Schr\"odinger Bridges (SB) have recently gained the attention of the ML
community as a promising extension of classic diffusion models which is also
interconnected to the Entropic Optimal Transport (EOT). Recent solvers for SB
exploit the pervasive bridge matching procedures. Such procedures aim to
recover a stochastic process transporting the mass between distributions given
only a transport plan between them. In particular, given the EOT plan, these
procedures can be adapted to solve SB. This fact is heavily exploited by recent
works giving rives to matching-based SB solvers. The cornerstone here is
recovering the EOT plan: recent works either use heuristical approximations
(e.g., the minibatch OT) or establish iterative matching procedures which by
the design accumulate the error during the training. We address these
limitations and propose a novel procedure to learn SB which we call the
\textbf{optimal Schr\"odinger bridge matching}. It exploits the optimal
parameterization of the diffusion process and provably recovers the SB process
\textbf{(a)} with a single bridge matching step and \textbf{(b)} with arbitrary
transport plan as the input. Furthermore, we show that the optimal bridge
matching objective coincides with the recently discovered energy-based modeling
(EBM) objectives to learn EOT/SB. Inspired by this observation, we develop a
light solver (which we call LightSB-M) to implement optimal matching in
practice using the Gaussian mixture parameterization of the Schr\"odinger
potential. We experimentally showcase the performance of our solver in a range
of practical tasks. The code for the LightSB-M solver can be found at
https://github.com/SKholkin/LightSB-Matching.

用最优 Schrödinger 桥匹配方法学习，借鉴能量建模目标，实现了与任意传输计划输入的单个桥匹配步骤下的 Schrödinger 桥过程的可靠恢复。

光与最优薛定谔桥匹配

Light and Optimal Schrödinger Bridge Matching

Schr\"odinger bridges (SBs) provide an elegant framework for modeling the
temporal evolution of populations in physical, chemical, or biological systems.
Such natural processes are commonly subject to changes in population size over
time due to the emergence of new species or birth and death events. However,
existing neural parameterizations of SBs such as diffusion Schr\"odinger
bridges (DSBs) are restricted to settings in which the endpoints of the
stochastic process are both probability measures and assume conservation of
mass constraints. To address this limitation, we introduce unbalanced DSBs
which model the temporal evolution of marginals with arbitrary finite mass.
This is achieved by deriving the time reversal of stochastic differential
equations with killing and birth terms. We present two novel algorithmic
schemes that comprise a scalable objective function for training unbalanced
DSBs and provide a theoretical analysis alongside challenging applications on
predicting heterogeneous molecular single-cell responses to various cancer
drugs and simulating the emergence and spread of new viral variants.

该研究提出一种新的神经参数化方法：非平衡扩散 Schrödinger Bridge（DSBs）。该方法可以模拟当端点是概率分布，且质量未被守恒时，人口的时间演化，并可应用于分析癌症药物的单细胞响应和病毒变异的传播过程。

不平衡扩散薛定谔桥

Unbalanced Diffusion Schrödinger Bridge

We propose a new framework to estimate the evolution of an ensemble of
indistinguishable agents on a hidden Markov chain using only aggregate output
data. This work can be viewed as an extension of the recent developments in
optimal mass transport and Schr\"odinger bridges to the finite state space
hidden Markov chain setting. The flow of the ensemble is estimated by solving a
maximum likelihood problem, which has a convex formulation at the
infinite-particle limit, and we develop a fast numerical algorithm for it. We
illustrate in two numerical examples how this framework can be used to track
the flow of identical and indistinguishable dynamical systems.

本研究提出了一种新的框架，通过使用聚合输出数据来估计不可区分代理的一组在隐藏的马尔可夫链上的演变，并将其视为近期发展在有限状态空间隐藏马尔可夫链环境中的最优质量传输和 Schrödinger 桥的扩展。通过解决最大似然估计问题来估计集合的流量，并为其开发了一种快速数值算法。我们在两个数值实例中说明了这个框架如何用于跟踪相同和不可区分的动态系统的流量。