# Re-Enrichment Learning: Metadata Saliency for the Evolutive Personalization of a Recommender System
## 3 Re-Enrichment Learning
### Graph-Based Domain Transfer  
An undirected weighted graph is assigned to every user in order to build a data structure that consists of item categories as the finite set of nodes and the similarities between categories as the finite set of edges. \[G_{k,t}=(V_{k,t},\ E_{k,t})\ \ s.t.\ k\in\{1,...,N\}\ \ \text{and}\ \ t\geq 0 \tag{1}\]  
\[V_{k,t}=\{v_{k,t}(1),\ v_{k,t}(2),\...,\ v_{k,t}(M)\} \tag{2}\]  
\[E_{k,t}=\{e_{k,t}(1,2),\ e_{k,t}(1,3),\...,\ e_{k,t}(M-1,M)\} \tag{3}\]  
As noted in Equations (1)-(3), the graph of the \(k^{th}\) user among \(N\) users at time \(t\), \(G_{k,t}\), is made up of \(M\) nodes \(V_{k,t}\) with their attributes \(v_{k,t}\) and \({}^{M}\)C\({}_{2}\) edges \(E_{k,t}\) with their attributes \(e_{k,t}\). The edge attribute between the two nodes \(v_{k,t}(i)\) and \(v_{k,t}(j)\) is \(e_{k,t}(i,j)\), which can be computed as in Equation (4), where \(i,j\in\{1,\...,\ M\}\). The attribute of an edge represents the combination of the influences and similarity of two nodes. A node is determined to be influential when its attribute is large, and two nodes are determined to be similar when they share similar levels of attributes to that of the same user. Here, the attribute of a node is the metadata saliency, which represents the attentional prominence of the node from the entire set of nodes. Equation (4) is composed of two terms: (i) the likelihood for the influence of a node and (ii) the weight for the similarity between two nodes. A softmax function has been adopted in the first term to produce the influence of a node as a probabilistic value within \([0,1]\) (see Equations (5) and (6)), where \(h\in\{1,\...,\ M\}\), and \(\alpha\) is the influence factor, which determines the degree of importance of an individual feedback (\(s.t.\ \alpha\geq 1\)). In the second term, the inversed substitution between two nodes' attributes is adopted to produce the similarity as a weight value (see Equations (7) and (8)). The attributes of nodes have been normalized to remove scale dependency. \[e_{k,t}(i,j)=\sigma(\alpha\cdot v^{\prime}_{k,t}(i))\cdot\big{(}1-\overset{ \triangle}{v}^{\prime}_{k,t}(i,j)\big{)} \tag{4}\]  
\[\sigma(\alpha\cdot v^{\prime}_{k,t}(i))=\frac{e^{\alpha\cdot v^{\prime}_{k,t}( i)}}{\sum_{h=1}^{M}e^{\alpha\cdot v^{\prime}_{k,t}(h)}} \tag{5}\]  
\[v^{\prime}_{k,t}(i)=\frac{v_{k,t}(i)-\underset{\forall h}{min}v_{k,t}(h)}{ \underset{\forall h}{max}v_{k,t}(h)-\underset{\forall h}{min}v_{k,t}(h)} \tag{6}\]  
\[\overset{\triangle}{v}^{\prime}_{k,t}(i,j)=\frac{\overset{\triangle}{v}^{ \triangle}_{k,t}(i,j)-\underset{\forall h}{min}\overset{\triangle}{v}^{ \triangle}_{k,t}(i,h)}{\underset{\forall h}{max}\overset{\triangle}{v}^{ \triangle}_{k,t}(i,h)-\underset{\forall h}{min}\overset{\triangle}{v}^{ \triangle}_{k,t}(i,h)} \tag{7}\]  
\[\overset{\triangle}{v}^{\triangle}_{k,t}(i,j)=|v_{k,t}(i)-v_{k,t}(j)| \tag{8}\]  
Then, the edges can be denoted by an \(|V_{k,t}|\times|V_{k,t}|\) matrix, i.e., adjacency matrix \(\mathbf{E_{k,t}}=e_{k,t}(i,j)\)[40], where each element indicates the attribute of an edge, and the elements can be stored in a triangular matrix as \(e_{k,t}(i,j)=e_{k,t}(j,i)\) (as shown in Figure 1). Finally, the attributes of nodes and edges, as well as the adjacent matrix that allows their sequential representation, are able to respond to every feedback event. In the end, a universal domain \(\mathbf{D_{t}}\) at time \(t\) can be determined based on the graphs of \(N\) users, where \(\mathbf{D_{t}}(\mathbf{h})\) is the \(h^{th}\) domain in which \(M\) nodes are sorted by the level of the attribute of the edge connected to the \(h^{th}\) node (see Equation (9)). That is to say, in \(\mathbf{D_{t}}(\mathbf{h})\), the \(h^{th}\) node itself appears first, the node with the second largest attribute of the edge connected to the \(h^{th}\) node appears second, and so on. Understandably, the domain changes over time as the individual graphs vary--e.g., with fluctuating attributes, newly inserted nodes starting with zero-level attributes, etc.