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.

翻译：本文研究从具有线性状态转移和双线性观测的部分观测动力系统中学习其实现的问题。在对过程噪声和测量噪声施加非常温和的假设条件下，我们为学习未知动力学矩阵（至多相差一个相似变换）提供了有限时间分析。我们的分析涉及一个具有重尾和相依数据的回归问题。此外，设计矩阵的每一行都包含当前输入与历史输入的克罗内克积，这使得难以保证激励的持续性。我们通过以下方式克服这些挑战：首先为任意但固定的输入提供数据依赖的高概率误差界；随后针对按简单随机设计选择的输入推导出数据独立的误差界。我们的主要结果给出了从单条有限长度的双线性观测轨迹中学习未知动力学矩阵的统计误差率上界与样本复杂度。