In this paper, we focus on learning a linear time-invariant (LTI) model with low-dimensional latent variables but high-dimensional observations. We provide an algorithm that recovers the high-dimensional features, i.e. column space of the observer, embeds the data into low dimensions and learns the low-dimensional model parameters. Our algorithm enjoys a sample complexity guarantee of order $\tilde{\mathcal{O}}(n/\epsilon^2)$, where $n$ is the observation dimension. We further establish a fundamental lower bound indicating this complexity bound is optimal up to logarithmic factors and dimension-independent constants. We show that this inevitable linear factor of $n$ is due to the learning error of the observer's column space in the presence of high-dimensional noises. Extending our results, we consider a meta-learning problem inspired by various real-world applications, where the observer column space can be collectively learned from datasets of multiple LTI systems. An end-to-end algorithm is then proposed, facilitating learning LTI systems from a meta-dataset which breaks the sample complexity lower bound in certain scenarios.

翻译：本文聚焦于学习具有低维潜在变量但高维观测的线性时不变模型。我们提出一种算法，能够恢复高维特征（即观测器列空间），将数据嵌入低维空间，并学习低维模型参数。该算法具有$\tilde{\mathcal{O}}(n/\epsilon^2)$阶的样本复杂度保证，其中$n$为观测维度。我们进一步建立了理论下界，表明该复杂度界限在忽略对数因子和维度无关常数的情况下是最优的。我们证明这种不可避免的$n$线性因子源于高维噪声存在时观测器列空间的学习误差。通过扩展研究结果，我们考察了受现实应用启发的元学习问题——多个LTI系统的数据集可共同学习观测器列空间。据此提出端到端算法，通过元数据集学习LTI系统，该方案在特定场景下突破了样本复杂度的理论下界。