Digraph Signal Processing With Generalized Boundary Conditions

Seifert, Bastian; Püschel, Markus

doi:10.1109/tsp.2021.3051267

Cited by 12 publications

(9 citation statements)

References 43 publications

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“…It can be solved using a combinatorial approach (exhaustive search), but only for a small graph size N . Another recently proposed approach uses matrix perturbation theory [25]. Here, we will propose a simple and fast, albeit sub-optimal strategy.…”

Section: General Dag Zero-paddingmentioning

confidence: 99%

“…Furthermore, we will consider the USA temperature map, commonly used as a benchmark case for directed graphs [17,22,25]. The results are shown in Fig.…”

Section: Examplesmentioning

confidence: 99%

“…The main contribution of this study is introducing an algorithm to address the gap identified in [25]. This is achieved through implementing two operations.…”

Section: Introductionmentioning

confidence: 99%

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

Stankovic,

Daković,

Bardi

et al. 2024

Preprint

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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 eigendecomposition 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 performed.

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Section: General Dag Zero-paddingmentioning

confidence: 99%

“…Furthermore, we will consider the USA temperature map, commonly used as a benchmark case for directed graphs [17,22,25]. The results are shown in Fig.…”

Section: Examplesmentioning

confidence: 99%

See 1 more Smart Citation

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

Stankovic,

Daković,

Bardi

et al. 2024

Preprint

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show abstract

“…To improve explanations on the impact of added edges, the emerging publications in directed graphs like [49] would be worthy to leverage. In addition, recent work in perturbation theory on directed graphs in [56] would interesting to explore to the change from a directed cycle to another graph configuration. Which ever one of these avenues one ultimately chooses to explore, the work in this dissertation lays out a clear plan to make these and other contributions in single-snapshot DOA estimation.…”

Section: Opportunities For Future Workmentioning

confidence: 99%

Direction-of-Arrival Estimation Using Signal Processing on Graphs

Alcantara

Atlas

Abadi

2021

2021 IEEE Statistical Signal Processing Workshop (SSP)

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“…One computes a Fourier basis that minimizes the sum of directed variations [8], or evenly spreads them [9], [10]. Other include changing the shift to the Hermitian Laplacian [11], using an approximation based on the Schur decomposition [12], or adding generalized boundary conditions, i.e., additional edges to the digraph [13]. All these are fundamentally different from our approach which is specific to acyclic digraphs and based on a very different notion of shift and Fourier basis.…”

Section: Introductionmentioning

confidence: 98%

Causal Fourier Analysis on Directed Acyclic Graphs and Posets

Seifert¹,

Wendler²,

Püschel³

2022

Preprint

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We present a novel form of Fourier analysis, and associated signal processing concepts, for signals (or data) indexed by edge-weighted directed acyclic graphs (DAGs). This means that our Fourier basis yields an eigendecomposition of a suitable notion of shift and convolution operators that we define. DAGs are the common model to capture causal relationships between data and our framework is causal in that shift, convolution, and Fourier transform are computed only from predecessors in the DAG. The Fourier transform requires the transitive closure of the DAG for which several forms are possible depending on the interpretation of the edge weights. Examples include level of influence, distance, or pollution distribution. Our framework is different from prior GSP: it is specific to DAGs and leverages, and extends, the classical theory of Moebius inversion from combinatorics. For a prototypical application we consider DAGs modeling dynamic networks in which edges change over time. Specifically, we model the spread of an infection on such a DAG obtained from real-world contact tracing data and learn the infection signal from samples assuming sparsity in the Fourier domain.

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Digraph Signal Processing With Generalized Boundary Conditions

Cited by 12 publications

References 43 publications

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

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

Direction-of-Arrival Estimation Using Signal Processing on Graphs

Causal Fourier Analysis on Directed Acyclic Graphs and Posets

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