Enhance explainability of manifold learning

The explainability of manifold learning is rarely investigated though there is an urgent need from both AI theory and practice. In this study, we propose a novel degree of locality preservation (DLP) approach to study the interpretability of manifold learning. We estimate the DLPs of the state-of-the-art manifold learning methods: t-SNE and UMAP as well as related methods: LLE, HLLE, and LTSA along with widely used PCA across benchmark datasets classified as low-dimensional and high-dimensional data. Our study provides well-founded explanations of the manifold learning methods in terms of the DLPs. The order of their DLPs follows t-SNE> UMAP> LLE> HLLE/PCA/LTSA, though it may have some exceptions for some high-dimensional data. Both t-SNE and UMAP demonstrate an embedding distance amplification mechanism under the Euclidean distance that forces the latent local data geometry to stand out in dimension reduction. It not only explains why t-SNE and UMAP have higher DLPs than other peers, but also indicates they are not locally isometric under the Euclidean distance. Furthermore, it discovers that t-SNE and UMAP embeddings demonstrate similar nonlinear nature in dimension reduction, besides larger (smaller) data variances for low (high)-dimensional data. To the best of our knowledge, this study is the first work about the explainability of manifold learning. The proposed methods and corresponding results can be also extended to other dimension reduction techniques.(c) 2022 Published by Elsevier B.V.