Quaternion softmax classifier

International audienceFor the feature extraction of red-blue-green (RGB) colour images, researchers usually deal with R, G and B channels separately to obtain three feature vectors, and then combine them together to obtain a long real feature vector. This approach does not exploit the relationships between the three channels of the colour images. Recently, attention has been paid to quaternion features, which take the relationships between channels into consideration and seem to be more suitable for representing colour images. However, there are only a few quaternion classifiers for dealing with quaternion features. To meet this requirement, a new quaternion classifier, namely, the quaternion softmax classifier is proposed, which is an extended version of the conventional softmax classifier generally defined in the complex (or real) domain. The proposed quaternion softmax classifier is applied to two of the most common quaternion features, that is, the quaternion principal components analysis feature and the colour image pixel feature. The experimental results show that the proposed method performs better than the quaternion back propagation neural network in terms of accuracy and convergence rate

Quaternion Denoising Encoder-Decoder for Theme Identification of Telephone Conversations

International audienceIn the last decades, encoder-decoders or autoencoders (AE) have received a great interest from researchers due to their capability to construct robust representations of documents in a low dimensional subspace. Nonetheless, autoencoders reveal little in way of spoken document internal structure by only considering words or topics contained in the document as an isolate basic element, and tend to overfit with small corpus of documents. Therefore, Quaternion Multi-layer Perceptrons (QMLP) have been introduced to capture such internal latent dependencies , whereas denoising autoencoders (DAE) are composed with different stochastic noises to better process small set of documents. This paper presents a novel autoencoder based on both hitherto-proposed DAE (to manage small corpus) and the QMLP (to consider internal latent structures) called "Quater-nion denoising encoder-decoder" (QDAE). Moreover, the paper defines an original angular Gaussian noise adapted to the speci-ficity of hyper-complex algebra. The experiments, conduced on a theme identification task of spoken dialogues from the DE-CODA framework, show that the QDAE obtains the promising gains of 3% and 1.5% compared to the standard real valued de-noising autoencoder and the QMLP respectively

Real to H-space Encoder for Speech Recognition

International audienceDeep neural networks (DNNs) and more precisely recurrent neural networks (RNNs) are at the core of modern automatic speech recognition systems, due to their efficiency to process input sequences. Recently, it has been shown that different input representations, based on multidimensional algebras, such as complex and quaternion numbers, are able to bring to neural networks a more natural, compressive and powerful representation of the input signal by outperforming common real-valued NNs. Indeed, quaternion-valued neural networks (QNNs) better learn both internal dependencies, such as the relation between the Mel-filter-bank value of a specific time frame and its time derivatives, and global dependencies, describing the relations that exist between time frames. Nonetheless, QNNs are limited to quaternion-valued input signals, and it is difficult to benefit from this powerful representation with real-valued input data. This paper proposes to tackle this weakness by introducing a real-to-quaternion encoder that allows QNNs to process any one dimensional input features, such as traditional Mel-filter-banks for automatic speech recognition

Understanding Vector-Valued Neural Networks and Their Relationship with Real and Hypercomplex-Valued Neural Networks

Despite the many successful applications of deep learning models for multidimensional signal and image processing, most traditional neural networks process data represented by (multidimensional) arrays of real numbers. The intercorrelation between feature channels is usually expected to be learned from the training data, requiring numerous parameters and careful training. In contrast, vector-valued neural networks are conceived to process arrays of vectors and naturally consider the intercorrelation between feature channels. Consequently, they usually have fewer parameters and often undergo more robust training than traditional neural networks. This paper aims to present a broad framework for vector-valued neural networks, referred to as V-nets. In this context, hypercomplex-valued neural networks are regarded as vector-valued models with additional algebraic properties. Furthermore, this paper explains the relationship between vector-valued and traditional neural networks. Precisely, a vector-valued neural network can be obtained by placing restrictions on a real-valued model to consider the intercorrelation between feature channels. Finally, we show how V-nets, including hypercomplex-valued neural networks, can be implemented in current deep-learning libraries as real-valued networks

Neural Computing in Quaternion Algebra

兵庫県立大学201

New Challenges in Neutrosophic Theory and Applications

Neutrosophic theory has representatives on all continents and, therefore, it can be said to be a universal theory. On the other hand, according to the three volumes of “The Encyclopedia of Neutrosophic Researchers” (2016, 2018, 2019), plus numerous others not yet included in Encyclopedia book series, about 1200 researchers from 73 countries have applied both the neutrosophic theory and method. Neutrosophic theory was founded by Professor Florentin Smarandache in 1998; it constitutes further generalization of fuzzy and intuitionistic fuzzy theories. The key distinction between the neutrosophic set/logic and other types of sets/logics lies in the introduction of the degree of indeterminacy/neutrality (I) as an independent component in the neutrosophic set. Thus, neutrosophic theory involves the degree of membership-truth (T), the degree of indeterminacy (I), and the degree of non-membership-falsehood (F). In recent years, the field of neutrosophic set, logic, measure, probability and statistics, precalculus and calculus, etc., and their applications in multiple fields have been extended and applied in various fields, such as communication, management, and information technology. We believe that this book serves as useful guidance for learning about the current progress in neutrosophic theories. In total, 22 studies have been presented and reflect the call of the thematic vision. The contents of each study included in the volume are briefly described as follows. The first contribution, authored by Wadei Al-Omeri and Saeid Jafari, addresses the concept of generalized neutrosophic pre-closed sets and generalized neutrosophic pre-open sets in neutrosophic topological spaces. In the article “Design of Fuzzy Sampling Plan Using the Birnbaum-Saunders Distribution”, the authors Muhammad Zahir Khan, Muhammad Farid Khan, Muhammad Aslam, and Abdur Razzaque Mughal discuss the use of probability distribution function of Birnbaum–Saunders distribution as a proportion of defective items and the acceptance probability in a fuzzy environment. Further, the authors Derya Bakbak, Vakkas Uluc¸ay, and Memet S¸ahin present the “Neutrosophic Soft Expert Multiset and Their Application to Multiple Criteria Decision Making” together with several operations defined for them and their important algebraic properties. In “Neutrosophic Multigroups and Applications”, Vakkas Uluc¸ay and Memet S¸ahin propose an algebraic structure on neutrosophic multisets called neutrosophic multigroups, deriving their basic properties and giving some applications to group theory. Changxing Fan, Jun Ye, Sheng Feng, En Fan, and Keli Hu introduce the “Multi-Criteria Decision-Making Method Using Heronian Mean Operators under a Bipolar Neutrosophic Environment” and test the effectiveness of their new methods. Another decision-making study upon an everyday life issue which empowered us to organize the key objective of the industry developing is given in “Neutrosophic Cubic Einstein Hybrid Geometric Aggregation Operators with Application in Prioritization Using Multiple Attribute Decision-Making Method” written by Khaleed Alhazaymeh, Muhammad Gulistan, Majid Khan, and Seifedine Kadry