Quaternion Generative Adversarial Networks

Hypercomplex Image-to-Image Translation

“…Indeed, state-of-the-art generative models usually comprise tens of million parameters and are often employed with multidimensional inputs such as color images or multichannel audio signals [17], [31]. The quaternion-valued variational autoencoder and the family of quaternion generative adversarial networks have demonstrated to obtain comparable performance while reducing the storage memory amount due to the parameters reduction [24]- [26], [32]. Encouraged by these results, we propose to expolit novel PHNNs methods to define a more advanced generative model for image-to-image translation.…”

“…This is an approximation to the optimal variance that is, however, computationally expensive to be calculated due to the particular form of quaternion covariance matrix [24], [35]- [37]. Finally, the hypercomplex instance normalization (HIN) can be applied following Eq.…”

“…Indeed, they process RGB images as a single entity, thus exploiting relations within input dimensions, building a more suitable representation [23]. Both GANs and variational autoencoders (VAEs) have been defined in the quaternion domain demonstrating encouraging scores and an improved generation ability [24]- [26]. However, QNNs accept 4D inputs only, thus a padding channel has to be annexed to RGB images to compose a pure quaternion.…”

Tessarine and Quaternion-Valued Deep Neural Networks for Image Classification

“…Recent developments are mainly made with real ANNs. Nonetheless, there is evidence that using other algebras such as complex numbers [Aizenberg 2011, Trabelsi et al 2017 and quaternions [Arena et al 1997, Gaudet and Maida 2017, Zhu et al 2019, Kumar and Tripathi 2019, Parcollet et al 2020, Grassucci et al 2021] may improve networks performance without increasing (and sometimes reducing) the number of parameters. Reducing the number of parameters is particularly advantageous for applications with memory and other constraints.…”

“…Quaternions have been conceived by Hamilton in 1844 [Hamilton 1844] while the tessarines have been proposed by Cockle in 1848 [Cockle 1848]. Quaternions have been successfully used in deep neural networks models achieving the state of the art performances [Gaudet and Maida 2017, Zhu et al 2019, Grassucci et al 2021. Although there are examples of tessarines usage for signal processing [Navarro-Moreno et al 2020, Navarro-Moreno andRuiz-Molina 2021], to the best of our knowledge, no tessarinevalued neural network has been proposed yet.…”