Multi-point dimensionality reduction to improve projection layout reliability
In ordinary Dimensionality Reduction (DR), each data instance in a high dimensional space (original space), or on a distance matrix denoting original space distances, is mapped to (projected onto) one point in a low dimensional space (visual space), building a layout of projected points trying to preserve as much as possible some property of data such as distances, neighbourhood relationships, and/or topology structures, with the ultimate goal of approximating semantic properties of data with preserved geometric properties or topology structures in visual space. In this paper, the concept of Multi-point Dimensionality Reduction is elaborated on where each data instance can be mapped to (projected onto) possibly more than one point in visual space by providing the first general solution (algorithm) for it as a move in the direction of improving reliablity, usability and interpretability of dimensionality reduction. Furthermore by allowing the points in visual space to be split into two layers while maintaining the possibility of having more than one projection (mapping) per data instance , the benefit of separating more reliable points from less reliable points is dicussed notwithstanding the effort to improve less reliable points. The proposed solution (algorithm) in this paper, named Layered Vertex Splitting Data Embedding (LVSDE), is built upon and extends a combination of ordinary DR and graph drawing techniques. Based on the experiments of this paper on some data sets, the particular proposed algorithm (LVSDE) practically outperforms popular ordinary DR methods visually (semantics, group separation, subgroup detection or combinational group detection) in a way that is easily explainable.