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Graph Signal Processing (GSP), as the field concerned with the extension of classical signal processing concepts to the graph domain, is still at the beginning on the path toward providing a generalized theory of signal processing. As such, this thesis aspires to conceive the theory of sparse representations on graphs by traversing the cornerstones of wavelet and sampling theory on graphs. Beginning with the novel topic of graph spline wavelet theory, we introduce families of spline and e-spline wavelets, and associated filterbanks on circulant graphs, which lever- age an inherent vanishing moment property of circulant graph Laplacian matrices (and their parameterized generalizations), for the reproduction and annihilation of (exponen- tial) polynomial signals. Further, these families are shown to provide a stepping stone to generalized graph wavelet designs with adaptive (annihilation) properties. Circulant graphs, which serve as building blocks, facilitate intuitively equivalent signal processing concepts and operations, such that insights can be leveraged for and extended to more complex scenarios, including arbitrary undirected graphs, time-varying graphs, as well as associated signals with space- and time-variant properties, all the while retaining the focus on inducing sparse representations. Further, we shift from sparsity-inducing to sparsity-leveraging theory and present a novel sampling and graph coarsening framework for (wavelet-)sparse graph signals, inspired by Finite Rate of Innovation (FRI) theory and directly building upon (graph) spline wavelet theory. At its core, the introduced Graph-FRI-framework states that any K-sparse signal residing on the vertices of a circulant graph can be sampled and perfectly reconstructed from its dimensionality-reduced graph spectral representation of minimum size 2K, while the structure of an associated coarsened graph is simultaneously inferred. Extensions to arbitrary graphs can be enforced via suitable approximation schemes. Eventually, gained insights are unified in a graph-based image approximation framework which further leverages graph partitioning and re-labelling techniques for a maximally sparse graph wavelet representation.

The approach to solving problems in communications from the perspective of lattice coding and decoding has received much attention in recent years as simple and elegant solutions can often be obtained. In this thesis, we develop several new techniques for the lattice problems involved, that allow the performance limit of lattice coding and the complexity-performance trade-off of lattice decoding to be better understood. First, we propose a greedy technique for lattice reduction (LR) which is usually taken as a preprocessing step in lattice decoding. Realizing the size reduction operations in LR can possibly increase the lengths of basis vectors, it motivates us to replace the size reduction steps through defining length-reduction operations. The principle of such operations is to be greedy because we need to pursuit the shortest possible candidates. The length-reduction technique is incorporated into the popular Korkine-Zolotarev (KZ) and Lenstra-Lenstra-Lov\'asz (LLL) algorithms. The resulted boosted KZ reduction is shown to have the tightest bounds on lengths in all LR algorithms. We apply the boosted KZ and LLL algorithms to designing integer-forcing linear receivers for multiple-input and multiple-output (MIMO) communications. Our simulations confirm their rate and complexity advantages. Second, we consider a graphical technique for the closest vector problem in vector perturbation (VP) precoding. The approximate message passing (AMP) algorithm based on factor graphs has become perhaps the most popular low-complexity algorithm for solving convex problems in compressed sensing. It is however unclear whether AMP can be beneficial for a lattice decoding problem which is NP-hard. We propose a hybrid framework to improve the performance of LR aided decoding, which particularly suits VP. Our work shows that the AMP algorithm can be beneficial for a lattice decoding problem whose data symbols lie in integers $\mathbb$ and entries of the lattice basis may bot be i.i.d. Gaussian. Numerical results confirm the low-complexity AMP algorithm can improve the symbol error rate performance of LR aided precoding significantly. Lastly, the hybrid scheme is also proved effective when solving the data detection problem of massive MIMO systems without using LR. Third, we examine algebraic lattice reduction in Compute-and-Forward that employs lattice codes based on the rings of complex quadratic fields. As the underlying lattices are complex, we first examine Hermite's constant and Minkowski's theorems in this context. Then we present an algebraic LLL algorithm to reduce these complex lattices. We show that the lower bound of Lov\'asz's constant in algebraic LLL depends on the covering radius of rings, while the upper bound of it implies that only norm-Euclidean domains can be proper. The rotation step in algebraic LLL can be simplified after introducing quaternions. Finally, we propose a novel scheme for Compute-and-Forward in block-fading channels, which is referred to as Ring Compute-and-Forward because the fading coefficients are quantized to the canonical embedding of a ring of algebraic integers. Thanks to the multiplicative closure of the algebraic lattices employed, a relay is able to decode an algebraic-integer linear combination of lattice codewords. We analyze its achievable computation rates and show it outperforms conventional Compute-and-Forward based on $\mathbb{Z}$-lattices. By investigating the effect of Diophantine approximation by algebraic conjugates, we prove that the degrees-of-freedom (DoF) of the computation rate is $n/L$, while the DoF of the sum-rate is $n$, where $n$ is the number of blocks and $L$ is the number of users.

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