ENTROPIC AND DISPLACEMENT INTERPOLATION: A COMPUTATIONAL APPROACH USING THE HILBERT METRIC
ENTROPIC AND DISPLACEMENT INTERPOLATION: A COMPUTATIONAL APPROACH USING THE HILBERT METRIC
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DOI：
10.1137/16m1061382
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发表时间：
2016-01-01
影响因子：
1.9
通讯作者：
Pavon, Michele
中科院分区：
文献类型：
--
作者：
Chen, Yongxin;Georgiou, Tryphon;Pavon, Michele
Monge-Kantorovich最优质量输运（OMT）为正密度空间中的几何图形提供了一个蓝图，它量化了将一个质量分布输运到另一个质量分布的成本。特别是，它为分布插值（位移插值）和流建模提供了自然的选择。因此，它已成为物理学、概率论、图像处理、时间序列分析和其他几个领域最近发展的基石。尽管有大量的工作和理论发展，大规模问题的OMT计算仍然是一项具有挑战性的任务。另一个插值分布的框架，植根于统计力学和大偏差，是薛定谔桥问题（SBP），它导致熵插值。SBP可以看作是OMT的随机正则化，可以看作是动态系统状态向量在两个边缘之间的概率密度的随机控制问题。然而，熵流的实际计算几乎没有得到任何关注。在我们最近关于马尔可夫链和量子通道的薛定谔桥的研究中，我们证明了从希尔伯特度量中收缩映射的不动点可以有效地得到解。因此，本文的目的是表明，在扩散过程的背景下可以采取类似的方法，这(i)导致关于SBP的经典结果的新证明，（ii）为SBP和OMT提供了有效的计算方案。我们通过在图像插值等代表性示例中获得密度插值来说明这种新的计算方法。
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 the Schrodinger bridge problem (SBP), which leads to entropic interpolation. SBP 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. The actual computation of entropic flows, however, has received hardly any attention. In our recent work on Schrodinger bridges for Markov chains and quantum channels, we showed 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 SBP and (ii) provides an efficient computational scheme for both SBP and OMT. We illustrate this new computational approach by obtaining interpolation of densities in representative examples such as interpolation of images.