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：本文研究从高维观测数据中学习具有低维潜在变量的线性时不变（LTI）模型。我们提出一种算法，该算法能够恢复高维特征（即观测器的列空间），将数据嵌入低维空间并学习低维模型参数。该算法具有阶数为 $\tilde{\mathcal{O}}(n/\epsilon^2)$ 的样本复杂度保证，其中 $n$ 为观测维度。我们进一步建立了基本下界，表明该复杂度在忽略对数因子和维度无关常数的情况下是最优的。研究揭示，这一不可避免的 $n$ 线性因子源于高维噪声环境下观测器列空间的学习误差。为拓展研究，我们受实际应用启发考虑元学习问题，在该问题中观测器列空间可从多个LTI系统的数据集联合学习。最终提出一种端到端算法，该算法能够从元数据集中学习LTI系统，并在特定场景下突破了样本复杂度下界。