I was reading the following paper here and had a question about the paragraph on page 1 (in the introduction). The equation being referred to is:

$$ \mathcal{L} = \mathcal{L}_0 + \lambda \mathcal{L}_{\text{reg}}$$ where $$ \mathcal{L}_{\text{reg}} = \sum_{i,j} A_{ij} ||f(X_i) - f(X_j)||^2 = f(X)^T \Delta f(X) $$

and then it goes on to say:

Here, $\mathcal{L}_0$ denotes the supervised loss w.r.t. the labeled part of the graph, $f(\cdot)$ can be a neural network- like differentiable function, $\lambda$ is a weighing factor and X is a matrix of node feature vectors $X_i$. $\Delta = D − A$ denotes the unnormalized graph Laplacian of an undirected graph $\mathcal{G} = (\mathcal{V}, \mathcal{E})$ with $N$ nodes $ v_i \in \mathcal{V}$, edges $(v_i, v_j) \in \mathcal{E}$, an adjacency matrix $A \in > \mathbb{R}^{N\times N}$ (binary or weighted) and a degree matrix $D_{ii} = 􏰂\sum_{j} A_{ij} $. The formulation of Eq. 1 relies on the assumption that connected nodes in the graph are likely to share the same label. This assumption, however, might restrict modeling capacity, as graph edges need not necessarily encode node similarity, but could contain additional information. In this work, we encode the graph structure directly using a neural network model $f(X,A)$ and train on a supervised target $\mathcal{L}_{0}$ for all nodes with labels, thereby avoiding explicit graph-based regularization in the loss function. Conditioning $f(\cdot)$ on the adjacency matrix of the graph will allow the model to distribute gradient information from the supervised loss $\mathcal{L}_0$ and will enable it to learn representations of nodes both with and without labels.

I don't really understand the last sentence, nor do I have an idea where I could go to find more (I haven't managed to find anything useful).


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