Inverting Multidimensional Scaling Projections Using Data Point Multilateration

Abstract

Current inverse projection methods are often complex, hard to predict, and may require extensive parametrization. We present a new technique to compute inverse projections of Multidimensional Scaling (MDS) projections with minimal parametrization. We use mutilateration, a method used for geopositioning, to find data values for unknown 2D points, i.e., locations where no data point is projected. Being based on a geometrical relationship, our technique is more interpretable than comparable machine learning-based approaches and can invert 2-dimensional projections up to |D|− 1 dimensional spaces given a minimum of |D| data points. We qualitatively and quantitatively compare our technique with existing inverse projection techniques on synthetic and real-world datasets using mean-squared errors (MSEs) and gradient maps. When MDS captures data distances well, our technique shows performance similar to existing approaches. While our method may show higher MSEs when inverting projected data samples, it produces smoother gradient maps, indicating higher predictability when inverting unseen points.