Wrapped Gaussian process neural network model for manifold-valued data

In modern regression analysis tasks, an increasing amount of data exhibits non-Euclidean characteristics. Modeling methods based on traditional Euclidean space are no longer effective in adapting to the intrinsic geometric features of manifold-structured data. To address this issue, this paper proposes a neural network modeling framework for manifold-valued response variables. The proposed method, on the one hand, achieves a local correspondence between manifold space and Euclidean space through exponential mapping and its inverse mapping, constructing a geometrically compatible connection bridge for the association between predictor variables and manifold-valued response variables. On the other hand, considering the complex nonlinearity and unknown nature of this association, it utilizes the powerful nonlinear fitting capability of neural networks to accurately model it, breaking through the limitations of traditional linear assumptions. In addition, for potential random effects in the data, a wrapped Gaussian process prior is introduced to probabilistically characterize them, enhancing the model's adaptability to random fluctuations. In the parameter estimation stage, this paper adopts a penalized maximum likelihood estimation method to simultaneously solve neural network parameters and kernel function parameters, ensuring the global effectiveness and computational stability of the estimation process. The proposed model also has extensibility: (1) by designing a reasonable metric kernel function, it can directly handle complex scenarios where predictor variables are also in non-Euclidean space; (2) it can achieve uncertainty quantification of manifold-valued response variables based on a probabilistic framework. Simulation studies and real data analysis show that the proposed method is more competitive than Euclidean regression methods that do not consider the intrinsic structure of Riemannian manifolds.