# A Discriminative-Based Geometric Deep Learning Model for Cross Domain Recommender Systems
## 3 Proposed Method
In our approach, the local-sensitivity adaptor is introduced into the structure of the original convolutional deep belief Network (CDBN) as an extension to support recommended objects. Deep belief network is a suitable probabilistic principle for modelling joint probabilistic distribution over pixels and labels. Nonetheless, it is a challenging task to extend the model from 2D pixel data to 3D voxel data. This is because the volume size of a 3D voxel will be much greater in size than an original image. In addition, a fully connected DBN involves a lot of parameters, which makes the training of the model very challenging. As a result, we propose the LDCBN by weight sharing to lessen the number of parameters used. Again, unlike the conventional DL techniques, the pooling layers are ignored in our approach due to uncertainties that may arise when features are being generated. The local-sensitivity convolutional layer's energy in the proposed model is as in Equation (8):\[E(f,g)=-\sum_{j=1}^{l}\sum_{i=1}^{N_{i}}(\left(g_{i}^{j}\right)D_{j}\left(x_{i}^{ j}\right)f+c^{j}g_{i}^{j})-\sum_{l}^{j}b_{l}f_{l}+\sum_{j=1}^{l}\sum_{i=1}^{N_{i}}||B_{i }^{j}\bigodot x_{i}^{j}||_{2}^{2}, \tag{8}\]  
where \(x_{i}^{j}\) is any point (hidden unit) \(i\) on the manifold belonging to the group or feature space j, the patch operator is denoted by \(D_{j}\left(x_{i}^{j}\right)f\), which is mapped into high dimensional feature space, and the index of the visible units is denoted by \(l\). The local sensitivity adaptor is represented by \(B_{i}^{j}\), the dimensionality of the extracted patch and the element-wise multiplicative operator is represented by \(\bigodot\). Each hidden unit in feature space j is denoted by \(g_{i}^{j}\), the visible unit in the 3D voxel input is denoted by \(f_{l}\), the convolutional filter is represented by \(D_{j}\), \(c^{j}\) and \(b_{l}\) represents bias terms in hidden and visible units, respectively. The local sensitive adaptor is used to enforce discriminability by helping in the capturing of essential local geometric information, which may be hiding in the structure of recommended objects, for efficient and effective recognition results. The proposed approach is modeled around the theories of [44]. We could set a voxel grid of \(45\times 45\times 45\) to a 3D shape with three extra padding cells in all dimensions to lessen the boundaries of the convolution objects. The labels are then put forward to ascertain the standards of the variables. There are two (2) steps followed in the training process of the proposed 3D LDCBN. These are layer by layer before-training and generative refined-tuning processes. The initial four (4) layers are separately trained during the before-training phase with a divergence algorithm described in [45], and a fast persistence divergence is used to train the topmost layer [46]. The underlining hidden activations serve as inputs into the next layer once the bottom layer is learned with their weights settled. In the refined-tuning step, methods from past studies such as the wake-sleep algorithm were adopted, where the wake phase utilizes the activations for learning of positive signals and by propagating the data bottom-up. For the sleep phase, a tenacious chain on the uppermost layer is maintained, and the data top-down section is propagated to assemble the deleterious learning signal. This refined-tuning technique copycats the generation and recognition performance of the model and functions well in principle. As long as the weights of the entire networks have been learned, forward computation is utilized to engender the geometric descriptor applying the input data of voxelization. In analyzing recommended objects from the angle of view-based descriptors, the recommended model is transformed into varied images through various perspectives. In principle, the images encompass as much information as possible from the recommended model.