# Multimodal Tucker Decomposition for Gated RBM Inference
## 6 Results and Discussion
In addition, note that different configurations of the core tensor explore different configuration of the energy loss, which is explained by the fact that the core tensor modulates the level of interaction between the different components of each layer, resulting in different configurations of energy. In Figure 12, we show the training time (in seconds) for the same core tensor configurations. As was expected, this figure confirms that the training speed is a function of the core tensor dimensionality. Although it could be inferred from Figures 11 and 12 that the best model configuration is the one with the smallest core tensor, it should be noted that the core tensor ranks should be calibrated according to the specifics of the problem to solve and evaluated according to the desired performance of the model. As a last evaluation, we compared the Tucker refactored gated RBM proposed in this research against the unfactored model presented in [10], which uses a full three-way tensor. The filters learned by the input, output, and hidden layers in combination with the filters learned by the core tensor are highly structural and in fact are very similar to the ones that are applied to the output images in [10], as shown in Figure 13b. The latter produces a canvas-like effect on the image. In both cases, the filters are learned in an unsupervised fashion. We should underline that there is no available code accompanying the research  
Figure 10: Gated RBM energy loss comparison with different learning rate configurations. in [10] and that we implemented their unfactored model using the update rules provided in the paper using PyTorch. Although the filters produced both for the Tucker refactored and unfactored models might be similar, the training for both approaches has different performance metrics. To better understand this, we trained both models on the same random shifts dataset while holding all the variables constant. The input and output images have a size of \(13\times 13\) pixels each, which determines the number of neurons in the input and output layers to be 169 in both models. The number of hidden units and learning rate are kept the same in both implementations and have values of 121 and \(1\times 10^{-3}\), respectively. Figure 14 shows the energy loss for the Tucker refactored, while Figure 15 shows the energy loss for the unfactored model. Figure 11: Gated RBM energy loss comparison with different core tensor ranks. Figure 12: Gated RBM training time comparison with different core tensor ranks. The only difference between the two models is that the unfactored gated RBM uses a full tensor \(169\times 121\times 169\), and the Tucker refactored model factorizes the full tensor into a core tensor \(80\times 80\times 80\), input and output factor matrices \(169\times 80\) and a hidden factor matrix \(121\times 80\). Note that the unfactored gated RBM starts with a much higher energy configuration. On the other hand, the [PERSON] refactored model starts at a much smaller energy configuration as a result of the core tensor modulating the interactions of the three layers. The relationship between the core tensor configuration and the resulting energy configuration explored is also observable in Figure 14. From Figures 14 and 15, we see that the Tucker refactored gated RBM reaches convergence in fewer iterations than the unfactored gated RBM while having the same learning rate and hidden units. Figure 14: Energy loss when training the Tucker refactored model. Figure 13: (**a**): Filters learned in by the input, output and hidden layers. (**b**): Filters applied from [PERSON] and [PERSON] [10]. Reproduced with permission from [PERSON] and [PERSON], Unsupervised Learning of Image Transformations; published by IEEE, 2007. Finally, in Figure 16, we show a comparison of the training time (in seconds) for both the Tucker refactored and unfactored gated RBMs. It is easy to see that the unfactored model using a full tensor takes much longer than the Tucker decomposed factored model proposed in this research.