Uncovering the Euclidean Geometry of Data

Suppose you have a collection of objects in some exotic metric space and you’d like to see what they would look like if they were instead in Euclidean space. Or suppose these objects aren’t even points in a metric space, they are just entities for which you have some rough, intuitive notion of distance between each pair of them—one that need not satisfy any official mathematical properties like the triangle inequality. There is a linear algebraic optimization procedure called multidimensional scaling that embeds these objects in Euclidean space in a manner that approximates their original distances. This uncovers the Euclidean geometry hidden in data. This article explores how it works and what it’s useful for, particularly in a data science context.