论文标题
线性动力学系统的学习混合物
Learning Mixtures of Linear Dynamical Systems
论文作者
论文摘要
我们研究了从未标记的短样品轨迹中学习多个线性动力系统（LDS）的混合物的问题，每种轨迹都是由一个LDS模型产生的。尽管混合模型在时间序列数据中具有广泛的适用性，但在现有文献中基本上缺乏端到端性能保证的学习算法。技术挑战有多种来源，包括但不限于（1）存在潜在变量（即轨迹的未知标签）； （2）样品轨迹的长度可能比LDS型号的尺寸$ d $小得多； （3）时间序列数据固有的复杂时间依赖性。为了应对这些挑战，我们开发了一个两阶段的元算法，可以保证有效地恢复每个地面lds型号，直至错误$ \ tilde {o}（\ sqrt {d/t}）$，其中$ t $是总样本尺寸。我们通过数值实验来验证我们的理论研究，证实了所提出的算法的功效。
We study the problem of learning a mixture of multiple linear dynamical systems (LDSs) from unlabeled short sample trajectories, each generated by one of the LDS models. Despite the wide applicability of mixture models for time-series data, learning algorithms that come with end-to-end performance guarantees are largely absent from existing literature. There are multiple sources of technical challenges, including but not limited to (1) the presence of latent variables (i.e. the unknown labels of trajectories); (2) the possibility that the sample trajectories might have lengths much smaller than the dimension $d$ of the LDS models; and (3) the complicated temporal dependence inherent to time-series data. To tackle these challenges, we develop a two-stage meta-algorithm, which is guaranteed to efficiently recover each ground-truth LDS model up to error $\tilde{O}(\sqrt{d/T})$, where $T$ is the total sample size. We validate our theoretical studies with numerical experiments, confirming the efficacy of the proposed algorithm.
