Related papers: Beyond Fully-Connected Layers with Quaternions: Pa…
Simplicial complexes form an important class of topological spaces that are frequently used in many application areas such as computer-aided design, computer graphics, and simulation. Representation learning on graphs, which are just 1-d…
Since their first applications, Convolutional Neural Networks (CNNs) have solved problems that have advanced the state-of-the-art in several domains. CNNs represent information using real numbers. Despite encouraging results, theoretical…
We computationally resolve an open problem concerning the expressibility of $4 \times 4$ full-rank matrices as Hadamard products of two rank-2 matrices. Through exhaustive search over $\mathbb{F}_2$, we identify 5,304 counterexamples among…
We construct the quaternion algebra [10] "geometrically" by a three dimensional analogue of the classic two dimensional geometric description of the complex field. The algebraic description of the multiplication operation in three…
This paper proposes a novel matrix rank-one decomposition for quaternion Hermitian matrices, which admits a stronger property than the previous results in (sturm2003cones,huang2007complex,ai2011new). The enhanced property can be used to…
Compact neural network offers many benefits for real-world applications. However, it is usually challenging to train the compact neural networks with small parameter sizes and low computational costs to achieve the same or better model…
Fast algorithms for arithmetic on real or complex polynomials are well-known and have proven to be not only asymptotically efficient but also very practical. Based on Fast Fourier Transform (FFT), they for instance multiply two polynomials…
Hypercomplex neural networks are gaining increasing interest in the deep learning community. The attention directed towards hypercomplex models originates from several aspects, spanning from purely theoretical and mathematical…
In recent years, hyper-complex deep networks (such as complex-valued and quaternion-valued neural networks) have received a renewed interest in the literature. They find applications in multiple fields, ranging from image reconstruction to…
The training process of neural networks is known to be time-consuming, and having a deep architecture only aggravates the issue. This process consists mostly of matrix operations, among which matrix multiplication is the bottleneck. Several…
Over the past decade, deep hypercomplex-inspired networks have enhanced feature extraction for image classification by enabling weight sharing across input channels. Recent works make it possible to improve representational capabilities by…
Optimization models involving quaternion matrices are widely used in color image process and other engineering areas. These models optimize real functions of quaternion matrix variables. In particular, $\ell_0$-norms and rank functions of…
Learning high-quality feature embeddings efficiently and effectively is critical for the performance of web-scale machine learning systems. A typical model ingests hundreds of features with vocabularies on the order of millions to billions…
The use of quaternions as a novel tool for color image representation has yielded impressive results in color image processing. By considering the color image as a unified entity rather than separate color space components, quaternions can…
Large language models (LLMs) have shown impressive performance on language tasks but face challenges when deployed on resource-constrained devices due to their extensive parameters and reliance on dense multiplications, resulting in high…
The goal of this paper is to test three classes of neural network (NN) architectures based on four-dimensional (4D) hypercomplex algebras for time series prediction. We evaluate different architectures, varying the input layers to include…
Quaternion contains one real part and three imaginary parts, which provided a more expressive hypercomplex space for learning knowledge graph. Existing quaternion embedding models measure the plausibility of a triplet either through…
Deep Neural Networks (DNNs) have already become a crucial computational approach to revealing the spatial patterns in the human brain; however, there are three major shortcomings in utilizing DNNs to detect the spatial patterns in…
In the present discussion, we have studied the Z2-grading of quaternion algebra (H). We have made an attempt to extend the quaternion Lie algebra to the graded Lie algebra by using the matrix representations of quaternion units. The…
Topological phase classifications have been intensively studied via machine-learning techniques where different forms of the training data are proposed in order to maximize the information extracted from the systems of interests. Due to the…