Geometric Machine Learning Research Articles

Fast evaluation of spherical harmonics with sphericart.

Spherical harmonics provide a smooth, orthogonal, and symmetry-adapted basis to expand functions on a sphere, and they are used routinely in physical and theoretical chemistry as well as in different fields of science and technology, from geology and atmospheric sciences to signal processing and computer graphics. More recently, they have become a key component of rotationally equivariant models in geometric machine learning, including applications to atomic-scale modeling of molecules and materials. We present an elegant and efficient algorithm for the evaluation of the real-valued spherical harmonics. Our construction features many of the desirable properties of existing schemes and allows us to compute Cartesian derivatives in a numerically stable and computationally efficient manner. To facilitate usage, we implement this algorithm in sphericart, a fast C++ library that also provides C bindings, a Python API, and a PyTorch implementation that includes a GPU kernel.

Geometric learning for computational mechanics Part II: Graph embedding for interpretable multiscale plasticity

The history-dependent behaviors of classical plasticity models are often driven by internal variables evolved according to phenomenological laws. The difficulty to interpret how these internal variables represent a history of deformation, the lack of direct measurement of these internal variables for calibration and validation, and the weak physical underpinning of those phenomenological laws have long been criticized as barriers to creating realistic models. In this work, geometric machine learning on graph data (e.g. finite element solutions) is used as a means to establish a connection between nonlinear dimensional reduction techniques and plasticity models. Geometric learning-based encoding on graphs allows the embedding of rich time-history data onto a low-dimensional Euclidean space such that the evolution of plastic deformation can be predicted in the embedded feature space. A corresponding decoder can then convert these low-dimensional internal variables back into a weighted graph such that the dominating topological features of plastic deformation can be observed and analyzed.

Geometric machine learning: research and applications

Geometric machine learning: research and applications

Geometric Machine Learning Over Riemannian Manifolds for Wireless Link Scheduling

In this paper, we propose two novel geometric machine learning (G-ML) methods for the wireless link scheduling problem in device-to-device (D2D) networks. In dynamic D2D networks (e.g., vehicular networks), obtaining a large number of training samples is time-consuming for real-time response, and acquiring accurate instantaneous channel state information (CSI) is challenging due to high mobility. Our goal is to efficiently represent D2D networks on Riemannian manifold and use G-ML with few training wireless network layouts and no CSI to approach the sum rate maximization performance as state-of-the-arts which is much needed in dynamic networks. To this aim, we first model the local graph around each D2D pair as a point through a set of regularized Laplacian matrices on the Riemannian manifold. We compute the Riemannian metric, e.g., Log-Euclidean metric (LEM), among D2D pairs, which are suitable measures of interference among these pairs. We use LEM in the geometric support vector machine (G-SVM) method to classify the link scheduling decisions in a supervised learning manner. Then we propose geometric <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-means clustering for unsupervised scheduling method for the case when no labeled training is available in some dynamic networks. Simulation results demonstrate that the proposed methods achieve promising performance for sum rate maximization against the existing state-of-the-arts approaches with only around a hundred training wireless network layouts for training and without using CSI.

Selecting XFEL single-particle snapshots by geometric machine learning.

A promising new route for structural biology is single-particle imaging with an X-ray Free-Electron Laser (XFEL). This method has the advantage that the samples do not require crystallization and can be examined at room temperature. However, high-resolution structures can only be obtained from a sufficiently large number of diffraction patterns of individual molecules, so-called single particles. Here, we present a method that allows for efficient identification of single particles in very large XFEL datasets, operates at low signal levels, and is tolerant to background. This method uses supervised Geometric Machine Learning (GML) to extract low-dimensional feature vectors from a training dataset, fuse test datasets into the feature space of training datasets, and separate the data into binary distributions of “single particles” and “non-single particles.” As a proof of principle, we tested simulated and experimental datasets of the Coliphage PR772 virus. We created a training dataset and classified three types of test datasets: First, a noise-free simulated test dataset, which gave near perfect separation. Second, simulated test datasets that were modified to reflect different levels of photon counts and background noise. These modified datasets were used to quantify the predictive limits of our approach. Third, an experimental dataset collected at the Stanford Linear Accelerator Center. The single-particle identification for this experimental dataset was compared with previously published results and it was found that GML covers a wide photon-count range, outperforming other single-particle identification methods. Moreover, a major advantage of GML is its ability to retrieve single particles in the presence of structural variability.