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>

Effect of Isolated Fracture on the Carbonate Acidizing Process

Carbonate reservoirs are one of the most important fossil fuel sources, and the acidizing stimulation is a practical technique for improving the recovery of carbonate reservoirs. In this study, the improved two-scale continuum model, including the representative elementary volume (REV) scale model and the upscaling model, is used to study the acidizing process with an isolated fracture. Based on this model, a comprehensive discussion is presented to study the effect of the physical parameters of the isolated fracture on the acidizing results and dissolution images, including the isolated fracture geometry, location, and morphology. Results show that the isolated fracture system is still the target system for the acidizing stimulation. The isolated fracture provides a limited contribution to the core porosity. The permeability of the core sample with fracture can be obviously increased only when the fracture penetrates through the whole sample. The existence of the isolated fracture reduces the consumption of acid solution to achieve a breakthrough. The acidizing curve is sensitive to the change of the length, aperture, and position of the isolated fracture. The acidizing curve difference corresponding to different rotation angles has not changed significantly for clockwise rotation and anticlockwise rotation groups.

Impacts of Public Health Insurance on Occupational Upgrading

Using data from the Current Population Survey’s Merged Outgoing Rotation Groups, the authors examine whether greater Medicaid generosity encourages people to switch toward better quality occupations. Exploiting variation in Medicaid eligibility expansions for children across states during the 1990s and early 2000s, they find that a one standard deviation increase in Medicaid infant income thresholds increased the likelihood that working parents move to a new occupation by 1.6 percentage points or 3.3%. Findings show that these effects are larger for those below 150% of the poverty line and for married parents who were not benefiting from Medicaid prior to the expansions. In addition, findings indicate that Medicaid generosity also increased mobility toward occupations with higher average wages and higher educational requirements. This article contributes to the literature on job lock by showing that access to public health insurance not only increases employment and job switches but also encourages occupational upgrading.

Designing Cyclic Job Rotations to Reduce the Exposure to Ergonomics Risk Factors

Job rotation is an administrative solution to prevent work-related musculoskeletal disorders that has become widespread. However, job rotation schedules development is a complex problem. This is due to the multi-factorial character of the disorders and to the productive and organizational constraints of the real working environments. To avoid these problems, this work presents an evolutionary algorithm to generate rotation schedules in which a set of workers rotate cyclically over a small number of jobs while reducing the potential for injury. The algorithm is able to generate rotation schedules that optimize multiple ergonomics criteria by clustering the tasks into rotation groups, selecting the workers for each group, and determining the sequence of rotation of the workers to minimize the effects of fatigue. The algorithm reduces prolonged exposure to risks related to musculoskeletal injuries and simplifies the assignment of workers to different tasks in each rotation. The presented procedure can be an effective tool for the design of job-rotation schedules that prevent work-related musculoskeletal disorders while simplifying scheduled changeovers at each rotation and facilitating job monitoring.

Model Formulation Over Lie Groups and Numerical Methods to Simulate the Motion of Gyrostats and Quadrotors

The present paper recalls a formulation of non-conservative system dynamics through the Lagrange–d’Alembert principle expressed through a generalized Euler–Poincaré form of the system equation on a Lie group. The paper illustrates applications of the generalized Euler–Poincaré equations on the rotation groups to a gyrostat satellite and a quadcopter drone. The numerical solution of the dynamical equations on the rotation groups is tackled via a generalized forward Euler method and an explicit Runge–Kutta integration method tailored to Lie groups.