Manifold learning aims to extract information of high-dimensional data and provide low-dimensional representations while preserving nonlinear structures of the input data. Numerous manifold learning algorithms have been proposed in the literature. We develop a model averaging procedure to combine different manifold learning algorithms for enhancing the robustness of the result. Toward this goal, we propose a new quality metric that is tuning-free and scale-invariant by utilizing the Mahalanobis distance. The quality metric can also be used for selection of tuning parameters. By optimizing the weights and ensembling different approaches, our model averaging manifold learning method is shown to achieve more robust results, because no single method in the ensemble can dominate others in all scenarios. Through synthetic and real data examples, we show that the new metric outperforms existing ones and the model averaging outcome provides a unanimously superior outcome that is always competitive with respect to visualization or classification under different contexts.

We study the causal effect with a functional treatment variable, where practical applications often arise in neuroscience, biomedical sciences, etc. Previous research concerning the effect of a functional variable on an outcome is typically restricted to exploring correlation rather than causality. The generalized propensity score, which is often used to calibrate the selection bias, is not directly applicable to a functional treatment variable due to a lack of definition of probability density function for functional data. We propose three estimators for the average dose-response functional based on the functional linear model, namely, the functional stabilized weight estimator, the outcome regression estimator and the doubly robust estimator, each of which has its own merits. We study their theoretical properties, which are corroborated through extensive numerical experiments. A real data application on electroencephalography data and disease severity demonstrates the practical value of our methods.