Fourier Analysis of Signals on Directed Acyclic Graphs (DAG) Using Graph Zero-Padding

Directed acyclic graphs (DAGs) are used for modeling causal relationships, dependencies, and flows in various systems. However, spectral analysis becomes impractical in this setting because the eigen-decomposition of the adjacency matrix yields all eigenvalues equal to zero. This inherent property of DAGs results in an inability to differentiate between frequency components of signals on such graphs. This problem can be addressed by alternating the Fourier basis or adding edges in a DAG. However, these approaches change the physics of the considered problem. To address this limitation, we propose a graph zero-padding approach. This approach involves augmenting the original DAG with additional vertices that are connected to the existing structure. The added vertices are characterized by signal values set to zero. The proposed technique enables the spectral evaluation of system outputs on DAGs (in almost all cases), that is the computation of vertex-domain convolution without the adverse effects of aliasing due to changes in a graph structure, with the ultimate goal of preserving the output of the system on a graph as if the changes in the graph structure were not done.

Improved Coherence Index-Based Bound in Compressive Sensing

Within the Compressive Sensing (CS) paradigm, sparse signals can be reconstructed based on a reduced set of measurements. Reliability of the solution is determined by the uniqueness condition. With its mathematically tractable and feasible calculation, coherence index is one of very few CS metrics with a considerable practical importance. In this paper, we propose an improvement of the coherence based uniqueness relation for the matching pursuit algorithms. Starting from a simple and intuitive derivation of the standard uniqueness condition based on the coherence index, we derive a less conservative coherence index-based lower bound for signal sparsity. The results are generalized to the uniqueness condition of the $l_0$-norm minimization for a signal represented in two orthonormal bases.