We address the problem of learning linear system models from observing
multiple trajectories from different system dynamics. This framework
encompasses a collaborative scenario where several systems seeking to estimate
their dynamics are partitioned into clusters according to their system
similarity. Thus, the systems within the same cluster can benefit from the
observations made by the others. Considering this framework, we present an
algorithm where each system alternately estimates its cluster identity and
performs an estimation of its dynamics. This is then aggregated to update the
model of each cluster. We show that under mild assumptions, our algorithm
correctly estimates the cluster identities and achieves an approximate sample
complexity that scales inversely with the number of systems in the cluster,
thus facilitating a more efficient and personalized system identification
process.

该研究提出了一种从观察多种系统动态下的轨迹中学习线性系统模型的算法，其中系统根据相似性被分为不同的簇，并推导出一种通过估计系统簇身份并估计动态来更新每个簇模型的方法，从而实现更高效和个性化的系统识别过程。

利用集群系统识别学习个性化模型

Learning Personalized Models with Clustered System Identification

Approximate message passing (AMP) is a low-cost iterative signal recovery
algorithm for linear system models. When the system transform matrix has
independent identically distributed (IID) Gaussian entries, the performance of
AMP can be asymptotically characterized by a simple scalar recursion called
state evolution (SE). However, SE may become unreliable for other matrix
ensembles, especially for ill-conditioned ones. This imposes limits on the
applications of AMP.
In this paper, we propose an orthogonal AMP (OAMP) algorithm based on
de-correlated linear estimation (LE) and divergence-free non-linear estimation
(NLE). The Onsager term in standard AMP vanishes as a result of the
divergence-free constraint on NLE. We develop an SE procedure for OAMP and show
numerically that the SE for OAMP is accurate for general unitarily-invariant
matrices, including IID Gaussian matrices and partial orthogonal matrices. We
further derive optimized options for OAMP and show that the corresponding SE
fixed point coincides with the optimal performance obtained via the replica
method. Our numerical results demonstrate that OAMP can be advantageous over
AMP, especially for ill-conditioned matrices

本文提出一种基于去相关线性估计和无发散非线性估计的正交 AMP 算法，并推导出适用于各种酉不变矩阵的状态演化过程，数值结果表明该算法在病态矩阵情况下比 AMP 更有优势。