[PR 2026]Hypergraph-based semantic and topological self-supervised learning for brain disease diagno-优快云博客

①Hyper graph is represented by $\mathcal{H}=\{\mathcal{V},\mathcal{E}\}$ , where $\mathcal{V}$ denotes brain region set and $\mathcal{E}$ is hyper edge set.

②The subset $\overline{\mathcal{V}}\supset\mathcal{V}$ is marked by $[MASK]$ , so the node feature matrix can be noted by:

$\widehat{\mathbf{X}}_{v}= \begin{cases} \mathbf{X}[M], & v\in\mathcal{V}, \\ \mathbf{X}_{v}, & v\notin\mathcal{V}. \end{cases}$

where ${\widehat{\mathbf{X}}}\in\mathbb{R}^{N\times C}$

③The hypergraph encoder processes node feature matrix and hyperedge index to high order semantic embedding $\mathbf{C}_{v}\in\mathbb{R}^{N\times D}$ :

$\mathbf{C}_v=\mathrm{Encoder}\left(\widehat{\mathbf{X}}_v,\mathbf{H}\right)$

（2）Semantic reconstruction decoding

①To avoid over smooth, they define $[DMASK]$ token for node subset $\overline{\mathcal{V}}$ :

$\widehat{\mathbf{C}}_v= \begin{cases} \mathbf{C}\left[DM\right],\quad & v\in\overline{\mathcal{V}}, \\ \mathbf{C}_v,\quad & v\not\in\overline{\mathcal{V}}. \end{cases}$

where ${\widehat{\mathbf{C}}}\in\mathbb{R}^{N\times D}$

②The embedding needs decoding:

$\mathbf{X}_v^\prime=\mathrm{Decoder}\left(\widehat{\mathbf{C}}_v,\mathbf{H}\right)$

（3）Loss function design

①Reonstruction loss:

$\begin{aligned} & \cos\left(\mathbf{x}_{v},\mathbf{x}_{v}^{\prime}\right)=\frac{\mathbf{x}_{v}^{\top}\mathbf{x}_{v}^{\prime}}{\|\mathbf{x}_{v}\|\|\mathbf{x}_{v}^{\prime}\|}, \\ & \mathcal{L}_{\mathrm{seman}}=\frac{1}{|\mathcal{Y}|}\sum_{v\in\mathcal{Y}}\left[1-\cos\left(x_{v},x_{v}^{\prime}\right)\right]^{\lambda_{s}},\quad\lambda_{s}\geq1. \end{aligned}$

2.4.3. High-order topology-aware SSL

（1）Topology augmentation and encoding

①“具体来说，对于每个超边，其中的每个节点都会被随机保留或丢弃，概率为 $p$ ，从而产生新的超边子集。随后，计算此子集与原始超边之间的差集，以形成增强的超边。为避免产生重复的超边（即具有相同节点组合的冗余超边），我们向这些冗余超边随机添加顶点以确保它们的唯一性。”

②They generate 2 enhanced hypergraph $\mathcal{H}_{a}=\{\mathcal{V},\mathcal{E}_{a}\}$ and $\mathcal{H}_{b}=\{\mathcal{V},\mathcal{E}_{b}\}$ with the same number of hyperedge

（2）Structural distance measurement

①Brain network hypergraph can be represented by $\mathcal{H}=(\mathcal{V},\mu,\mathcal{E},\eta,\kappa)$ , where $\mu$ and $\eta$ are probability measures, $\kappa:\mathcal{V}\times\mathcal{E}\to\mathbb{R}$ is relationship between brain regions and functional connections

②Regularize the distribution of hypergraph:

$\mu_{a}\left(v\right)=\frac{\deg_{\mathcal{H}_{a}}\left(v\right)}{\sum_{v\in\mathcal{Y}}\deg_{\mathcal{H}_{a}}\left(v\right)},\quad\mu_{b}\left(v\right)=\frac{\deg_{\mathcal{H}_{b}}\left(v\right)}{\sum_{v\in\mathcal{Y}}\deg_{\mathcal{H}_{b}}\left(v\right)}$

③The structural distance between corresponding hyperedges:

$d_{e,\rho}^{\mathbf{H}}=\inf_{\pi\in\Pi(\mu_{a},\mu_{b})}\int_{|\mathcal{V}|\times|\mathcal{V}|}|\kappa\left(v,e_{a}\right)-\kappa\left(v^{\prime},e_{b}\right)|\pi\left(dv\times dv^{\prime}\right)$

where $\pi$ is a coupling function between the node distributions

（3）Topology-aware contrastive learning

①Similarity weight of hyperedges:

$\gamma_e=e^{-\lambda_t\cdot d_{H,\rho}}$

where $\lambda_{t}$ denotes hyperparameters

②Contrastive loss of edge pairs:

$\mathcal{L}_{topol}\left(e\right)=-\gamma_{e}\cdot\log\frac{\exp(\mathbf{c}_{e}\cdot\mathbf{c}_{e^{\prime}}/\tau)}{\sum_{e^{\prime}\in E_{b}}\exp(\mathbf{c}_{e}\cdot\mathbf{c}_{e^{\prime}}/\tau)}$