Hypercomplex Widely Linear Processing: Fundamentals for Quaternion Machine Learning
Numerous attempts have been made to replicate the success of complex-valued algebra in engineering and science to other hypercomplex domains such as quaternions, tessarines, biquaternions, and octonions. Perhaps, none have matched the success of quaternions. The most useful feature of quaternions lies in their ability to model three-dimensional rotations which, in turn, have found various industrial applications such as in aeronautics and computergraphics. Recently, we have witnessed a renaissance of quaternions due to the rise of machine learning. To equip the reader to contribute to this emerging research area, this chapter lays down the foundation for: - augmented statistics for modelling quaternion-valued random processes, - widely linear models to exploit such advanced statistics, - quaternion calculus and algebra for algorithmic derivations, - mean square estimation for practical considerations. For ease of exposure, several examples are offered to facilitate the learning, understanding, and(hopefully) the adoption of this multidimensional domain.
Code (0)
등록된 구현이 없습니다.
Similar Papers 제목 키워드 기반
Algebraic foundations of split hypercomplex nonlinear adaptive filtering
A split hypercomplex learning algorithm for the training of nonlinear finite impulse response adaptive filters for the processing of hypercomplex signals of any dimension is proposed. The derivation strictly takes into a…
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 fu…
Sound Event DetectionBeyond Fully-Connected Layers with Quaternions: Parameterization of Hypercomplex Multiplications with $1/n$ Parameters
Recent works have demonstrated reasonable success of representation learning in hypercomplex space. Specifically, "fully-connected layers with Quaternions" (4D hypercomplex numbers), which replace real-valued matrix mult…
Machine TranslationNatural Language InferenceRepresentation LearningStyle Transfer+2Lightweight and Efficient Neural Natural Language Processing with Quaternion Networks
Many state-of-the-art neural models for NLP are heavily parameterized and thus memory inefficient. This paper proposes a series of lightweight and memory efficient neural architectures for a potpourri of natural language…
An Invitation to Hypercomplex Phase Retrieval: Theory and Applications
Hypercomplex signal processing (HSP) provides state-of-the-art tools to handle multidimensional signals by harnessing intrinsic correlation of the signal dimensions through Clifford algebra. Recently, the hypercomplex re…
Retrieval