Recommendation System with Biclustering

Biclustering can be considered as the combination of column and row clusterings. Item clustering is performed with the standard FA model $F{A}_i$ ^{[ 15 ]}. The inputs are item vectors $M_{tr}^n(:,j)$ ( $j=1,2,\mathrm{\dots },n$ ), vigilance parameter ${\rho }_i$ , choice parameter ${\alpha }_i$ , learning parameter ${\beta }_i$ , and maximal epochs $M{E}_i$ . Finally, the item cluster set $CI=\{C{I}_1,C{I}_2,\mathrm{\dots },C{I}_{n_1}\}$ composed of $n_1$ item clusters can be obtained.

User clustering is performed with $F{A}_u$ , a modified FA. The pseudocode of user clustering is displayed in Algorithm 1. The modification lies in that when an input pattern (user vector) passes the vigilance test, subsequently, the bicluster test needs to be passed before assigning to the winning neuron (user cluster). The bicluster test is that if the correlation between the input pattern and the bicluster set is not smaller than a predetermined threshold $T_c$ , then the input pattern is assigned to the winning neuron; otherwise, increase ${\rho }_u$ (the vigilance parameter, and ${\rho }_s$ is its increment) and rerun $F{A}_u$ with the updated ${\rho }_u$ . Given that high vigilance threshold leads to narrow generalization and the neuron (cluster) represents fewer input patterns, if no existing committed neuron (cluster) passes the bicluster test, ${\rho }_u$ must be enlarged. With $F{A}_u$ , user cluster set $CU=\{C{U}_1,C{U}_2,\mathrm{\dots },C{U}_{n_2}\}$ composed of $n_2$ user clusters can be obtained. When an input pattern is presented to the $F_1$ layer of $F{A}_u$ , the following steps are performed:

(1) The input $U$ is complement-coded to avoid category proliferation. The complement coding is performed with $A=(U,U^c)$ , where $U_c$ is the complement of $U$ , and $U_i^c=1-U_i,\forall i=1,\mathrm{\dots },n$ .

(2) Each neuron in $F_2$ calculates a value for the cluster choice function, which selects the neuron ( $n_c$ ) having maximal function value, as shown in the following:

(10) $$T_j=\frac{\left|A\Lambda W_j\right|}{\left|W_j\right|+{\alpha }_u}$$

where $W_j$ is the weight of the $j$ -th neuron, “ $\Lambda$ ” is the fuzzy MIN operation defined by ${(A\Lambda W_j)}_k=\min (A_k,W_{j,k})$ , $W_{j,k}$ is the value of the $j$ -th neuron’s $k$ -th weight. “ $\left|\right|$ ” means taking the first-order norm, and ${\alpha }_u$ is a very small positive real number.

(3) Calculate the match function value of $n_c$ as follows:

(11) $$M_j=\frac{\left|W_j\Lambda A\right|}{\left|A\right|}$$

if $M_j⩾{\rho }_u$ , $n_c$ passes the vigilance test and becomes the potential winning neuron $n_{pw}$ . If $M_j⩽{\rho }_u$ , then apparently $n_c$ cannot be the potential winning neuron and is shut off by the orienting subsystem. The competition among the remaining neurons is rerun until the potential winning neuron $n_{pw}$ is found.

(4) Combine $C{U}_u$ (the user cluster that contains $n_{pw}$ ) and each item cluster in $CI$ to construct biclusters. Then, calculate the correlation ( $AC$ ) between the bicluster and input pattern as follows:

(12) $$S_l(u,v)=$$ $${\displaystyle \frac{\underset{i\in S}{\sum }(r_{u,i}-{\bar{r}}_u)(r_{v,i}-{\bar{r}}_v)}{(1+{\text{e}}^{-\frac{n}{2}})\sqrt{\underset{i\in S}{\sum }{(r_{u,i}-{\bar{r}}_u)}^2}\sqrt{\underset{i\in S}{\sum }{(r_{u,j}-{\bar{r}}_v)}^2}}}$$

(13) $$AC(B_k,U)=\frac{1}{\left|R\right|}\times \underset{j=1}{\overset{\left|R\right|}{\sum }}{S}_l(B_k(j,:),U)$$

where $S=I_u\cap I_v$ denotes the items co-rated by users $u$ and $v$ . $N_S=\left|S\right|$ is the cardinality of $S$ . $S_l$ is the local similarity measure of two users and is the modified Pearson correlation coefficient. The difference lies in that $S_l$ considers the number of co-rated items $n_s$ . The bigger $n_s$ becomes, the more items two users rate simultaneously, and the higher their similarity becomes. $B_k$ is the $k$ -th bicluster, $B_k(j,:)$ represents the $j$ -th row of $B_k$ , and $\left|R\right|$ is the number of rows (users) in $B_k$ .

If at least one of these passes the bicluster test ( $AC⩾T_c$ ), then set $n_{pw}$ as the final winning neuron $n_w$ . Otherwise, increase the vigilance with ${\rho }_u={\rho }_u+{\rho }_s$ and jump to the second step to rerun.

(5) Update the weight of the winning neuron $n_w$ for learning the information contained in the input pattern as follows:

(14) $$W_j={\beta }_u(W_j\Lambda A)+(1-{\beta }_u)W_j$$

where ${\beta }_u\in [0,1]$ is the learning rate.

With the above item and user clustering, the $n_1$ item clusters and $n_2$ user clusters are generated, $n_1\times n_2$ biclusters can be obtained by pairwise coupling $CU$ and $CI$ . The next step is to obtain local predictions with the mined biclusters and BBCF from $M_{tr}^n$ .