A speed preserving Hilbert gradient flow for generalized integral Menger curvature

We establish long-time existence for a projected Sobolev gradient flow of generalized integral Menger curvature in the Hilbert case and provide C 1 , 1 C^{1,1} -bounds in time for the solution that only depend on the initial curve. The self-avoidance property of integral Menger curvature guarantees that the knot class of the initial curve is preserved under the flow, and the projection ensures that each curve along the flow is parametrized with the same speed as the initial configuration. Finally, we describe how to simulate this flow numerically with substantially higher efficiency than in the corresponding numerical L 2 L^{2} gradient descent or other optimization methods.

Feature selection method based on Menger curvature and LDA theory for a P300 brain-computer interface

Brain–computer interface (BCI) systems decode electroencephalogram signals to establish a channel for direct interaction between the human brain and the external world without the need for muscle or nerve control. The P300 speller, one of the most widely used BCI applications, presents a selection of characters to the user and performs character recognition by identifying P300 event-related potentials from the EEG. Such P300-based BCI systems can reach good levels of accuracy but are difficult to use in day-to-day life due to redundancy and noisy signal. A room for improvement should be considered. We propose a novel hybrid feature selection method for the P300-based BCI system to address the problem of feature redundancy, which combines the Menger curvature and linear discriminant analysis. First, selected strategies are applied separately to a given dataset to estimate the gain for application to each feature. Then, each generated value set is ranked in descending order and judged by a predefined criterion to be suitable in classification models. The intersection of the two approaches is then evaluated to identify an optimal feature subset. The proposed method is evaluated using three public datasets, i.e., BCI Competition III dataset II, BNCI Horizon dataset, and EPFL dataset. Experimental results indicate that compared with other typical feature selection and classification methods, our proposed method has better or comparable performance. Additionally, our proposed method can achieve the best classification accuracy after all epochs in three datasets. In summary, our proposed method provides a new way to enhance the performance of the P300-based BCI speller.