Nonlinear sufficient dimension reduction for distribution-on-distribution regression

We introduce a new approach to nonlinear sufficient dimension reduction in cases where both the predictor and the response are distributional data, modeled as members of a metric space. Our key step is to build universal kernels (cc-universal) on the metric spaces, which results in reproducing kernel Hilbert spaces for the predictor and response that are rich enough to characterize the conditional independence that determines sufficient dimension reduction. For univariate distributions, we construct the universal kernel using the Wasserstein distance, while for multivariate distributions, we resort to the sliced Wasserstein distance. The sliced Wasserstein distance ensures that the metric space possesses similar topological properties to the Wasserstein space, while also offering significant computation benefits. Numerical results based on synthetic data show that our method outperforms possible competing methods. The method is also applied to several data sets, including fertility and mortality data and Calgary temperature data.

Unsupervised domain adaptation of bearing fault diagnosis based on Join Sliced Wasserstein Distance

Deep neural networks have been successfully utilized in the mechanical fault diagnosis, however, a large number of them have been based on the same assumption that training and test datasets followed the same distributions. Unfortunately, the mechanical systems are easily affected by environment noise interference, speed or load change. Consequently, the trained networks have poor generalization under various working conditions. Recently, unsupervised domain adaptation has been concentrated on more and more attention since it can handle different but related data. Sliced Wasserstein Distance has been successfully utilized in unsupervised domain adaptation and obtained excellent performances. However, most of the approaches have ignored the class conditional distribution. In this paper, a novel approach named Join Sliced Wasserstein Distance (JSWD) has been proposed to address the above issue. Four bearing datasets have been selected to validate the practicability and effectiveness of the JSWD framework. The experimental results have demonstrated that about 5% accuracy is improved by JSWD with consideration of the conditional probability than no the conditional probability, in addition, the other experimental results have indicated that JSWD could effectively capture the distinguishable and domain-invariant representations and have a has superior data distribution matching than the previous methods under various application scenarios.