arXiv research

QRPNNs use quaternion-valued recurrent correlation neural networks to solve cross-talk issues.

Proposes QNN to protect input privacy in neural networks.

Quaternion neural networks improve distant speech recognition.

This paper proposes a set of rules to revise various neural networks for 3D point cloud processing to rotation-equivariant quaternion neural networks (REQNNs). We find that when a neural network uses quaternion features under certain conditions, the network feature naturally has the rotation-equivariance property. Rota…

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 coeffic…

QGNN uses Quaternion space for better graph and node classification.

Neural network architectures are at the core of powerful automatic speech recognition systems (ASR). However, while recent researches focus on novel model architectures, the acoustic input features remain almost unchanged. Traditional ASR systems rely on multidimensional acoustic features such as the Mel filter bank en…

Recurrent neural networks (RNNs) are powerful architectures to model sequential data, due to their capability to learn short and long-term dependencies between the basic elements of a sequence. Nonetheless, popular tasks such as speech or images recognition, involve multi-dimensional input features that are characteriz…

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 3D audio processing. Similar to their real-valued counterparts, quaternion neura…

Quaternion self-attention reduces computational cost and improves performance.

Recurrent neural networks (RNN) are at the core of modern automatic speech recognition (ASR) systems. In particular, long-short term memory (LSTM) recurrent neural networks have achieved state-of-the-art results in many speech recognition tasks, due to their efficient representation of long and short term dependencies …

Convolutional neural networks (CNN) have recently achieved state-of-the-art results in various applications. In the case of image recognition, an ideal model has to learn independently of the training data, both local dependencies between the three components (R,G,B) of a pixel, and the global relations describing edge…

Extends RCNNs to handle hypercomplex-valued data.

Unified entropy formula for real, complex, and quaternionic DLNs.

In neural networks, it is often desirable to work with various representations of the same space. For example, 3D rotations can be represented with quaternions or Euler angles. In this paper, we advance a definition of a continuous representation, which can be helpful for training deep neural networks. We relate this t…

AlgebraNets use alternative algebras for neural networks, improving performance on image and language tasks.

Algorithm designs neural group actions for symmetric transformations.

In this paper, we address the stability of a broad class of discrete-time hypercomplex-valued Hopfield-type neural networks. To ensure the neural networks belonging to this class always settle down at a stationary state, we introduce novel hypercomplex number systems referred to as real-part associative hypercomplex nu…

Study on quaternionic bisectional curvature for quaternion-Kähler manifolds.

In this paper, we give the definitions and characterizations of quaternionic Salkowski, quaternionic anti-Salkowski and quaternionic similar curves in the Euclidean spaces E^3 and E^4. We obtain relationships between these curves and some special quaternionic curves such as quaternionic slant helices and quaternionic B…

We construct explicit left invariant quaternionic contact structures on Lie groups with zero and non-zero torsion, and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of quaternionic contact manifolds not locally quaternionic contact conformal to the quaternionic sphere. W…

The study finds conditions for quaternionic structures on symmetric spaces.

We call a quaternionic Kaehler manifold with non-zero scalar curvature, whose quaternionic structure is trivialized by a hypercomplex structure, a hyper-Hermitian quaternionic Kaehler manifold. We prove that every locally symmetric hyper-Hermitian quaternionic Kaehler manifold is locally isometric to the quaternionic p…

We introduce a natural notion of quaternionic map between almost quaternionic manifolds and we prove the following, for maps of rank at least one: 1) A map between quaternionic manifolds endowed with the integrable almost twistorial structures is twistorial if and only if it is quaternionic. 2) A map between quaternion…

Defines quaternionic k-vector fields on quaternionic Kähler manifolds.

The paper proves quaternion projective space is unstable.

The paper studies quaternionic structures on GKM graphs and their relation to torus actions on quaternionic projective spaces.

Penrose's two-spinor notation for $4$-dimensional Lorentzian manifolds can be extended to two-component notation for quaternionic manifolds, which is a very useful tool for calculation. We construct a family of quaternionic complexes over unimodular quaternionic manifolds by elementary calculation. On complex quaternio…

We introduce the notion of CR quaternionic map and we prove that any such real-analytic map, between CR quaternionic manifolds, is the restriction of a quaternionic map between quaternionic manifolds. As an application, we prove, for example, that for any submanifold $M$, of dimension $4k-1$, of a quaternionic manifold…

The paper extends Gray's result to quaternion-Kähler manifolds.

Quaternionic curves with specific torsion properties don't exist.

The conformal infinity of a quaternionic-Kahler metric on a 4n-manifold with boundary is a codimension 3-distribution on the boundary called quaternionic contact. In dimensions 4n-1 greater than 7, a quaternionic contact structure is always the conformal infinity of a quaternionic-Kahler metric. On the contrary, in dim…

Modelled on a real hypersurface in a quaternionic manifold, we introduce a quaternionic analogue of CR structure, called quaternionic CR structure. We define the strong pseudoconvexity of this structure as well as the notion of quaternionic pseudohermitian structure. Following the construction of the Tanaka-Webster con…

Quaternionic differential geometry expands geometric concepts using quaternions.

The paper explores quaternionic curves using differential geometry.

Quaternionic Brownian motion on flag manifold linked to sphere diffusion.

3D capsule network for point clouds handles rotations and translations.

Study cohomology of quaternionic foliations and orbifolds.

Motivated by the quaternionic geometry corresponding to the homogeneous complex manifolds endowed with (holomorphically) embedded spheres, we introduce and initiate the study of the `quaternionic-like manifolds'. These contain, as particular subclasses, the CR quaternionic and the $ρ$-quaternionic manifolds. Moreover, …

Study counts and equidistributes rational points in quaternionic Heisenberg groups.

We apply the general theory of codimension one integrability conditions for $G$-structures developed in arXiv:1306.6817v3 [math.DG] to the case of quaternionic CR geometry. We obtain necessary and sufficient conditions for an almost CR quaternionic manifold to admit local immersions as an hypersurface of the quaternion…

A partial solution of the quaternionic contact Yamabe problem on the quaternionic sphere is given. It is shown that the torsion of the Biquard connection vanishes exactly when the trace-free part of the horizontal Ricci tensor of the Biquard connection is zero and this occurs precisely on 3-Sasakian manifolods. All con…

A tensor invariant is defined on a quaternionic contact manifold in terms of the curvature and torsion of the Biquard connection involving derivatives up to third order of the contact form. This tensor, called quaternionic contact conformal curvature, is similar to the Weyl conformal curvature in Riemannian geometry an…

In this paper we study 3-submersions from a QR-hypersurface of a quaternionic Kaehler manifold onto an almost quaternionic hermitian manifold. We also prove the non-existence of quaternionic submersions between quaternionic Kaehler manifolds which are not locally hyper-Kaehler.

We construct left invariant quaternionic contact (qc) structures on Lie groups with zero and non-zero torsion and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of non-flat quaternionic contact manifolds. We prove that the product of the real line with a seven dimensional…

Study calculates Kulkarni limit sets for quaternionic projective groups.

A quaternionic Kähler manifold M is called {\it positive} if it has positive scalar curvature. The main purpose of this paper is to prove several connectedness theorems for quaternionic immersions in a quaternionic Kähler manifold, e.g. the Barth-Lefschetz type connectedness theorem for quaternionic submanifolds in a p…

Study various submanifolds in quaternionic skew-Hermitian spaces.