  Different unsupervised models for dimensionality reduction like PCA, LLE,
Shannon's mapping, tSNE, UMAP, etc. work on different principles, hence, they
are difficult to compare on the same ground. Although they are usually good for
visualisation purposes, they can produce spurious patterns that are not present
in the original data, losing its trustability (or credibility). On the other
hand, information about some response variable (or knowledge of class labels)
allows us to do supervised dimensionality reduction such as SIR, SAVE, etc.
which work to reduce the data dimension without hampering its ability to
explain the particular response at hand. Therefore, the reduced dataset cannot
be used to further analyze its relationship with some other kind of responses,
i.e., it loses its generalizability. To make a better dimensionality reduction
algorithm with a better balance between these two, we shall formally describe
the mathematical model used by dimensionality reduction algorithms and provide
two indices to measure these intuitive concepts such as trustability and
generalizability. Then, we propose a Localized Skeletonization and
Dimensionality Reduction (LSDR) algorithm which approximately achieves
optimality in both these indices to some extent. The proposed algorithm has
been compared with state-of-the-art algorithms such as tSNE and UMAP and is
found to be better overall in preserving global structure while retaining
useful local information as well. We also propose some of the possible
extensions of LSDR which could make this algorithm universally applicable for
various types of data similar to tSNE and UMAP.
