G2BFNN: Generalized geodesic basis function neural network

Real-world data is typically distributed on low-dimensional manifolds embedded in high-dimensional Euclidean spaces. Accurately extracting spatial distribution features on general manifolds that reflect the intrinsic characteristics of data is crucial for effective feature representation. Therefore, we propose a generalized geodesic basis function neural network (G${}^2$BFNN) architecture. The generalized geodesic basis functions (G${}^2$BF) are defined based on generalized geodesic distances. The generalized geodesic distance metric (G${}^2$DM) is obtained by learning the manifold structure. To implement this architecture, a specific G${}^2$BFNN, named discriminative local preserving projection-based G${}^2$BFNN (DLPP-G${}^2$BFNN) is proposed. DLPP-G${}^2$BFNN mainly contains two modules, namely the manifold structure learning module (MSLM) and the network mapping module (NMM). In the MSLM module, a supervised adjacency graph matrix is constructed to constrain the learning of the manifold structure. This enables the learned features in the embedding subspace to maintain the manifold structure while enhancing the discriminability. The features and G${}^2$DM learned in the MSLM are fed into the NMM. Through the G${}^2$BF in the NMM, the spatial distribution features on manifold are obtained. Finally, the output of the network is obtained through the fully connected layer. Compared with the local response neural network based on Euclidean distance, the proposed network can reveal more essential spatial structure characteristics of the data. Meanwhile, the proposed G${}^2$BFNN is a generalized network architecture that can be combined with any manifold learning method, showcasing high scalability. The experimental results demonstrate that the proposed DLPP-G${}^2$BFNN outperforms existing methods by utilizing fewer kernels while achieving higher recognition performance.