Manifold Discovery for High-Dimensional Data Using Deep Method

It is a challenge for manifold discovery from the data in the high-dimensional space, since the data in the high-dimensional space is sparsely distributed, which hardly provides rich information for manifold discovery so as to be possible to obtain deformed manifolds. To address this issue, this paper designed a deep model based on Brenier theorem for manifold discovery. Since Brenier theorem can find the optimal transportation mass distance between the reconstructed data distribution and the original data distribution, the manifold discovered from in the reconstructed data distribution can be as close to the original manifold as possible. Results not only show that the proposed method wins over competing methods both the precision of manifold discovery and resistance to data sparsity, but also show those non-linear architectures with deep paradigms outperform those architectures without deep paradigms in terms of manifold discovery. We find that the loss function derived by Brenier theorem can help that models minimize the error between the reconstructed manifold and the original manifold. Moreover, the manner constraining neurons with norm-2 is better than that of with norm-1 in both easing data sparsity and improving the precision of for manifold discovery.