Conformal dimensionality reduction / increase

Abstract

We give two low-complexity algorithms, one for dimensionality reduction and one for dimensionality increase, which are applicable to any dataset, regardless of whether the set has an intrinsic dimension or not. The corresponding methods introduce chains of compositions of conformal homeomorphisms that transform any data set $\mathbb{X}$ in a Euclidean space $\mathbb{R}^{D+1}$ into an isopleth dataset $ \mathbb{Y}$ within a Euclidean space $\mathbb{R}^{\mathfrak{D}+1}$ of arbitrarily smaller or of arbitrarily larger dimension $\mathfrak{D}+1$ and preserve all angles, in the sense that all angles formed between points in the original dataset $ \mathbb{X}$ are equal to the angles formed between the images of these points in the new dataset $\mathbb{Y}$. Because they preserve angles, the two methods also preserve shapes locally, although, in general, the overall sizes and shapes are distorted away from a center point.