Fast Quaternion Product Units for Learning Disentangled Representations in SO(3)

Real-world 3D structured data like point clouds and skeletons often can be represented as data in a 3D rotation group (denoted as $\mathbb{SO}(3)$). However, most existing neural networks are tailored for the data in the Euclidean space, which makes the 3D rotation data not closed under their algebraic operations and leads to sub-optimal performance in 3D-related learning tasks. To resolve the issues caused by the above mismatching between data and model, we propose a novel non-real neuron model called \textit{quaternion product unit} (QPU) to represent data on 3D rotation groups. The proposed QPU leverages quaternion algebra and the law of the 3D rotation group, representing 3D rotation data as quaternions and merging them via a weighted chain of Hamilton products. We demonstrate that the QPU mathematically maintains the $\mathbb{SO}(3)$ structure of the 3D rotation data during the inference process and disentangles the 3D representations into ``rotation-invariant'' features and ``rotation-equivariant'' features, respectively. Moreover, we design a fast QPU to accelerate the computation of QPU. The fast QPU applies a tree-structured data indexing process, and accordingly, leverages the power of parallel computing, which reduces the computational complexity of QPU in a single thread from $\mathcal{O}(N)$ to $\mathcal {O}(\log N)$. Taking the fast QPU as a basic module, we develop a series of quaternion neural networks (QNNs), including quaternion multi-layer perceptron (QMLP), quaternion message passing (QMP), and so on. In addition, we make the QNNs compatible with conventional real-valued neural networks and applicable for both skeletons and point clouds. Experiments on synthetic and real-world 3D tasks show that the QNNs based on our fast QPUs are superior to state-of-the-art real-valued models, especially in the scenarios requiring the robustness to random rotations.<br>

Fast Quaternion Product Units for Learning Disentangled Representations in SO(3)

Real-world 3D structured data like point clouds and skeletons often can be represented as data in a 3D rotation group (denoted as $\mathbb{SO}(3)$). However, most existing neural networks are tailored for the data in the Euclidean space, which makes the 3D rotation data not closed under their algebraic operations and leads to sub-optimal performance in 3D-related learning tasks. To resolve the issues caused by the above mismatching between data and model, we propose a novel non-real neuron model called \textit{quaternion product unit} (QPU) to represent data on 3D rotation groups. The proposed QPU leverages quaternion algebra and the law of the 3D rotation group, representing 3D rotation data as quaternions and merging them via a weighted chain of Hamilton products. We demonstrate that the QPU mathematically maintains the $\mathbb{SO}(3)$ structure of the 3D rotation data during the inference process and disentangles the 3D representations into ``rotation-invariant'' features and ``rotation-equivariant'' features, respectively. Moreover, we design a fast QPU to accelerate the computation of QPU. The fast QPU applies a tree-structured data indexing process, and accordingly, leverages the power of parallel computing, which reduces the computational complexity of QPU in a single thread from $\mathcal{O}(N)$ to $\mathcal {O}(\log N)$. Taking the fast QPU as a basic module, we develop a series of quaternion neural networks (QNNs), including quaternion multi-layer perceptron (QMLP), quaternion message passing (QMP), and so on. In addition, we make the QNNs compatible with conventional real-valued neural networks and applicable for both skeletons and point clouds. Experiments on synthetic and real-world 3D tasks show that the QNNs based on our fast QPUs are superior to state-of-the-art real-valued models, especially in the scenarios requiring the robustness to random rotations.<br>

On the Asymptotics of the Schrödinger Equation Solutions and the Euler Model of an Ideal Fluid

In this paper we analyze the asymptotics of the Schrödinger equation solutions with respect to a small parameter ~. It is well known, that short- waveasymptoticstosolutionsofthisequationleadstothepairofequations— the Hamilton–Jacobi equation for the phase and the continuity equation. These equations coincide with the ones for the potential flows of an ideal fluid. The wave function is invariant with respect to the complex plane rotations group, and the asymptotics is constructed as a point-dependent action of this group on some function that is found by solving the transfer equation. It is shown in the paper, that if the Heisenberg group is used instead of the rotation group, then the limit of the equations solutions with ~ tending to zero leads to the equations for vortex flows of an ideal fluid in a potential field of forces. If the original Schrödinger equation is nonlinear, then equations for barotropic processes in an ideal fluid are obtained.

Associations between facet tropism and vertebral rotation in patients with degenerative lumbar disease

Background Vertebral rotation and facet tropism are very common in various lumbar degenerative diseases. Facet tropism means the presence of asymmetric angles on both sides of the facet joints. Studies have shown that facet tropism contributes to lumbar degenerative disease, and also inevitably leads to the asymmetry of movement and the imbalance of force, which may be possible to rotate the vertebral body. The aim of this study was to explore the correlation between lumbar vertebral rotation and facet tropism in patients with lumbar degenerative diseases. Methods A total of 198 patients with lumbar degenerative diseases from 2018 to 2019 were enrolled. Five hundred and seventy vertebral rotation angles and 1140 facet angles were measured. The vertebral bodies are divided into non-rotation group (Group A) and rotation group (Group B) with the vertebral rotation angle of 3° as the boundary. The information including gender, age, BMI (body mass index), bone mineral density, history of smoking, drinking, hypertension, diabetes, diagnosis, segment distribution, and degree of facet degeneration were also counted. Using inter-class correlation coefficients (ICC) to test the reliability of measurement results. Univariate and multivariate logistic regression analysis were used to analyze the relationship between vertebral rotation and facet tropism. Results The consistency of the ICC within the groups of the observers is above 0.8, with good agreement. The results of univariate analysis showed that facet tropism was significantly different between group A and group B (OR (odds ratio) = 3.30, 95% CI = 2.03–5.35, P < 0.0001). Other significant factors were included as adjustment variables into the multivariate regression model. Three models were analyzed separately (Model 1: non-adjusted. Model 2: adjust for age; facet degeneration; Model 3: adjust for age; disease distribution; segment distribution; facet degeneration). The results showed that after adjusting the confounders, the correlation between facet tropism and vertebral rotation did not change (Model 1: OR = 3.30, 95% CI = 2.03–5.35, P < 0.0001; Model 2: adjusted OR = 2.87, 95% CI = 1.66–4.97, P = 0.0002, Model 3: adjusted OR = 2.84, 95% CI = 1.56–5.17, P = 0.0006). Conclusion Current research demonstrates that there is an association between vertebral rotation and facet tropism, suggesting that vertebral rotation may also have a certain degree of correlation with lumbar degenerative diseases.

SU(N) gauge theories in 3+1 dimensions: glueball spectrum, string tensions and topology

We calculate the low-lying glueball spectrum, several string tensions and some properties of topology and the running coupling for SU(N) lattice gauge theories in 3 + 1 dimensions. We do so for 2 ≤ N ≤ 12, using lattice simulations with the Wilson plaquette action, and for glueball states in all the representations of the cubic rotation group, for both values of parity and charge conjugation. We extrapolate these results to the continuum limit of each theory and then to N = ∞. For a number of these states we are able to identify their continuum spins with very little ambiguity. We calculate the fundamental string tension and k = 2 string tension and investigate the N dependence of the ratio. Using the string tension as the scale, we calculate the running of a lattice coupling and confirm that g2(a) ∝ 1/N for constant physics as N → ∞. We fit our calculated values of a√σ with the 3-loop β-function, and extract a value for $$ {\Lambda}_{\overline{MS}} $$ Λ MS ¯ , in units of the string tension, for all our values of N, including SU(3). We use these fits to provide analytic formulae for estimating the string tension at a given lattice coupling. We calculate the topological charge Q for N ≤ 6 where it fluctuates sufficiently for a plausible estimate of the continuum topological susceptibility. We also calculate the renormalisation of the lattice topological charge, ZQ(β), for all our SU(N) gauge theories, using a standard definition of the charge, and we provide interpolating formulae, which may be useful in estimating the renormalisation of the lattice θ parameter. We provide quantitative results for how the topological charge ‘freezes’ with decreasing lattice spacing and with increasing N. Although we are able to show that within our typical errors our glueball and string tension results are insensitive to the freezing of Q at larger N and β, we choose to perform our calculations with a typical distribution of Q imposed upon the fields so as to further reduce any potential systematic errors.

Interpolation on the special orthogonal group with high-dimensional Kuramoto model

The construction of smooth interpolation trajectories in different non-Euclidean spaces finds application in robotics, computer graphics, and many other engineering fields. This paper proposes a method for generating interpolation trajectories on the special orthogonal group SO(3), called the rotation group. Our method is based on a high-dimensional generalization of the Kuramoto model which is a well-known mathematical description of self-organization in large populations of coupled oscillators. We present the method through several simulations and visualize each simulation as trajectories on unit spheres S2. In addition, we applied our method to the specific problem of object rotation interpolation.

Existence and Stability of Kayaking Orbits for Nematic Liquid Crystals in Simple Shear Flow

We use geometric methods of equivariant dynamical systems to address a long-standing open problem in the theory of nematic liquid crystals, namely a proof of the existence and asymptotic stability of kayaking periodic orbits in response to steady shear flow. These are orbits for which the principal axis of orientation of the molecular field (the director) rotates out of the plane of shear and around the vorticity axis. With a small parameter attached to the symmetric part of the velocity gradient, the problem can be viewed as a symmetry-breaking bifurcation from an orbit of the rotation group $$\mathrm{SO}(3)$$ SO ( 3 ) that contains both logrolling (equilibrium) and tumbling (periodic rotation of the director within the plane of shear) regimes as well as a continuum of neutrally stable kayaking orbits. The results turn out to require expansion to second order in the perturbation parameter.

A multiple-relaxation-time collision model by Hermite expansion

The Bhatnagar–Gross–Krook (BGK) single-relaxation-time collision model for the Boltzmann equation serves as the foundation of the lattice BGK (LBGK) method developed in recent years. The description of the collision as a uniform relaxation process of the distribution function towards its equilibrium is, in many scenarios, simplistic. Based on a previous series of papers, we present a collision model formulated as independent relaxations of the irreducible components of the Hermite coefficients in the reference frame moving with the fluid. These components, corresponding to the irreducible representation of the rotation group, are the minimum tensor components that can be separately relaxed without violating rotation symmetry. For the 2nd, 3rd and 4th moments, respectively, two, two and three independent relaxation rates can exist, giving rise to the shear and bulk viscosity, thermal diffusivity and some high-order relaxation process not explicitly manifested in the Navier–Stokes-Fourier equations. Using the binomial transform, the Hermite coefficients are evaluated in the absolute frame to avoid the numerical dissipation introduced by interpolation. Extensive numerical verification is also provided. This article is part of the theme issue ‘Progress in mesoscale methods for fluid dynamics simulation’.

2D ELASTICITY TENSOR INVARIANTS, INVARIANTS DEFINITE POSITIVE CRITERIA

In this paper, we constructed relationships with the differents 2D elasticity tensor invariants. Indeed, let ${\bf A}$ be a 2D elasticity tensor. Rotation group action leads to a pair of Lax in linear elasticity. This pair of Lax leads to five independent invariants chosen among six. The definite positive criteria are established with the determined invariants. We believe that this approach finds interesting applications, as in the one of elastic material classification or approaches in orbit space description.