In this work, we develop a new complexity metric for an important class of low-rank matrix optimization problems, where the metric aims to quantify the complexity of the nonconvex optimization landscape of each problem and the success of local search methods in solving the problem. The existing literature has focused on two complexity measures. The RIP constant is commonly used to characterize the complexity of matrix sensing problems. On the other hand, the sampling rate and the incoherence are used when analyzing matrix completion problems. The proposed complexity metric has the potential to unify these two notions and also applies to a much larger class of problems. To mathematically study the properties of this metric, we focus on the rank-1 generalized matrix completion problem and illustrate the usefulness of the new complexity metric from three aspects. First, we show that instances with the RIP condition have a small complexity. Similarly, if the instance obeys the We note that a similar complexity metric based on a special case of instances in Section 3.3 was proposed in our conference paper [55]. However, the complexity metric in this work has a different form and is proved to work on a broader set of applications. In addition, we prove several theoretical properties of the metric in this work, which are not included in [55].
We study the identification of a linear timeinvariant dynamical system affected by large-and-sparse disturbances modeling adversarial attacks or faults. Under the assumption that the states are measurable, we develop necessary and sufficient conditions for the recovery of the system matrices by solving a constrained lasso-type optimization problem. In addition, we provide an upper bound on the estimation error whenever the disturbance sequence is a combination of small noise values and large adversarial values. Our results depend on the null space property that has been widely used in the lasso literature, and we investigate under what conditions this property holds for linear time-invariant dynamical systems. Lastly, we further study the conditions for a specific probabilistic model and support the results with numerical experiments.
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