Multidimensional Scaling for Big Data

Abstract: We present a set of algorithms implementing multidimensional scaling (MDS) for large data sets. MDS is a family of dimensionality reduction techniques using a $n \times n$ distance matrix as input, where $n$ is the number of individuals, and producing a low dimensional configuration: a $n\times r$ matrix with $r<<n$. When $n$ is large, MDS is unaffordable with classical MDS algorithms because of their extremely large memory and time requirements. We compare six non-standard algorithms intended to overcome these difficulties. They are based on the central idea of partitioning the data set into small pieces, where classical MDS methods can work. Two of these algorithms are original proposals. In order to check the performance of the algorithms as well as to compare them, we have done a simulation study. Additionally, we have used the algorithms to obtain an MDS configuration for EMNIST: a real large data set with more than $800000$ points. We conclude that all the algorithms are appropriate to use for obtaining an MDS configuration, but we recommend using one of our proposals since it is a fast algorithm with satisfactory statistical properties when working with big data. An R package implementing the algorithms has been created.