| 讲座简介： | Exploring the regression relationship in a general metric space is still a fundamental and difficult issue in statistics and machine learning. As a famous and special regression relationship in a metric space, the Fréchet regression is actually defined within a linear framework, since the weight function is linearly defined, and the resulting Fréchet regression function is identified to be a linear model when the random object belongs to a Hilbert space. Even for nonparametric and semiparametric Fr\'echet regressions, the existing methods handle them by local linear (or local polynomial) technique, and the resulting Fr\'echet regressions are also (locally) linear. We in this work introduce a framework of nonlinear Fréchet regression. As an exploratory work, we first suggest some motivating methods for learning and defining the nonlinearity, and then propose the methods for parameter estimation, and finally establish the asymptotic theories. The proposed framework can be utilized to fit the essentially nonlinear models in a general metric space and uniquely identify the nonlinear structure in a Hilbert space. Particularly, its generalized linear form can return to the standard linear Fréchet regression through a special choice of the weight function. The nonlinear learning methods and favorable theoretical properties, along with the comprehensive simulation studies and real data analysis, demonstrate that the new strategy is easy to use and significantly outperforms the competitors. |
