[NeurIPS]Exploitation of a Latent Mechanism in Graph Contrastive Learning: Representation Scattering

①Graph setting: $\mathcal{G}=(\mathcal{V},\mathcal{E})$ , where $\mathcal{V}=\{v_{1},v_{2},\ldots,v_{N}\}$ is node set, $\varepsilon\subseteq\mathcal{V}\times\mathcal{V}$ is edge set

②Feature matrix: $\mathbf{X}\in\mathbb{R}^{N\times D}$ , $N$ denotes the number of nodes, $D$ is the feature dimension

③Adjacency matrix: binarized $\mathbf{A}\in\mathbb{R}^{N\times N}$

④Degree matrix: $\mathbf{D}=\mathrm{diag}(d_{1},d_{2},\ldots,d_{N})$ , each element $d_{i}=\sum_{i\in\mathcal{V}}\mathbf{A}_{ij}$

⑤Degree normalized adjacency matrix: $\mathbf{A}_{\mathrm{sym}}=\mathbf{D}^{-1/2}(\mathbf{A}+\mathbf{I})\mathbf{D}^{-1/2}$

2.4. Representation Scattering in GCL

①Definition of representation scattering: in a $d$ -dimensional embedding space $\mathbb{R}^d$ comprising $n$ vectors organized into a matrix $\mathbf{V}\in\mathbb{R}^{n\times d}$ , consider a subspace $\mathbb{S}^{k}(1\leq k\leq d)$ of $\mathbb{R}^{d}$ and a scatter center $\mathbf{c}$ . there are 2 constraint:

Center-Away Constraint: Node representations are encouraged to be distant from the scattered center $\mathbf{c}$

Uniformity Constraint: Node representations are uniformly distributed over the subspace $\mathbb{S}^{k}$

2.4.1. DGI-like methods

①DGI-like methods employ mutual information discriminator $\mathcal{D}\left ( \cdot \right )$ to maximizes the mutual information between nodes and their source graphs to train the model

②Assumption: a) for normalized propagation matrix $\mathbf{\tilde{A}}=\mathbf{D}^{-1}\mathbf{\hat{A}}$ where $\mathbf{\hat{A}}=\mathbf{{A}}+\mathbf{{I}}$ , b) DGI generates the corrupted graph by randomly shuffling the entities in the feature matrix $\mathbf{X}$ and keeping adjacency matrix $\mathbf{A}$ unchanged, c) the orginal data are class-balanced (the numbers are SAME)

③Theorem: At the node level, minimizing the DGI loss is equivalent to maximizing the Jensen Shannon (JS) divergence between the local semantic distribution in the original graph $\mathcal{G}$ and its average distribution (corrupted grpah $\tilde{\mathcal{G}}$ ), i.e.

$Min\left ( \mathcal{L}_{DGI}\right )\Leftrightarrow Max\left ( JS\left ( \mathcal{G}|| \tilde{\mathcal{G}}\right ) \right )$

④Single layer GNN:

$\mathbf{H}=\mathrm{GNN}(\mathbf{A},\mathbf{X})=\tilde{\mathbf{A}}\mathbf{X}\mathbf{W}^{\prime},\mathbf{h}_{i}=\sum_{j\in\mathcal{N}_{i}}\alpha_{ij}\mathbf{x}_{j}\cdot\mathbf{W}^{\prime}=\sum_{j\in\mathcal{N}_{i}}\frac{1}{d_{i}}\mathbf{x}_{j}\cdot\mathbf{W}^{\prime}\\ \forall j\in\mathcal{N}_{i},\mathbf{x}_{j}\sim p_{i}(\mu_{i},\sigma_{i}^{2})$

where $\mathbf{W}^{\prime}=\xi(\mathbf{W})$ , $\mathbf{W}\in\mathbb{R}^{D\times K}$ with ${{\xi}}$ is activation function, $\mathcal{N}_{i}$ is first order neighbors of node $v_i$

⑤Mean of aggregated representation $\mathbf{h}_i$ :

$\mathbb{E}[\mathbf{h}_i]=\mathbb{E}\left[\sum_{j\in\mathcal{N}(i)}\alpha_{ij}\mathbf{x}_j\cdot\mathbf{W}^{\prime}\right]=\sum_{j\in\mathcal{N}(i)}\alpha_{ij}\mathbb{E}[\mathbf{x}_j]\cdot\mathbf{W}^{\prime}=\sum_{j\in\mathcal{N}(i)}\alpha_{ij}\mu_i\cdot\mathbf{W}^{\prime}$

⑥Variance of aggregated representation $\mathbf{h}_i$ :

$\mathrm{Var}(\mathbf{h}_{i})=\mathrm{Var}\left(\sum_{j\in\mathcal{N}(i)}\alpha_{ij}\mathbf{x}_{j}\cdot\mathbf{W}^{\prime}\right)=\sum_{j\in\mathcal{N}(i)}\alpha_{ij}\mathrm{Var}(\mathbf{x}_{j})\cdot\mathbf{W}^{\prime2}=\sigma_{i}^{2}\sum_{j\in\mathcal{N}(i)}\alpha_{ij}^{2}\cdot\mathbf{W}^{\prime2}$

⑦The mean and variance of representation $\mathbf{h}_i$ 's distribution: $\mu_i\cdot \mathbf{W}^{\prime}$ / $\sigma^{2}\sum_{j\in\mathcal{N}(i)}\alpha_{ij}^{2}\cdot\mathbf{W}^{\prime2}$ , $\mathbf{h}_{i}\sim p_{i}^{\prime}(\mu_{i}\cdot\mathbf{W}^{\prime},\sigma_{i}^{2}\cdot\mathbf{W}^{\prime2})$

⑧Randomly extract feature vectors $\tilde{\mathbf{x}}_i$ from original graph $\mathbf{X}$ with distribution $p_{\mathrm{data}}(\mu,\sigma^{2})$ , $\mathbf{\hat{h}}_{i}\sim p_{\mathrm{data}}^{\prime}(\mu\cdot\mathbf{W}^{\prime},\sigma^{2}\cdot\mathbf{W}^{\prime2})$

⑨Corollary: the mean of original graph as center $\mathbf{c}$ , original representation space as subspace $\mathbb{S}^{k}$

⑩t-SNE embedding of DGI on Co.CS dataset:

where blue and red points are negative and positive samples

Corollary n. 推论；必然的结果

2.4.2. InfoNCE-based methods

①Theorem: define $\mathbf{\bar{h}}=1/n\sum_{i=1}^{n}\mathbf{h}_{i}$ , $sim\left ( \cdot \right )$ be the cosine similarity function. For node $v_i$ , the lower bound of InfoNCE loss is:

$\mathcal{L}_{InfoNCE}(\mathbf{h}_{i})\geq sim(\mathbf{h}_{i},\mathbf{\bar{h}})+\ln(2n)$

however, it's inefficient due to calculating of every representation pair

2.4.3. BGRL-like methods

①Batch normalization for a feature vector $\mathbf{x}_i$ :

$\mathrm{BN}(\mathbf{x}_{i})=\gamma(\mathbf{x}_{i}-\mu)/\sqrt{\sigma^{2}+\epsilon}+\beta$

where $\mu$ and $\sigma ^2$ are mean and variance, $\gamma$ and $\beta$ are learnable scale and shift parameters, $\epsilon$ is a small constant

②The impact of Batch Normalization in BGRL:

2.5. Methodology

①The overall framework of SGRL:

2.5.1. Representation Scattering Mechanism (RSM)

①Apply $\text{Trans}\left ( \cdot \right )$ to transform $\mathbb{R}^d$ to $\mathbb{S}^k$ with L2 norm each row:

$\tilde{\mathbf{h}}_{i}=\mathrm{Trans}_{\mathbb{R}^{d}\to\mathbb{S}^{k}}(\mathbf{h}_{i})=\frac{\mathbf{h}_{i}}{\mathrm{Max}(\|\mathbf{h}_{i}\|_{2},\varepsilon)},\quad\mathbb{S}^{k}=\{\tilde{\mathbf{h}}_{i}:\|\tilde{\mathbf{h}}_{i}\|_{2}=1\}$

where $\mathbf{\mathbf{}h}_i$ generated by target encoder, $\|\tilde{\mathbf{h}}_{i}\|_{2}=(\sum_{j=1}^{k}\tilde{\mathbf{h}}_{ij}^{2})^{\frac{1}{2}}$ , $\epsilon$ is a small vale to avoid division by zero

②Scattering center:

$\mathcal{L}_{\text{scattering}}=-\frac{1}{n}\sum_{i=1}^{n}\|\tilde{\mathbf{h}}_{i}-\mathbf{c}\|_{2}^{2},\quad\mathbf{c}=\frac{1}{n}\sum_{i=1}^{n}\tilde{\mathbf{h}}_{i}$

2.5.2. Topology-based Constraint Mechanism(TCM)

①For connected $\forall v_{i},v_{j}\in V,\mathbf{h}_{i},\mathbf{h}_{j}\in\mathbb{S}^{k}$ , there is a threshold $d:\left \| \mathbf{h}_i-\mathbf{h}_j \right \|^2_2< d,\, \text{if}\left ( i,j \right )< \mathcal{V}$ , and $\|\mathbf{h}_{i}-\mathbf{h}_{j}\|_{2}^{2}>d,\mathrm{if}(i,j)\notin\mathcal{V}$

②Topology connection:

$\mathbf{H_{target}^{topology}}=\mathbf{AH_{target}}$

③Topology-based Constraint Mechanism (TCM)

$\mathbf{H_{online}^{topology}}=\mathbf{\hat{A}^{k}H_{online}}+\mathbf{H_{online}}$

where $\text{ k}$ denotes the order of neighbors

④Prediction:

$\mathbf{Z_{online}}=q_{\theta}(\mathbf{H_{online}^{topology}})$

⑤Aim: align $\mathbf{Z_{online}}$ with $\mathbf{H_{target}}$

⑥Alignment loss:

$\mathcal{L}_{\mathrm{alignment}}=-\frac{1}{N}\sum_{i=1}^{N}\frac{\mathbf{Z}_{(\mathrm{online},i)}^{\top}\mathbf{H}_{(\mathrm{target},i)}}{\|\mathbf{Z}_{(\mathrm{online},i)}\|\|\mathbf{H}_{(\mathrm{target},i)}\|}$

⑦Employ Exponential Moving Average at the end of each training epoch:

$\phi\leftarrow\tau\phi+(1-\tau)\theta$

where $\tau \in \left [ 0,1 \right ]$ is target decay rate

2.6. Experiment

①5 datasets: AmazonPhoto (Photo) and Amazon-Computers (Computers), WikiCS, Coauthor-CS (Co.CS), Coauthor-Physics (Co.Physics)

②⭐Data split: 10% for training and 90% for testing

③Running times: 20

2.6.1. Performance Analysis

①Node classification performance table:

②t-SNE visualization on CS dataset:

③Performance on Clustering in terms of NMI and homogeneity. Optimal results are shown in bold and suboptimal results are underlined:

2.6.2. Model Analysis

①The ablation of $\text{k}$ :

②Module ablation:

2.7. Conclusion

3. Reference

He, D. et al. (2024) 'Exploitation of a Latent Mechanism in Graph Contrastive Learning: Representation Scattering', NeurIPS.

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