# Analysis of Hypergraph Signals via High-Order Total Variation

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## Abstract

**:**

## 1. Introduction

#### 1.1. Related Works

#### 1.2. Main Works

- We propose an HOTV over hypergraphs, by which we obtain a hypergraph Laplacian and present an orthonormal basis reflecting distinct spectral information. The HOTV aggregates the HGS groupwise instead of pairwise, which describes the dissimilarity of the HGS over the topology in a more comprehensive way;
- We propose a novel signal transformation (a new HGFT) by the orthonormal basis which bridges the vertex domain and the spectral domain of an HGS. We then can process the HGS in the two domains, clearly provide spectral interpretations for all processing of the HGS and put forward a framework for the analysis and processing of the HGS;
- We present hypergraph filtering tasks in the two domains and discuss two specific forms of hypergraph filters, which do provide a new idea for the HGS filtering.

## 2. Hypergraph Signals

**Example**

**1.**

**Example**

**2.**

#### 2.1. Total Variation of Hypergraph Signals

**Example**

**3.**

**Example**

**4.**

#### 2.2. Tensor-Based Representations

#### 2.2.1. Representations of Topologies

**Definition**

**1**

#### 2.2.2. Representations of Signals

## 3. Hypergraph Fourier Transform

#### 3.1. Construction of an Orthonormal Basis

**Example**

**5.**

#### 3.2. Hypergraph Fourier Transform

#### 3.3. Spectral Form of Total Variation

**Example**

**6.**

## 4. Hypergraph Filters

**Example**

**7.**

#### 4.1. Polynomial Filters Based on the Hypergraph Laplacians

#### 4.2. Reducible Filters

**Proposition**

**2.**

## 5. Application

#### 5.1. Hypergraph Label Learning Model

#### 5.2. Experimental Setups and Results

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CP | CANDECOMP/PARAFAC |

GFT | Graph Fourier transform |

GSP | Graph signal processing |

HGFT | Hypergraph Fourier transform |

HGLM | Hypergraph Laplacian matrix |

HGS | Hypergraph signal |

HGSP | Hypergraph signal processing |

HOTV | High-order total variation |

IHGFT | Inverse hypergraph Fourier transform |

ROA | Rank-one approximation |

## Appendix A. Solution of Problem (13)

## Appendix B

**Proposition**

**A1.**

**Proof.**

## Appendix C. Proof of (28)

## Appendix D. Proof of Proposition 2

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**Figure 3.**Three basis vectors of the example hypergraph $\mathcal{H}$. The signal value at each vertex is represented by the color of the vertex. (

**a**) $\mathrm{TV}\left({\mathit{u}}_{1}\right)=0$; (

**b**) $\mathrm{TV}\left({\mathit{u}}_{2}\right)=0.0063$; (

**c**) $\mathrm{TV}\left({\mathit{u}}_{8}\right)=0.4153$.

**Figure 6.**Examples of 15-vertex 4-uniform sub-hypergraphs of the two datasets. Each sub-hypergraph consists of a vertex set, a hyperedge set and vertex indices. (

**a**) The acute inflammations dataset; (

**b**) The iris dataset.

**Figure 7.**Accuracy of a disease diagnosis example using hypergraph label learning with different numbers of known-label agents.

**Figure 8.**Accuracy of a 3-class simulation example using hypergraph label learning with different numbers of known-label agents.

Notation | Description |
---|---|

$\mathcal{H}=(\mathcal{V},\mathcal{E},\mathbf{W})$ | a c-uniform weighted undirected hypergraph |

${\mathbf{L}}_{\left(c\right)}$ | the Laplacian tensor of $\mathcal{H}$ |

$\mathit{s}$ | an HGS |

${\mathit{s}}_{\left(M\right)}$ | an Mth-order HGS |

${\mathcal{S}}_{\left(M\right)}$ | the Mth-order HGS set |

$\widehat{\mathit{s}}$ | a spectral-domain HGS |

$\mathbf{F}$ | a vertex-domain hypergraph filter |

$\widehat{\mathbf{F}}$ | a spectral-domain hypergraph filter |

⊗ | the Kronecker product |

⊠ | the outer product |

${\times}_{n}$ | the n-mode product |

⊙ | the Khatri–Rao product |

$\left[n\right]$ | $\{1,\cdots ,n\}$ |

$\mathrm{mat}(\xb7)$ | tensor matricization |

$\mathrm{vec}(\xb7)$ | tensor vectorization |

$\mathrm{ten}(\xb7)$ | tensorization of a matrix or a vector |

$\mathrm{ROA}(\xb7)$ | the rank-one approximation of any form of a tensor |

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**MDPI and ACS Style**

Qu, R.; Feng, H.; Xu, C.; Hu, B.
Analysis of Hypergraph Signals via High-Order Total Variation. *Symmetry* **2022**, *14*, 543.
https://doi.org/10.3390/sym14030543

**AMA Style**

Qu R, Feng H, Xu C, Hu B.
Analysis of Hypergraph Signals via High-Order Total Variation. *Symmetry*. 2022; 14(3):543.
https://doi.org/10.3390/sym14030543

**Chicago/Turabian Style**

Qu, Ruyuan, Hui Feng, Chongbin Xu, and Bo Hu.
2022. "Analysis of Hypergraph Signals via High-Order Total Variation" *Symmetry* 14, no. 3: 543.
https://doi.org/10.3390/sym14030543