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Whenever I look for papers involving semi-supervised learning, I always find some that talk about graph semi-supervised learning (e.g. A Unified Framework for Data Poisoning Attack to Graph-based Semi-supervised Learning).

What is the difference between graph semi-supervised learning and normal semi-supervised learning?

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The authors of your cited paper use the term graph-based semi-supervised learning (G-SSL) to refer to semi-supervised learning techniques which take graph structured data as their input.

Given their main example, the MNIST dataset, is not graph structured, they detail a method for converting the raw Euclidean data $X$ into said form (represented by its adjacency matrix $S$), and then compute the Laplacian $L$ of this graph:

We consider the graph-based semi-supervised learning (G-SSL) problem. The input include labeled data $X_{l} ∈ \mathbb{R}^{n_{l}×d}$ and unlabeled data $X_{u} \in \mathbb{R}^{n_{u}×d}$, we define the whole features $X = [X_{l}; X_{u}]$. Denoting the labels of $X_{l}$ as $y_{l}$, our goal is to predict the labels of test data $y_{u}$. The learner applies algorithm $A$ to predict $y_{u}$ from available data $\{X_{l}, y_{l}, X_{u}\}$. Here we restrict $A$ to label propagation method, where we first generate a graph with adjacency matrix $S$ from Gaussian kernel: $S_{ij} = \exp(−γ\lVert x_i − x_j\rVert ^{2})$, where the subscripts $x_{i(j)}$ represents the $i(j)$-th row of $X$. Then the graph Laplacian is calculated by $L = D − S$, where $D = \text{diag}\{\sum_{k=1}^{n} S_{ik}\}$ is the degree matrix.

This is consistent with the terminology as used in other literature:

Semi-supervised learning for node-level classification. Given a single network with partial nodes being labeled and others remaining unlabeled, ConvGNNs can learn a robust model that effectively identifies the class labels for the unlabeled nodes [22]. To this end, an end-to-end framework can be built by stacking a couple of graph convolutional layers followed by a softmax layer for multi-class classification.

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