\chapter{Conclusions and Discussions}
\label{chp:conclusion}
In this dissertation, we proposes a novel vector-valued neuron which employs arbitrary bilinear products for feedforward and backpropagation. The proposed architecture generalizes and extends all existing vector-valued neurons and is useful for datasets where each training sample is a multidimensional vector. Through bilinear products, the vector neural network captures the associations among different entries in the same position in each vector. The model can be trained efficiently by using vector error backpropagation through the Adam algorithm. 
Experimental results on multispectral denoising and singing voice separation show that our proposed model performs better than conventional NNs by using less number of parameters. On the other hand, we have also presented a new type of CNN architecture consisting of $N$-dimensional vector-valued neurons endowed with the cyclic group algebra, where $N$ can be an arbitrary positive integer. In the convolution procedure, scalar multiplications in conventional CNNs are changed to vector multiplications through circulant matrices.
During the feedforward and backpropagation, the circular convolution is formulated as a vector-matrix multiplication with a circulant matrix. The learning process is performed with a backpropagation algorithm and optimized through Adam. Experimental results on color image inpainting and multispectral denoising show that the proposed model exhibit better performance than conventional CNNs and deep quaternion networks.


Future work involves three directions. First, 
% we will apply the technique of vector neural learning with bilinear product on convolutional neural network (CNN) to deal with spatial data such as images. Next, we will compare the performance of vector neural learning on CNN to other deep models with classification or regression tasks. Finally, 
we observe that the training time of the proposed network grows quadratically as a function of the dimensionality $N$. We make the training more scalable to higher values of $N$. Next, we would like to test it on classification problems, and datasets with even larger dimensions, such as EEG-based emotion recognition. Finally, 
since our proposed architecture could operates with arbitrary biliner products, it is similar to the concept of object orientation, where each neuron could be seen an object encapsulating all bilinear products and operating with other neurons. 
In the future, we will extend the proposed architecture to graph neural networks to investigate more powerful models.