This article investigates the distributed estimation problem of an unknown high-dimensional sparse state vector for a stochastic dynamic system. The communication topology randomly switches, and the switching law is governed by a time-homogeneous Markovian chain. By means of the compressed sensing (CS) theory and a diffusion strategy, we propose a compressed distributed Kalman filter (CDKF). That is, each sensor first compresses the original high-dimensional regression data. Then, the covariance intersection fusion rule is utilized to obtain a distributed Kalman filter (DKF) estimate in the compressed low-dimensional space. Afterward, the original high-dimensional sparse state vector can be well recovered by a reconstruction technique. In terms of stability analysis, one of the main difficulties lies in analyzing the product of nonindependent and nonstationary random matrices in the context of time-varying communication topologies. Relying on the stochastic stability theory, the Markov chain theory, and the CS theory, we establish the upper bound for the estimation error under the compressed cooperative excitation condition, which is much weaker than the traditional uncompressed collective observability conditions used in the existing literature. Finally, we provide a simulation example to illustrate the performance of the proposed algorithm.

This paper considers Stackelberg equilibrium (SE) and Nash equilibrium (NE) computation in a class of non-convex network aggregative games with one leader and multiple followers. The cost function of each follower is influenced by its strategy, the leader’s strategy, and its neighbors’ aggregative strategies. Also, the structured non-convex cost function of the leader is the composition of a canonical function and a vector-valued geometrical operator that relies on its strategy and followers’ strategies. In the leader-follower scheme, when the leader has knowledge of the best responses of the followers in a closed form, the SE strategy will be the optimal choice due to its relatively low cost. When the leader does not know the exact expression of followers’ best responses or the leader’s dominance is threatened, NE will be what all players are committed to achieving. The widespread existence of nonconvexity creates a significant challenge for computing the above equilibria in different circumstances. The results in existing convex games are not directly applicable to such a non-convex case, as they get trapped in local equilibria or stationary points rather than global equilibria. Here, we adopt the canonical transformation to reformulate the non-convex games and present the existence condition based on the canonical duality theory. Then two projection gradient algorithms are designed to pursue the SE and the NE, followed by proving the convergence of the algorithms.

In this article, a distributed stochastic gradient (SG) algorithm is proposed where the estimators are aimed to collectively estimate an unknown time-invariant parameter from a set of noisy measurements obtained by distributed sensors. The proposed distributed SG algorithm combines the consensus strategy of the estimation of neighbors with the diffusion of regression vectors. For the theoretical investigation of the proposed algorithm, the main challenge lies in analyzing the influence of the Laplacian matrix on the state transition matrix and the properties of the product of nonindependent and nonstationary random matrices. Some analysis techniques such as graph theory and martingale theory are used to deal with the above issues. A cooperative excitation condition is introduced, under which the convergence of the distributed SG algorithm can be obtained without relying on the independency or stationarity assumptions of regression vectors which are commonly used in the existing literature. Furthermore, the convergence rate of the algorithm can be established. Finally, we show that all the sensors can cooperate to fulfill the estimation task even though any individual sensor cannot by a simulation example.