# Entropic and displacement interpolation: a computational approach using the Hilbert metric

arXiv: https://arxiv.org/abs/1506.04255 · HTML: https://mlsanity.com/paper/1506.04255/

## Key facts

- Cited by 34 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Jun 2015.

## Built on by (most cited first, 30 of 34)

- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [Mirror Descent with Relative Smoothness in Measure Spaces, with application to Sinkhorn and EM](https://mlsanity.com/paper/2206.08873/) (2022)
- [Prior Does Matter: Visual Navigation via Denoising Diffusion Bridge Models](https://mlsanity.com/paper/2504.10041/) (2025)
- [Reflected Schr"odinger Bridge for Constrained Generative Modeling](https://mlsanity.com/paper/2401.03228/) (2024)
- [Schr"odinger Bridge with Quadratic State Cost is Exactly Solvable](https://mlsanity.com/paper/2406.00503/) (2024)
- [Quantitative Uniform Stability of the Iterative Proportional Fitting Procedure](https://mlsanity.com/paper/2108.08129/) (2021)
- [Riemannian Diffusion Schr"odinger Bridge](https://mlsanity.com/paper/2207.03024/) (2022)
- [Neural Schr"odinger Bridge with Sinkhorn Losses: Application to Data-driven Minimum Effort Control of Colloidal Self-assembly](https://mlsanity.com/paper/2307.14442/) (2023)
- [A Physics-informed Deep Learning Approach for Minimum Effort Stochastic Control of Colloidal Self-Assembly](https://mlsanity.com/paper/2208.09182/) (2022)
- [Hilbert's projective metric for functions of bounded growth and exponential convergence of Sinkhorn's algorithm](https://mlsanity.com/paper/2311.04041/) (2023)
- [Soft-constrained Schrodinger Bridge: a Stochastic Control Approach](https://mlsanity.com/paper/2403.01717/) (2024)
- [Non-asymptotic convergence bounds for Sinkhorn iterates and their gradients: a coupling approach](https://mlsanity.com/paper/2304.06549/) (2023)
- [On the Hopf-Cole Transform for Control-affine Schr"odinger Bridge](https://mlsanity.com/paper/2503.17640/) (2025)
- [Matching for causal effects via multimarginal unbalanced optimal transport](https://mlsanity.com/paper/2112.04398/) (2021)
- [Simplified Diffusion Schr"odinger Bridge](https://mlsanity.com/paper/2403.14623/) (2024)
- [A semiconcavity approach to stability of entropic plans and exponential convergence of Sinkhorn's algorithm](https://mlsanity.com/paper/2412.09235/) (2024)
- [Discrete Adjoint Schr"odinger Bridge Sampler](https://mlsanity.com/paper/2602.08243/) (2026)
- [Data Assimilation for Sign-indefinite Priors: A generalization of Sinkhorn's algorithm](https://mlsanity.com/paper/2308.11791/) (2023)
- [On the Contraction Coefficient of the Schr"odinger Bridge for Stochastic Linear Systems](https://mlsanity.com/paper/2309.06622/) (2023)
- [Solution of the Probabilistic Lambert Problem: Connections with Optimal Mass Transport, Schr"odinger Bridge and Reaction-Diffusion PDEs](https://mlsanity.com/paper/2401.07961/) (2024)
- [Weyl Calculus and Exactly Solvable Schr"odinger Bridges with Quadratic State Cost](https://mlsanity.com/paper/2407.15245/) (2024)
- [A Generalized Sinkhorn Algorithm for Mean-Field Schr"odinger Bridge](https://mlsanity.com/paper/2604.06531/) (2026)
- [Schr"odinger Bridge Over A Compact Connected Lie Group](https://mlsanity.com/paper/2603.14049/) (2026)
- [Sinkhorn Flow: A Continuous-Time Framework for Understanding and Generalizing the Sinkhorn Algorithm](https://mlsanity.com/paper/2311.16706/) (2023)
- [Forward Reverse Kernel Regression for the Schr"odinger bridge problem](https://mlsanity.com/paper/2507.00640/) (2025)
- [Incorporating Pre-trained Diffusion Models in Solving the Schr"odinger Bridge Problem](https://mlsanity.com/paper/2508.18095/) (2025)
- [Robust Sublinear Convergence Rates for Iterative Bregman Projections](https://mlsanity.com/paper/2602.01372/) (2026)
- [Rectified Schr"odinger Bridge Matching for Few-Step Visual Navigation](https://mlsanity.com/paper/2604.05673/) (2026)

## Abstract (arXiv)

Monge-Kantorovich optimal mass transport (OMT) provides a blueprint for geometries in the space of positive densities -- it quantifies the cost of transporting a mass distribution into another. In particular, it provides natural options for interpolation of distributions (displacement interpolation) and for modeling flows. As such it has been the cornerstone of recent developments in physics, probability theory, image processing, time-series analysis, and several other fields. In spite of extensive work and theoretical developments, the computation of OMT for large scale problems has remained a challenging task. An alternative framework for interpolating distributions, rooted in statistical mechanics and large deviations, is that of Schroedinger bridges (entropic interpolation). This may be seen as a stochastic regularization of OMT and can be cast as the stochastic control problem of steering the probability density of the state-vector of a dynamical system between two marginals. In this approach, however, the actual computation of flows had hardly received any attention. In recent work on Schroedinger bridges for Markov chains and quantum evolutions, we noted that the solution can be efficiently obtained from the fixed-point of a map which is contractive in the Hilbert metric. Thus, the purpose of this paper is to show that a similar approach can be taken in the context of diffusion processes which i) leads to a new proof of a classical result on Schroedinger bridges and ii) provides an efficient computational scheme for both, Schroedinger bridges and OMT. We illustrate this new computational approach by obtaining interpolation of densities in representative examples such as interpolation of images.