An improved HLLE algorithm based on the midpoint-nearest neighborhood selection

The tangent spaces of data points play an important role in HLLE. It is based on the tangent spaces of data points that HLLE defines and calculates the Hessian matrices of data points. However, the proof presented in this paper shows that the space commonly used to calculate the Hessian matrix of a data point in HLLE algorithm is not the tangent space of the data point, but the tangent space of the midpoint of the data point's neighborhood. When a data point is far away from the midpoint of its neighborhood, HLLE will break down. This defect of HLLE algorithm has never been pointed out in previous literatures. Based on this fact, an improvement to the original HLLE algorithm is proposed in this paper. The main idea of the improved HLLE algorithm is that the neighborhood of a data point must be chosen so as to make the midpoint of the data point's neighborhood as close to the data point itself as possible. The experimental results presented in this paper show that the improved HLLE algorithm outperforms the original HLLE algorithm on the manifolds such as Punctured Sphere, where the data are often unevenly sampled.