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.

本研究解决了在部分观察的动态系统中学习线性状态转移和双线性观察的难题。通过引入数据依赖的高概率误差界限和简单随机设计的独立误差界限，本文为从单一有限轨迹中学习未知动态矩阵的统计误差率和样本复杂度提供了上界，从而拓展了对动态系统建模的理解。

从双线性观察中学习线性动态