On the Selection of Tuning Parameters for Patch-Stitching Embedding Methods

In the general problem known as multidimensional scaling (MDS), the primary objective is to represent a set of items as points within a Euclidean space of a specified dimension. This representation should ideally preserve the given pairwise dissimilarities as accurately as possible, by ensuring that the Euclidean distances between these points mirror the original dissimilarities. MDS is a extensively researched problem found in diverse fields such as psychometrics [16], mathematics, and computer science [9, 14, 57], engineering (where it is also known as network localization) [61], as well as statistics [3, 71] and machine learning [43, Ch 14]. Dimensionality reduction (DR) aims at embedding data points in a Euclidean space into a lower-dimensional Euclidean space while preserving, as much as possible, the geometry of the point cloud [36, 59]. When the data points are assumed to be on or near a smooth submanifold, a variant of DR known as manifold learning, this typically means preserving the pairwise intrinsic distances to the greatest extent.