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.

本文旨在展示一种新的计算框架，基于 Hilbert 距离的收缩映射得到的固定点解法，用于高效计算难以处理的大规模最优质量传输和 Schroedinger 桥等问题。

用 Hilbert 度量的计算方法进行熵和位移插值

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

We study the time scales associated to diffusion processes that take place on
multiplex networks, i.e. on a set of networks linked through interconnected
layers. To this end, we propose the construction of a supra-Laplacian matrix,
which consists of a dimensional lifting of the Laplacian matrix of each layer
of the multiplex network. We use perturbative analysis to reveal analytically
the structure of eigenvectors and eigenvalues of the complete network in terms
of the spectral properties of the individual layers. The spectrum of the
supra-Laplacian allows us to understand the physics of diffusion-like processes
on top of multiplex networks.

本研究探讨了在多层网络上进行的扩散过程所涉及的时间尺度，并提出了一个上拉普拉斯矩阵来建立多重网络的结构，采用微扰分析方法，从谱特性的角度揭示了完整网络的特征向量和特征值的结构，以理解在多重网络之上进行扩散样过程的物理学 。

多重网络上的扩散动力学

Diffusion dynamics on multiplex networks

Maximization of the entropy rate is an important issue to design diffusion
processes aiming at a well-mixed state. We demonstrate that it is possible to
construct maximal-entropy random walks with only local information on the graph
structure. In particular, we show that an almost maximal-entropy random walk is
obtained when the step probabilities are proportional to a power of the degree
of the target node, with an exponent $\alpha$ that depends on the degree-degree
correlations, and is equal to 1 in uncorrelated graphs.

通过局部信息，可以构建最大化熵随机游走，在无相关的情况下，步骤概率与目标节点的度数的幂成比例关系。

具有有限信息的复杂网络中的最大熵随机游走

Maximal-entropy random walks in complex networks with limited  information

We construct a new random probability measure on the sphere and on the unit
interval which in both cases has a Gibbs structure with the relative entropy
functional as Hamiltonian. It satisfies a quasi-invariance formula with respect
to the action of smooth diffeomorphism of the sphere and the interval
respectively. The associated integration by parts formula is used to construct
two classes of diffusion processes on probability measures (on the sphere or
the unit interval) by Dirichlet form methods. The first one is closely related
to Malliavin's Brownian motion on the homeomorphism group. The second one is a
probability valued stochastic perturbation of the heat flow, whose intrinsic
metric is the quadratic Wasserstein distance. It may be regarded as the
canonical diffusion process on the Wasserstein space.

本文构建了一个新的随机概率度量空间，并使用相对熵函数作为 Hamiltonian，使其在球面和单位区间上都具有 Gibbs 结构。我们利用其积分换先公式，通过 Dirichlet 形式方法构建了两类概率度量空间上的扩散过程，第一类与 Malliavin 的 Brownian motion 密切相关，第二类是热流的概率值随机扰动，其内部度量是二次 Wasserstein 距离，可视为 Wasserstein 空间上的经典扩散过程。