• Entropic interpolation is a framework that uses entropy minimization via Schrödinger bridge principles to construct smooth transitions between probability distributions.
• It bridges concepts from statistical mechanics, optimal transport, and regularization, offering insights for applications in computational imaging and quantum information.
• The method exhibits robust regularity and convexity properties and supports efficient computation through Sinkhorn-like algorithms in both continuous and discrete settings.

Entropic interpolation refers to a family of variational constructions and flows for probability measures, originally rooted in statistical mechanics and large deviations, that interpolate between two marginal distributions via an entropy-minimizing (Schrödinger bridge) principle. This paradigm provides a stochastic or entropy-regularized generalization of displacement interpolation in optimal mass transport (OMT), exhibits remarkable regularity and convexity properties, and has far-reaching implications in mathematical analysis, probability, quantum theory, and computational imaging.

1. Fundamental Principles and Formulations

The entropic interpolation problem arises from the dynamic Schrödinger problem: Given a reference path space measure $R$ (typically, stationary Brownian motion or a Markov process on a manifold or discrete graph), and two target probability measures $\mu_0, \mu_1$ on a state space $X$, the problem seeks a path-space measure $P$ minimizing relative entropy to $R$ under the constraint that its marginals at initial and terminal time match $\mu_0$ and $\mu_1$, respectively. Explicitly (Léonard, 2013, Ripani, 2017, Chen et al., 2016, Chen et al., 2015): $$\min_{P \ll R} H(P \mid R)\quad\text{subject to }P_0 = \mu_0,\, P_1 = \mu_1,$$ where $H(P \mid R)$ is the Kullback–Leibler divergence on path space.

The solution $P^*$ admits an $\mu_0, \mu_1$0-transform structure: $\mu_0, \mu_1$1 with Schrödinger potentials $\mu_0, \mu_1$2 determined by the marginal constraints. The associated time-marginal curve $\mu_0, \mu_1$3 is called the entropic interpolation between $\mu_0, \mu_1$4 and $\mu_0, \mu_1$5 (Léonard, 2013, Chen et al., 2015).

A Benamou–Brenier-type action characterizes the entropic cost (Ripani, 2017, Clerc et al., 2020): $\mu_0, \mu_1$6 subject to the continuity equation and endpoints $\mu_0, \mu_1$7.

2. Regularity, Convexity, and Displacement Connection

Entropic interpolations possess strong regularity: For diffusions on manifolds or graphs, entropic interpolations yield smooth curves in both space and time; the associated time-dependent densities solve hypoelliptic parabolic equations (Gentil et al., 2018). Unlike classical OMT/geodesic interpolation, which may only be Lipschitz, entropic interpolations enable pointwise formulas for derivatives and allow a rigorous calculus analogous to Otto–Villani's formalism (Léonard, 2013, Clerc et al., 2020).

A central property is the convexity of the entropy functional $\mu_0, \mu_1$8 along entropic interpolation. For reversible Markov semigroups with Bakry–Émery curvature-dimension conditions ($\mu_0, \mu_1$9), the second derivative satisfies (Léonard, 2013, Ripani, 2017): $X$0 establishing displacement convexity.

As the regularization parameter $X$1, entropic interpolation converges to the McCann displacement interpolation—that is, Wasserstein geodesics—both in path and cost (Chen et al., 2015, Chen et al., 2016, Clerc et al., 2020): $X$2 The interpolating densities satisfy Schrödinger-type PDEs with the regularization parameter $X$3 scaling the Laplacian, bridging regularized and non-regularized transport (Gentil et al., 2018, Clerc et al., 2020, Chen et al., 2015).

3. Functional Inequalities and Analytical Implications

The regularity and convexity along entropic interpolations provide new and unified routes to several functional inequalities. Differentiation of entropy along such flows yields, under curvature-dimension conditions ($X$4), direct proofs of logarithmic Sobolev inequalities and exponential convergence to equilibrium (Léonard, 2013, Gozlan et al., 2012, Ripani, 2017): $X$5 Furthermore, the entropic approach directly yields the HWI inequality linking entropy, quadratic transport cost, and Fisher information, as shown via a pathwise argument in (Gentil et al., 2018). In the discrete setting (e.g., graphs), entropic interpolation leads to displacement convexity results for entropy, Prekopa–Leindler and Talagrand-type inequalities, and strengthened log-Sobolev inequalities (Gozlan et al., 2012).

4. Computational and Algorithmic Developments

From an algorithmic viewpoint, entropic interpolation can be efficiently realized using Sinkhorn-like algorithms for discrete sample spaces (Chen et al., 2015). The Schrödinger bridge equations reduce, after discretization, to matrix scaling and iterative proportional fitting, with strong contractivity properties proven in the Hilbert metric. This approach grants fast computation of entropic interpolants, including in high-dimensional imaging tasks, and scales well due to linear convergence guarantees.

The entropic cost structure supports parallel computation and naturally handles noisy or high-dimensional data, as illustrated in applications to image morphing and density flows (Chen et al., 2015, Shamsolmoali et al., 2023). Entropic interpolations also appear in learning frameworks such as Entropy Transformer Networks, which interpolate feature maps along the data manifold with entropy regularization for improved gradient stability and data-consistent transformations (Shamsolmoali et al., 2023).

5. Asymptotic Regimes and Long-Time Behavior

Entropic interpolations interpolate not only in space but in time scales between different regimes of dynamics. In the short-time limit ($X$6 or small noise), the interpolation converges to displacement interpolation (Wasserstein geodesics). In the long-time limit ($X$7) under Bakry–Émery curvature-dimension conditions, the entropic interpolation converges to the standard heat flow (gradient flow of entropy), and the entropic cost grows only logarithmically (Clerc et al., 2020): $X$8 Moreover, the time-windowed interpolants approach the heat semigroup trajectory $X$9, with convergence rates controlled in the Wasserstein distance and for the Fisher information (Clerc et al., 2020).

6. Extensions and Applications in Quantum and Statistical Physics

Entropic interpolations have structurally significant roles in quantum information, statistical mechanics, and fluid dynamics. In quantum theory, entropic interpolation appears in improved entropic uncertainty relations for coarse-grained position and momentum, producing stronger lower bounds that interpolate between quantum and classical measurement regimes (Schürmann, 2010). In statistical mechanics, entropy interpolation/integration schemes enable powerful methods for merging high- and low-temperature expansions to accurately reconstruct thermodynamic quantities across regimes (Schmidt et al., 2017).

In incompressible fluids, entropic interpolation ("Bredinger problem") generalizes Brenier’s Euler-geodesic variational principle to viscid (Brownian) paths, linking entropy minimization and stochastic control to generalized Navier–Stokes equations, with noise-regularization interpreted as viscosity (Arnaudon et al., 2017).

7. Discrete Spaces and Graph-Theoretic Generalizations

In discrete contexts, such as finite graphs, where true OMT geodesics may not exist, binomial- or Markov-driven entropic interpolations provide a meaningful analogue (Gozlan et al., 2012). These interpolations enable rigorous calculus of entropy profiles, functional inequalities, and displacement convexity in non-geodesic metric spaces (Léonard, 2013).

A summary of key features across settings is shown below:

Setting	Reference Process	Characterization	Limiting Interpolation
Euclidean: $P$0	Brownian motion	Stochastic control / Schrödinger	Displacement (Wasserstein)
Manifold ($P$1)	Diffusion	Parabolic HJB / Fokker–Planck	OMT geodesic / Heat flow
Finite graph	Random walk	Binomial/geodesic averaging	Weak transport / Markov chain
Quantum phase space	Coarse-grained obs.	Convex optimization / Slepian O.	Quantum–classical crossover

Entropic interpolation thus serves as a unifying thread connecting entropy minimization, regularized optimal transport, PDE flows, stochastic control, and convexity theory, providing tools and insights at the interface of probability, geometry, analysis, and computation (Chen et al., 2015, Gentil et al., 2018, Léonard, 2013, Ripani, 2017, Gozlan et al., 2012).