A survey of quaternion neural networks

Universal approximation theorem for vector- and hypercomplex-valued neural networks

The universal approximation theorem states that a neural network with one hidden layer can approximate continuous functions on compact sets with any desired precision. This theorem supports using neural networks for various applications, including regression and classification tasks. Furthermore, it is valid for real-valued neural networks and some hypercomplex-valued neural networks such as complex-, quaternion-, tessarine-, and Clifford-valued neural networks. However, hypercomplex-valued neural networks are a type of vector-valued neural network defined on an algebra with additional algebraic or geometric properties. This paper extends the universal approximation theorem for a wide range of vector-valued neural networks, including hypercomplex-valued models as particular instances. Precisely, we introduce the concept of non-degenerate algebra and state the universal approximation theorem for neural networks defined on such algebras.

PHNNs: Lightweight Neural Networks via Parameterized Hypercomplex Convolutions

Hypercomplex neural networks have proven to reduce the overall number of parameters while ensuring valuable performance by leveraging the properties of Clifford algebras. Recently, hypercomplex linear layers have been further improved by involving efficient parameterized Kronecker products. In this article, we define the parameterization of hypercomplex convolutional layers and introduce the family of parameterized hypercomplex neural networks (PHNNs) that are lightweight and efficient large-scale models. Our method grasps the convolution rules and the filter organization directly from data without requiring a rigidly predefined domain structure to follow. PHNNs are flexible to operate in any user-defined or tuned domain, from 1-D to [Formula: see text] regardless of whether the algebra rules are preset. Such a malleability allows processing multidimensional inputs in their natural domain without annexing further dimensions, as done, instead, in quaternion neural networks (QNNs) for 3-D inputs like color images. As a result, the proposed family of PHNNs operates with 1/n free parameters as regards its analog in the real domain. We demonstrate the versatility of this approach to multiple domains of application by performing experiments on various image datasets and audio datasets in which our method outperforms real and quaternion-valued counterparts. Full code is available at: https://github.com/eleGAN23/HyperNets.

Quaternion Cross-Modality Spatial Learning for Multi-Modal Medical Image Segmentation

Recently, the Deep Neural Networks (DNNs) have had a large impact on imaging process including medical image segmentation, and the real-valued convolution of DNN has been extensively utilized in multi-modal medical image segmentation to accurately segment lesions via learning data information. However, the weighted summation operation in such convolution limits the ability to maintain spatial dependence that is crucial for identifying different lesion distributions. In this paper, we propose a novel Quaternion Cross-modality Spatial Learning (Q-CSL) which explores the spatial information while considering the linkage between multi-modal images. Specifically, we introduce to quaternion to represent data and coordinates that contain spatial information. Additionally, we propose Quaternion Spatial-association Convolution to learn the spatial information. Subsequently, the proposed De-level Quaternion Cross-modality Fusion (De-QCF) module excavates inner space features and fuses cross-modality spatial dependency. Our experimental results demonstrate that our approach compared to the competitive methods perform well with only 0.01061 M parameters and 9.95G FLOPs.

QMLS: quaternion mutual learning strategy for multi-modal brain tumor segmentation

Objective.Due to non-invasive imaging and the multimodality of magnetic resonance imaging (MRI) images, MRI-based multi-modal brain tumor segmentation (MBTS) studies have attracted more and more attention in recent years. With the great success of convolutional neural networks in various computer vision tasks, lots of MBTS models have been proposed to address the technical challenges of MBTS. However, the problem of limited data collection usually exists in MBTS tasks, making existing studies typically have difficulty in fully exploring the multi-modal MRI images to mine complementary information among different modalities.Approach.We propose a novel quaternion mutual learning strategy (QMLS), which consists of a voxel-wise lesion knowledge mutual learning mechanism (VLKML mechanism) and a quaternion multi-modal feature learning module (QMFL module). Specifically, the VLKML mechanism allows the networks to converge to a robust minimum so that aggressive data augmentation techniques can be applied to expand the limited data fully. In particular, the quaternion-valued QMFL module treats different modalities as components of quaternions to sufficiently learn complementary information among different modalities on the hypercomplex domain while significantly reducing the number of parameters by about 75%.Main results.Extensive experiments on the dataset BraTS 2020 and BraTS 2019 indicate that QMLS achieves superior results to current popular methods with less computational cost.Significance.We propose a novel algorithm for brain tumor segmentation task that achieves better performance with fewer parameters, which helps the clinical application of automatic brain tumor segmentation.

QCNN-H: Single-Image Dehazing Using Quaternion Neural Networks

Single-image haze removal is challenging due to its ill-posed nature. The breadth of real-world scenarios makes it difficult to find an optimal dehazing approach that works well for various applications. This article addresses this challenge by utilizing a novel robust quaternion neural network architecture for single-image dehazing applications. The architecture's performance to dehaze images and its impact on real applications, such as object detection, is presented. The proposed single-image dehazing network is based on an encoder-decoder architecture capable of taking advantage of quaternion image representation without interrupting the quaternion dataflow end-to-end. We achieve this by introducing a novel quaternion pixel-wise loss function and quaternion instance normalization layer. The performance of the proposed QCNN-H quaternion framework is evaluated on two synthetic datasets, two real-world datasets, and one real-world task-oriented benchmark. Extensive experiments confirm that the QCNN-H outperforms state-of-the-art haze removal procedures in visual quality and quantitative metrics. Furthermore, the evaluation shows increased accuracy and recall of state-of-the-art object detection in hazy scenes using the presented QCNN-H method. This is the first time the quaternion convolutional network has been applied to the haze removal task.

Pneumonia detection with QCSA network on chest X-ray

Worldwide, pneumonia is the leading cause of infant mortality. Experienced radiologists use chest X-rays to diagnose pneumonia and other respiratory diseases. The diagnostic procedure's complexity causes radiologists to disagree with the decision. Early diagnosis is the only feasible strategy for mitigating the disease's impact on the patent. Computer-aided diagnostics improve the accuracy of diagnosis. Recent studies established that Quaternion neural networks classify and predict better than real-valued neural networks, especially when dealing with multi-dimensional or multi-channel input. The attention mechanism has been derived from the human brain's visual and cognitive ability in which it focuses on some portion of the image and ignores the rest portion of the image. The attention mechanism maximizes the usage of the image's relevant aspects, hence boosting classification accuracy. In the current work, we propose a QCSA network (Quaternion Channel-Spatial Attention Network) by combining the spatial and channel attention mechanism with Quaternion residual network to classify chest X-Ray images for Pneumonia detection. We used a Kaggle X-ray dataset. The suggested architecture achieved 94.53% accuracy and 0.89 AUC. We have also shown that performance improves by integrating the attention mechanism in QCNN. Our results indicate that our approach to detecting pneumonia is promising.

Fixed-time and prescribed-time synchronization of quaternion-valued neural networks: A control strategy involving Lyapunov functions

A control strategy containing Lyapunov functions is proposed in this paper. Based on this strategy, the fixed-time synchronization of a time-delay quaternion-valued neural network (QVNN) is analyzed. This strategy is extended to the prescribed-time synchronization of the QVNN. Furthermore, an improved two-step switching control strategy is also proposed based on this flexible control strategy. Compared with some existing methods, the main method of this paper is a non-decomposition one, does not contain a sign function in the controller, and has better synchronization accuracy. Two numerical examples verify the above advantages.

Fast Algorithms for Deep Octonion Networks

This brief presents the results of a study of the possibilities of reducing the arithmetic complexity of computing basic operations in octonionic neural networks and also proposes new algorithmic solutions for efficiently performing these operations. Here, we primarily mean the operation of multiplying octonions, the operation of computing the dot product of two octonion-valued vectors, and the operation of multiple multiplications of an octonion by several other octonions. In order to reduce the computational complexity of these operations, it is proposed to use the fast Walsh-Hadamard transform, which is well known in digital signal processing. Using this transform reduces the number of multiplications and additions of real numbers required to perform computations. Thus, the use of the proposed algorithms will speed up computations in octonion-valued neural networks.

A novel non-linear neuron model based on multiplicative aggregation in quaternionic domain

The learning algorithm for a three-layered neural structure with novel non-linear quaternionic-valued multiplicative (QVM) neurons is proposed in this paper. The computing capability of non-linear aggregation in the cell body of biological neurons inspired the development of a non-linear neuron model. However, unlike linear neuron models, most non-linear neuron models are built on higher order aggregation, which is more mathematically complex and difficult to train. As a result, building non-linear neuron models with a simple structure is a difficult and time-consuming endeavor in the neurocomputing field. The concept of a QVM neuron model was influenced by the non-linear neuron model, which has a simple structure and the great computational ability. The suggested neuron’s linearity is determined by the weight and bias associated with each quaternionic-valued input. Non-commutative multiplication of all linearly connected quaternionic input-weight terms accommodates the non-linearity. To train three-layered networks with QVM neurons, the standard quaternionic-gradient-based backpropagation (QBP) algorithm is utilized. The computational and generalization capabilities of the QVM neuron are assessed through training and testing in the quaternionic domain utilizing benchmark problems, such as 3D and 4D chaotic time-series predictions, 3D geometrical transformations, and 3D face recognition. The training and testing outcomes are compared to conventional and root-power mean (RPM) neurons in quaternionic domain using training–testing MSEs, network topology (parameters), variance, and AIC as statistical measures. According to these findings, networks with QVM neurons have greater computational and generalization capabilities than networks with conventional and RPM neurons in quaternionic domain.

A T-S Fuzzy Quaternion-Value Neural Network-Based Data-Driven Generalized Predictive Control Scheme for Mecanum Mobile Robot

Four-mecanum-wheeled omnidirectional mobile robots (FMOMR) are widely used in many practical scenarios because of their high mobility and flexibility. However, the performance of trajectory tracking would be degenerated largely due to various reasons. To deal with this issue, this paper proposes a data-driven algorithm by using the T-S fuzzy quaternion-value neural network (TSFQVNN). TSFQVNN is tailored to obtain the controlled autoregressive integral moving average (CARIMA) model, and then the generalized predictive controller (GPC) is designed based on the CARIMA model. In this way, the spatial relationship between the three-dimensional pose coordinates can be preserved and training times can be reduced. Furthermore, the convergence of the proposed algorithm is verified by the Stone–Weierstrass theorem, and the convergence conditions of the algorithm are discussed. Finally, the proposed control scheme is applied to the three-dimensional (3D) trajectory tracking problem on the arc surface, and the simulation results prove the necessity and feasibility of the algorithm.

QMEDNet: A quaternion-based multi-order differential encoder-decoder model for 3D human motion prediction

In order to deal with the sequence information in the task of 3D human motion prediction effectively, many previous methods seek to predict the motion state of the next moment using the traditional recurrent neural network in Euclidean space. However human motion representation in Euclidean space has high distortion and shows a weak semantic expression when using deep learning models. In this work, we try to process human motion by mapping Euclidean space into a Hypercomplex vector space. We propose a novel model based on quaternion to predict the three-dimensional motion of a human body. The core idea of this study is to use the fusion information to understand and process the human motion state in quaternion space. The multi-order differential information is fused both in the encoder and decoder of feature extraction and mapped to the quaternion space, respectively. The encoder takes graph convolution as the basic unit and the decoder adopts gated recurrent units. Numerous experiments have been carried out to prove that the multi-order information in quaternion space can help build a more reasonable description for 3D human motion. The performance of the proposed QMEDNet is superior to most of the advanced short and long-term motion prediction methods in both public datasets, Human 3.6M and CMU Mocap.

Feedforward–Feedback Controller Based on a Trained Quaternion Neural Network Using a Generalised HR Calculus with Application to Trajectory Control of a Three-Link Robot Manipulator

This study derives a learning algorithm for a quaternion neural network using the steepest descent method extended to quaternion numbers. This applies the generalised Hamiltonian–Real calculus to obtain derivatives of a real–valued cost function concerning quaternion variables and designs a feedback–feedforward controller as a control system application using such a network. The quaternion neural network is trained in real-time by introducing a feedback error learning framework to the controller. Thus, the quaternion neural network-based controller functions as an adaptive-type controller. The designed controller is applied to the control problem of a three-link robot manipulator, with the control task of making the robot manipulator’s end effector follow a desired trajectory in the Cartesian space. Computational experiments are conducted to investigate the learning capability and the characteristics of the quaternion neural network used in the controller. The experimental results confirm the feasibility of using the derived learning algorithm based on the generalised Hamiltonian–Real calculus to train the quaternion neural network and the availability of such a network for a control systems application.

Weyl almost automorphic solutions in distribution sense of Clifford-valued stochastic neural networks with time-varying delays

In this paper, the existence and stability of Weyl almost automorphic solutions in distribution sense for a class of Clifford-valued stochastic neural networks with time-varying delays are studied by using the direct method. Firstly, the existence and uniqueness of Weyl almost automorphic solutions in distribution sense for this class of neural networks are studied by using the Banach fixed point theorem and the relationship between several different senses of random almost automorphy. Then, the global exponential stability in p th mean of the unique Weyl almost automorphic solution in distribution sense is proved by inequality technique and counter proof method. Even when this class of neural networks we consider is real-valued, our results are new. Meanwhile, the method proposed in this paper can be used to study the existence of Weyl almost automorphic solutions of other types of neural networks including stochastic and deterministic neural networks. Finally, an example is given to illustrate the feasibility of our results.

An Information-Theoretic Perspective on Proper Quaternion Variational Autoencoders

Variational autoencoders are deep generative models that have recently received a great deal of attention due to their ability to model the latent distribution of any kind of input such as images and audio signals, among others. A novel variational autoncoder in the quaternion domain H, namely the QVAE, has been recently proposed, leveraging the augmented second order statics of H-proper signals. In this paper, we analyze the QVAE under an information-theoretic perspective, studying the ability of the H-proper model to approximate improper distributions as well as the built-in H-proper ones and the loss of entropy due to the improperness of the input signal. We conduct experiments on a substantial set of quaternion signals, for each of which the QVAE shows the ability of modelling the input distribution, while learning the improperness and increasing the entropy of the latent space. The proposed analysis will prove that proper QVAEs can be employed with a good approximation even when the quaternion input data are improper.

A Quaternion Gated Recurrent Unit Neural Network for Sensor Fusion

Recurrent Neural Networks (RNNs) are known for their ability to learn relationships within temporal sequences. Gated Recurrent Unit (GRU) networks have found use in challenging time-dependent applications such as Natural Language Processing (NLP), financial analysis and sensor fusion due to their capability to cope with the vanishing gradient problem. GRUs are also known to be more computationally efficient than their variant, the Long Short-Term Memory neural network (LSTM), due to their less complex structure and as such, are more suitable for applications requiring more efficient management of computational resources. Many of such applications require a stronger mapping of their features to further enhance the prediction accuracy. A novel Quaternion Gated Recurrent Unit (QGRU) is proposed in this paper, which leverages the internal and external dependencies within the quaternion algebra to map correlations within and across multidimensional features. The QGRU can be used to efficiently capture the inter- and intra-dependencies within multidimensional features unlike the GRU, which only captures the dependencies within the sequence. Furthermore, the performance of the proposed method is evaluated on a sensor fusion problem involving navigation in Global Navigation Satellite System (GNSS) deprived environments as well as a human activity recognition problem. The results obtained show that the QGRU produces competitive results with almost 3.7 times fewer parameters compared to the GRU. The QGRU code is available at.

Interrogating the microenvironmental landscape of tumors with computational image analysis approaches

The tumor microenvironment is an interacting heterogeneous collection of cancer cells, resident as well as infiltrating host cells, secreted factors, and extracellular matrix proteins. With the growing importance of immunotherapies, it has become crucial to be able to characterize the composition and the functional orientation of the microenvironment. The development of novel computational image analysis methodologies may enable the robust quantification and localization of immune and related biomarker-expressing cells within the microenvironment. The aim of the review is to concisely highlight a selection of current and significant contributions pertinent to methodological advances coupled with biomedical or translational applications. A further aim is to concisely present computational advances that, to our knowledge, have currently very limited use for the assessment of the microenvironment but have the potential to enhance image analysis pipelines; on this basis, an example is shown for the detection and segmentation of cells of the microenvironment using a published pipeline and a public dataset. Finally, a general proposal is presented on the conceptual design of automation-optimized computational image analysis workflows in the biomedical and clinical domain.