从缺失观测中学习结构化线性动力系统
Learning structured linear dynamical systems from missing observations
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中文总结 AI 辅助
本文针对每个时间点仅含少量观测的场景，提出学习凸集上结构化线性动力系统的估计器，结合投影梯度下降算法，在$T$远小于无约束情形、$p = o(1)$时可有效恢复转移矩阵。
中文摘要 AI 辅助
我们研究在凸集$\boldsymbol{\textit{K}}$上学习结构化线性动力系统的问题，其中每个时间点仅能获取一小部分观测值。本文提出一种最小化经偏差校正的、可能非凸的目标函数的估计器，得到了统计误差的非渐近界，该界依赖于$\boldsymbol{\textit{K}}$的局部复杂度、轨迹长度$T$以及子采样概率$p$，同时还证明了投影梯度下降算法的收敛性。将该通用理论应用于三类场景：（i）$\boldsymbol{\textit{K}}$为子空间；（ii）$\boldsymbol{\textit{K}}$为双保序矩阵集合；（iii）$\boldsymbol{\textit{K}}$为行由采样Lipschitz函数构成的矩阵集合。研究表明，当$T$远小于无约束情形所需的长度且$p = o(1)$时，可实现转移矩阵的有效恢复。
英文摘要
We consider the problem of learning structured linear dynamical systems over convex sets $\mathcal{K}$, where only a small subset of the observations are available at each time point. An estimator which minimizes a bias-corrected, potentially non-convex objective function is proposed. Non-asymptotic bounds are obtained for the statistical error, which depend on the local complexity of $\mathcal{K}$, the trajectory length $T$, and the sub-sampling probability $p$. Convergence of the projected gradient descent algorithm is also established. The general theory is applied to settings where (i) $\mathcal{K}$ is a subspace, (ii) $\mathcal{K}$ is the set of bi-isotonic matrices, and (iii) $\mathcal{K}$ is the set of matrices whose rows are formed by sampling Lipschitz functions. We show meaningful recovery of the transition matrix is possible for values of $T$ much smaller than what is required in the unconstrained case, and for $p = o(1)$.
发表机构
- Division of Mathematical Sciences, SPMS, NTU Singapore（南洋理工大学数学科学分院，SPMS）
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