# Multimodal Tucker Decomposition for Gated RBM Inference
## 6 Results and Discussion
In fact, the core tensor also learns filters, as presented in Figure 9. Moreover, the filters learned by each layer of the model show a correspondence with the filters in the three-way core tensor. In other words, each unit in the model is connected to the core tensor with variable strengths, and this allows them to detect specific changes in frequency, orientation, and phase shift in the data. In Figure 9, we show the filters learned by a model with a core tensor of shape \(49\times 49\times 49\). In the image, we display each frontal slice of the core tensor. For simplicity, we present the frontal slices for this model with a smaller tensor, but the same behavior was observed regardless of the tensor shape. We also confirmed that the learning rate would affect convergence. The learning rate controls how much the parameters in the model are adjusted with respect to the energy loss gradient. Figure 10 shows energy loss during training for different learning rate configurations: \(1\times 10^{-2}\), \(1\times 10^{-3}\), and \(1\times 10^{-4}\). We limit the number of epochs to 10. The number of units in the hidden layer is 144, and the ranks selected for the core tensor is \(120\times 120\times 120\). As can be observed in the figure, the largest learning rate provides the fastest convergence in less than one epoch. On the other hand, the smallest learning rate decreases steadily on each epoch but does not arrive to a point of convergence. In each case, we observed that the corresponding model learns filters at different stages. In general, a smaller learning rate requires more training epochs given the smaller changes made to the model parameters in each update. However if the learning rate is too small, it can cause the learning process to get stuck and not converge. On the other hand, a larger learning rate results in rapid changes and requires fewer training epochs. However, it is possible that too large a learning rate causes the model to converge to a sub-optimal solution. The rate at which the learning takes place is an important hyper-parameter and will vary depending on the application of the model. Figure 9: Core tensor filters learned from shifted random images. To gain insight into the way the core tensor affects the learning speed, we also trained the gated RBM model on the random-shifts dataset under various configurations of the core tensor while holding the learning rate and the number of hidden units constant. The gated RBM evaluated has 169 units both in its input and output layer and 169 units in its hidden layer. With this configuration, the original weight tensor has a shape of \(169\times 169\times 169\). On the other hand, the core tensor was tested with ranks \(80\times 80\times 80\), \(120\times 120\times 120\), and \(169\times 169\times 169\). Note that in the last configuration, the core tensor provides no compression since it has the same shape as the original tensor. For simplicity, we only considered cubical tensors. The models were trained using the same learning rate of \(1\times 10^{-3}\) for 10 epochs and were not stopped via early stopping. The idea is to isolate the effect of the core tensor configurations. Figure 11 shows the energy loss when training under the three core tensor configurations. Note that in each case, the energy loss decreases steadily although with different gradients. We should remember that in a gated RBM as well as in an RBM, the concept of the energy function is analogous to the cost function. In fact, the configuration with the smallest core tensor has the highest energy decrease, while the core tensor configuration that is the exact shape of the original tensor has the lowest energy decrease. The reason is because the core tensor does not provide any compression but only functions as a change of basis.