从有限观测轨迹中学习非线性系统的动态矩阵，给出误差上界和样本复杂度分析。
Learning Linear Dynamics from Bilinear Observations
- 基于带重尾噪声的依赖数据，用克罗内克积设计回归矩阵。
- 在单条轨迹下给出统计误差与样本复杂度的上界。
- 适用于系统辨识、控制理论中的建模问题。
我们研究部分可观测动态系统中线性状态转移与双线性观测的建模问题。在对过程与测量噪声假设极为宽松的前提下，针对未知动态矩阵（至相似变换意义下）的学习，给出了有限时间分析。该分析涉及具有重尾与依赖性的回归问题，且设计矩阵的每一行均为当前输入与历史输入的克罗内克积，难以保证激励持续性。本文首先为任意但固定的输入提供数据依赖的高概率误差界；随后，对按简单随机设计选取的输入，推导出数据无关的误差界。主要结果给出了从单条有限轨迹中学习未知动态矩阵的统计误差率与样本复杂度上界。
原文摘要 · Abstract (English)
We consider the problem of learning a realization of a partially observed dynamical system with linear state transitions and bilinear observations. Under very mild assumptions on the process and measurement noises, we provide a finite time analysis for learning the unknown dynamics matrices (up to a similarity transform). Our analysis involves a regression problem with heavy-tailed and dependent data. Moreover, each row of our design matrix contains a Kronecker product of current input with a history of inputs, making it difficult to guarantee persistence of excitation. We overcome these challenges, first providing a data-dependent high probability error bound for arbitrary but fixed inputs. Then, we derive a data-independent error bound for inputs chosen according to a simple random design. Our main results provide an upper bound on the statistical error rates and sample complexity of learning the unknown dynamics matrices from a single finite trajectory of bilinear observations.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。