Due to their effectiveness in capturing similarities between different entities, graphical models are widely used to represent datasets that reside on irregular and complex manifolds. Graph signal processing offers support to handle such complex datasets. By extending the digital signal processing conceptual frame from time and frequency domain to graph domain, operators such as graph shift, graph filter and graph Fourier transform are defined. In this dissertation, two novel graph filter design methods are proposed. First, a graph filter with multiple shift matrices is applied to semi-supervised classification, which can handle features with uneven qualities through an embedded feature importance evaluation process. Three optimization solutions are provided: an alternating minimization method that is simple to implement, a convex relaxation method that provides a theoretical performance benchmark and a genetic algorithm, which is computationally efficient and better at configuring overfitting. Second, a graph filter with splitting-and-merging scheme is proposed, which splits the graph into multiple subgraphs. The corresponding subgraph filters are trained parallelly and in the last, by merging all the subgraph filters, the final graph filter is obtained. Due to the splitting process, the redundant edges in the original graph are dropped, which can save computational cost in semi-supervised classification. At the same time, this scheme also enables the filter to represent unevenly sampled data in manifold learning. To evaluate the performance of the proposed graph filter design approaches, simulation experiments with synthetic and real datasets are conduct. The Monte Carlo cross validation method is employed to demonstrate the need for the proposed graph filter design approaches in various application scenarios. Criterions, such as accuracy, Gini score, F1-score and learning curves, are provided to analyze the performance of the proposed methods and their competitors.

For a sensor array, part of its elements may fail to work due to hardware failures. Then the missing data may distort in the beam pattern or decrease the accuracy of direction-of-arrival (DOA) estimation. Therefore, considerable research has been conducted to develop algorithms that can estimate the missing signal information. On the other hand, through those algorithms, array elements can also be selectively turned off while the missed information can be successfully recovered, which will save power consumption and hardware cost.

Conventional approaches focusing on array element failures are mainly based on interpolation or sequential learning algorithm. Both of them rely heavily on some prior knowledge such as the information of the failures or a training dataset without missing data. In addition, since most of the existing approaches are developed for DOA estimation, their recovery target is usually the co-variance matrix but not the signal matrix.

In this thesis, a new signal recovery method based on matrix completion (MC) theory is introduced. It aims to directly refill the absent entries in the signal matrix without any prior knowledge. We proposed a novel overlapping reshaping method to satisfy the applying conditions of MC algorithms. Compared to other existing MC based approaches, our proposed method can provide us higher probability of successful recovery. The thesis describes the principle of the algorithms and analyzes the performance of this method. A few application examples with simulation results are also provided.