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.

This paper examines learning the optimal filtering policy, known as the Kalman gain, for a linear system with unknown noise covariance matrices using noisy output data. The learning problem is formulated as a stochastic policy optimiza- tion problem, aiming to minimize the output prediction error. This formulation provides a direct bridge between data-driven optimal control and, its dual, op- timal filtering. Our contributions are twofold. Firstly, we conduct a thorough convergence analysis of the stochastic gradient descent algorithm, adopted for the filtering problem, accounting for biased gradients and stability constraints. Secondly, we carefully leverage a combination of tools from linear system theory and high-dimensional statistics to derive bias-variance error bounds that scale logarithmically with problem dimension, and, in contrast to subspace methods, the length of output trajectories only affects the bias term.

From an optimization point of view, resource allocation is one of the cornerstones of research for addressing problems commonly arising in applications such as power outages and traffic jams. In this paper, we take a data-driven approach to estimate an optimal nodal restoration sequence for immediate recovery of infrastructure networks after natural disasters such as earthquakes. Data is generated from td-INDP, a high-fidelity simulator of optimal restoration strategies for interdependent networks, and employ artificial neural networks to approximate those strategies. Despite the fact that the underlying problem is NP-complete, the restoration sequences obtained by our method are observed to be nearly optimal. In addition, by training multiple models—the so-called estimators—for a variety of resource availability levels, our proposed method balances a tradeoff between resource utilization and restoration time. Decision-makers can use our trained models to allocate resources more efficiently before and after contingencies, and in turn, improve the community resilience. Besides their predictive power, such trained estimators unravel the effect of interdependencies among different nodal functionalities in the restoration strategies. We showcase our methodology by the real-world interdependent infrastructure of Shelby County, TN.