arXiv：cs.LG· Aravinda Kanchana Ruwanpathirana, Hemant Tyagi, Sunny G. W. Wang·· 4 小时前
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
AI 导读
研究提出一种偏差校正估计器，用于在每一时刻仅能观测到少量数据时学习凸集上的结构化线性动力系统，并给出统计误差的非渐近界，该界取决于凸集局部复杂度、轨迹长度 T 与子采样概率 p。研究还证明了投影梯度下降算法的收敛性，并将理论应用于子空间、双等调矩阵及行由 Lipschitz 函数采样构成的矩阵三类场景。结果显示，在 T 远小于无约束情形所需值、且 p = o(1) 时仍可有效恢复转移矩阵。
正文
Abstract: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 $p = o(1)$.
| Comments: | 62 pages, 3 figures |
| Subjects: | Machine Learning (stat.ML); Machine Learning (cs.LG); Systems and Control (eess.SY); Optimization and Control (math.OC); Statistics Theory (math.ST) |
| Cite as: | arXiv:2610.11869 [stat.ML] |
| (or arXiv:2610.11869v1 [stat.ML] for this version) | |
| https://doi.org/10.48550/arXiv.2610.11869 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Aravinda Kanchana Ruwanpathirana [view email]
[v1]
Thu, 8 Oct 2026 12:44:53 UTC (96 KB)
来源：arXiv：cs.LG · arxiv.org