We study the sparse recovery problem with an underdetermined linear system characterized by a Kronecker-structured dictionary and a Kronecker-supported sparse vector. We cast this problem into the sparse Bayesian learning (SBL) framework and rely on the expectation-maximization method for a solution. To this end, we model the Kronecker-structured support with a hierarchical Gaussian prior distribution parameterized by a Kronecker-structured hyperparameter, leading to a non-convex optimization problem. The optimization problem is solved using the alternating minimization (AM) method and a singular value decomposition (SVD)-based method, resulting in two algorithms. Further, we analytically guarantee that the AM-based method converges to the stationary point of the SBL cost function. The SVD-based method, though it adopts approximations, is empirically shown to be more efficient and accurate. We then apply our algorithm to estimate the uplink wireless channel in an intelligent reflecting surface-aided MIMO system and extend the AM-based algorithm to address block sparsity in the channel. We also study the SBL cost to show that the minima of the cost function are achieved at sparse solutions and that incorporating the Kronecker structure reduces the number of local minima of the SBL cost function. Our numerical results demonstrate the effectiveness of our algorithms compared to the state-of-the-art.

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Graphs can be models for many real-world systems, where nodes indicate the entities and edges indicate the pairwise connections in between. In various cases, it is important to detect informative subsets of nodes such that the nodes within the subsets are ’closer’ to each other. For example, in a cellular network, determining appropriate node subsets can reduce the operation costs. A subset is usually called a cluster. This leads to the graph clustering problem. Furthermore, plenty of systems in the real world are changing over time, and consequently, graphs as models vary with time as well. It is thus also important to update the clusters when the graph changes.

In this thesis work, we studied two problems from the cellular network background. We needed to partition graphs that have certain structures and cluster their nodes to minimize certain cost functions. In the first problem, we partitioned a bipartite graph by minimizing the so-called MinMaxCut cost function, while in the second problem, we partitioned a structured graph by minimizing the so-called Modified-MinMaxCut cost function. The structural property of the graph is incorporated in defining this new cost function. The solutions we proposed are under the framework of spectral clustering, where one relies on the eigenvectors of the graph matrices, e.g., the Laplacian matrix or the adjacency matrix, and any clustering algorithm, e.g., K-means, to partition nodes into disjoint clusters.

Furthermore, for the time-variant graph, we decomposed the problem into two steps. First, we transformed the variations in the graph topology into perturbations to the graph matrices. Then we transformed the update of the clusters into an update of the (generalized) eigenvectors of these graph matrices. We utilized matrix perturbation theory to update the generalized eigenvectors and then update the clusters. Our simulations showed that on synthetic data, the proposed method can efficiently track the eigenvectors and the clusters generated by the updated eigenvectors have almost the same cost function value as that of exact computation.