# Multiple Network Fusion with Low-Rank Representation for Image-Based Age Estimation
## 2 Multiple Network Learning with Low-Rank Representation
### Fusion with Low-Rank Representation  
As mentioned before, multi-modal feature fusion by computing semantic relationship is more reasonable than simple concatenation. The key to learning the semantic relationship is how to define and compute affinities among data. Many existing methods can be used, such as subspace learning and manifold learning. Recently, low-rank learning attracts plenty of attention. In low-rank representation, assume the data is clean and is drawn from independent subspaces, then there exists \(Q^{*}\), which is block-diagonal, and the rank of each block equals the dimension of the corresponding subspace. Giventhe \(i\)-th modal \(X^{(i)}\), computed in the previous sub-section, we can compute the affinities among feature vectors by solving the minimization problem:  
\[ \begin{array}{l}\underset{Q_{0},E_{0}}{\min}\parallel Q_{0}\parallel_{*}+ \lambda\parallel E_{0}\parallel_{2,1}\\ \text{s. t. }X^{(i)}=X^{(i)}Q_{0}+E_{0}\end{array}  \tag{15}\]  
where \(\parallel\bullet\parallel_{*}\) denotes the trace norm, and \(\parallel\bullet\parallel_{2,1}\) is the \(\ell_{2,1}\)-norm to characterize noise. \(\lambda>0\) is the parameter to balance the influences of the two parts. The optimal solution to Equation (15), which is denoted as \(Q_{0}^{*}\), naturally defines an affinity relationship that implies the pairwise similarities between features. In this way, the similarity \(\mathbb{S}_{kl}^{(i)}\) between two features \(x_{k}^{(i)}\) and \(x_{l}^{(i)}\) can be computed by  
\[\mathbb{S}_{kl}^{(i)}=|(Q_{0}^{*})_{lk}|+|(Q_{0}^{*})_{kl}| \tag{16}\]  
where \((\bullet)_{lk}\) is the \((l,k)\)-th element of a matrix. In previous methods, the above low-rank learning process can be used with only a single type of feature vectors. Therefore, we extend it to the multi-modal scenario and apply it to feature fusion. We define the multi-modal low-rank learning by  
\[ \begin{array}{l}\underset{\begin{subarray}{c}Q_{1}^{(1)},\ldots,Q_{n}^{(m) }\\ E_{1}^{(1)},\ldots,E_{n}^{(m)}\end{subarray}}{\min}\sum\limits_{i=1}^{m}( \parallel Q^{(i)}\parallel_{*}+\lambda\parallel E^{(i)}\parallel_{2,1})+ \alpha\parallel Q\parallel_{2,1}^{(i)}\\ \text{s. t. }X^{(i)}=X^{(i)}Q^{(i)}+E^{(i)},j=1,...,m\end{array}  \tag{17}\]  
where \(\alpha>0\) is a balanced parameter and \(m\) is the number of modals. In this way, we can infer a set of matrices \(Q^{(1)}\),\(Q^{(2)}\),..., \(Q^{(m))}\). In this set, each \(n\times n\) matrix \(Q^{(i)}\) corresponds to the \(i\)-th modal \(X^{(i)}\). The global solution defined by \(m\times n^{2}\) matrix \(Q\) is constructed by arranging \(Q^{1}\),\(Q^{2}\),..., \(Q^{m}\) as follows:  
\[Q=\left[ \begin{array}{cccc}Q_{11}^{1}&Q_{12}^{1}&...&Q_{nn}^{1}\\ Q_{11}^{1}&Q_{12}^{1}&...&Q_{nn}^{2}\\ \vdots&\vdots&\ddots&\vdots\\ Q_{11}^{m}&Q_{12}^{m}&...&Q_{nn}^{m}\end{array} \right]. \tag{18}\]  
\(Q_{*}^{1},Q_{*}^{2},...,Q_{*}^{m}\) is defined as the optimal solution to Equation (18). A universal affinity matrix can then obtained by quantifying the columns of \(Q\):  
\[\mathbb{S}_{kl}^{(i)}=\frac{1}{2}(\sqrt{\sum_{i=1}^{m}(Q_{lk}^{(i)})^{2}}+ \sqrt{\sum_{i=1}^{m}(Q_{kl}^{(i)})^{2}}). \tag{19}\]  
In manifold learning, the key to solving the manifold is computing the affinity matrix. Therefore, with the affinity matrix computed by Equation (19), we can construct the manifold in low-rank space to obtain fused feature descriptors. Specifically, we use affinity matrix \(Q\) to construct Laplacian matrix \(L\). There are a number of solutions to this problem. In the proposed method, we follow the spectral hypergraph clustering method [28] and use it in the low-rank space. In this way, we consider each feature vector as a vertex \(v\) in the low-rank feature space. If some vertices share the same property, they are connected by a hyperedge \(e\). In this method, \(L\) is defined as  
\[L=I-C \tag{20}\]  
where \(I\) denotes an \(n\times n\) identity matrix.