Question: What is the promise of the "Generic framework"?
Evidence: We propose a new DR method called SpaceMAP, which, in principle, can map data of any dimensionality onto a 2-dimensional space for visualization with the calculated space expansion. Different from established methods such as t-SNE or UMAP that perform implicit transformation of distances, we analytically derived a quantitative EED transformation of distances between high-dimensional space and low-dimensional spaces. We further show that the EED transformation fits in the classical framework of MLE of intrinsic dimension, effectively altering the intrinsic dimension thereby realizing low-dimensional mapping.
We argue that all successful DR methods, including among others t-SNE and UMAP, make use of the rationale of space expansion to enable data visualization in a space of drastically reduced dimension. However, previous methods did such transformations in a highly implicit manner, with the rules concealed in the self-defined similarity measure (e.g. t-distribution in t-SNE or inverse of  polynomial in UMAP, with different parameters). Despite their empirical success, we posit that an analytical form of distance transformation is desirable to deal with situations where we would like to take more control of the DR results or impose prior knowledge for DR.
SpaceMAP further differentiates different range of neighborhoods to model the hierarchical structure existent in many real-world datasets. SpaceMAP is generic and has a limited number of hyperparameters, with the most important ones related to the selection of number of nearest neighbors in the near field and middle field. We observed that the final results are not particularly sensitive to the selection of these parameters (Appendix Figure).
In conclusion, we have proposed a new DR method, SpaceMAP, which is based on a principled way to explicitly transform distances in high-and low-dimensional spaces, and models the hierarchical structure of data based on the intrinsic dimension of local and global manifolds. Our experiments on a diverse range of datasets demonstrated its excellent performance in comparison with other state-of-the-art DR methods.     Here we observed that SpaceMAP potentially better preserves the geometry of the manifold than UMAP. As the Swiss Roll has a denser distribution of points on the purple end than on the red end, UMAP resulted in a wider band on the purple end, promoted by the uniform approximation.
We argue that all successful DR methods, including among others t-SNE and UMAP, make use of the rationale of space expansion to enable data visualization in a space of drastically reduced dimension. However, previous methods did such transformations in a highly implicit manner, with the rules concealed in the self-defined similarity measure (e.g. t-distribution in t-SNE or inverse of polynomial in UMAP, with different parameters). Despite their empirical success, we posit that an analytical form of distance transformation is desirable to deal with situations where we would like to take more control of the DR results or impose prior knowledge for DR.
Answer: In this revised version, the promise of a generic framework has been now delivered in Discussion (2nd paragraph). We argue that all successful DR methods, including among others t-SNE and UMAP, make use of the rationale of space expansion to enable data visualization in a space of drastically reduced dimension. However, previous methods did such transformations in a highly implicit manner, with the rules concealed in the self-defined similarity measure (e.g. t-distribution in t-SNE or inverse of polynomial in UMAP, with different parameters). Despite their empirical success, we posit that an analytical form of distance transformation is desirable to deal with situations where we would like to take more control of the DR results or impose prior knowledge for DR.