The field of deep learning has seen significant advancement in recent years. However, much of the existing work has been focused on real-valued numbers. Recent work has shown that a deep learning system using the complex numbers can be deeper for a fixed parameter budget compared to its real-valued counterpart. In this work, we explore the benefits of generalizing one step further into the hyper-complex numbers, quaternions specifically, and provide the architecture components needed to build deep quaternion networks. We develop the theoretical basis by reviewing quaternion convolutions, developing a novel quaternion weight initialization scheme, and developing novel algorithms for quaternion batch-normalization. These pieces are tested in a classification model by end-to-end training on the CIFAR-10 and CIFAR-100 data sets and a segmentation model by end-to-end training on the KITTI Road Segmentation data set. These quaternion networks show improved convergence compared to real-valued and complex-valued networks, especially on the segmentation task, while having fewer parameters
One common use of quaternions is for representing rotation into a more compact form. PoseNet @cite_7 used a quaternion as the target output in their model where the goal was to recover the @math DOF camera pose from a single RGB image. The ability to encode rotations may make a quaternion network more robust to rotational variance.
@cite_7
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In this work, we move beyond the traditional complex-valued representations, introducing more expressive hypercomplex representations to model entities and relations for knowledge graph embeddings. More specifically, quaternion embeddings, hypercomplex-valued embeddings with three imaginary components, are utilized to represent entities. Relations are modelled as rotations in the quaternion space. The advantages of the proposed approach are: (1) Latent inter-dependencies (between all components) are aptly captured with Hamilton product, encouraging a more compact interaction between entities and relations; (2) Quaternions enable expressive rotation in four-dimensional space and have more degree of freedom than rotation in complex plane; (3) The proposed framework is a generalization of ComplEx on hypercomplex space while offering better geometrical interpretations, concurrently satisfying the key desiderata of relational representation learning (i.e., modeling symmetry, anti-symmetry and inversion). Experimental results demonstrate that our method achieves state-of-the-art performance on four well-established knowledge graph completion benchmarks. We propose a Convolutional Neural Network (CNN)-based model "RotationNet," which takes multi-view images of an object as input and jointly estimates its pose and object category. Unlike previous approaches that use known viewpoint labels for training, our method treats the viewpoint labels as latent variables, which are learned in an unsupervised manner during the training using an unaligned object dataset. RotationNet is designed to use only a partial set of multi-view images for inference, and this property makes it useful in practical scenarios where only partial views are available. Moreover, our pose alignment strategy enables one to obtain view-specific feature representations shared across classes, which is important to maintain high accuracy in both object categorization and pose estimation. Effectiveness of RotationNet is demonstrated by its superior performance to the state-of-the-art methods of 3D object classification on 10- and 40-class ModelNet datasets. We also show that RotationNet, even trained without known poses, achieves the state-of-the-art performance on an object pose estimation dataset. Recently, the connectionist temporal classification (CTC) model coupled with recurrent (RNN) or convolutional neural networks (CNN), made it easier to train speech recognition systems in an end-to-end fashion. However in real-valued models, time frame components such as mel-filter-bank energies and the cepstral coefficients obtained from them, together with their first and second order derivatives, are processed as individual elements, while a natural alternative is to process such components as composed entities. We propose to group such elements in the form of quaternions and to process these quaternions using the established quaternion algebra. Quaternion numbers and quaternion neural networks have shown their efficiency to process multidimensional inputs as entities, to encode internal dependencies, and to solve many tasks with less learning parameters than real-valued models. This paper proposes to integrate multiple feature views in quaternion-valued convolutional neural network (QCNN), to be used for sequence-to-sequence mapping with the CTC model. Promising results are reported using simple QCNNs in phoneme recognition experiments with the TIMIT corpus. More precisely, QCNNs obtain a lower phoneme error rate (PER) with less learning parameters than a competing model based on real-valued CNNs. This paper proposes a new class of unit quaternion curves in 3 . A general method is developed that transforms a curve in 3 (defined as a weighted sum of basis functions) into its unit quaternion analogue in 3 . Applying the method to well-known spline curves (such as B´ ezier, Hermite, and B-spline curves), we are able to construct various unit quaternion curves which share many important differential properties with their original curves. Many of our naive common beliefs in geometry break down even in the simple non-Euclidean space 3 or 3 . For example, the de Casteljau type construction of cubic B-spline quaternion curves does not preserve 2 -continuity [10]. Through the use of decomposition into simple primitive quaternion curves, our quaternion curves preserve most of the algebraic and differential properties of the original spline curves.