Conformal dimensionality reduction / increase

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.

💡 Research Summary

This paper presents a novel mathematical framework for dimensionality reduction and increase that operates independently of the intrinsic dimension of the dataset. The authors propose two low-complexity, deterministic algorithms based on chains of conformal homeomorphisms—specifically, compositions of generalized stereographic projections and their inverses.

The core idea leverages the angle-preserving (conformal) property of the generalized stereographic projection. This projection is a homeomorphism between a D-dimensional sphere (minus the North Pole) and a D-dimensional plane. Its key feature is that it preserves the angles between intersecting curves. By chaining this projection and its inverse with appropriate mappings between spheres of different dimensions, the authors construct a pathway to transform a dataset from an original Euclidean space R^{D+1} to a target space R^{𝔇+1}, where 𝔇 can be arbitrarily chosen to be smaller or larger than D.

The dimensionality reduction algorithm (Section 2) works as follows: First, each data point in the original high-dimensional space is mapped onto a point on a high-dimensional sphere S^D using the inverse stereographic projection. Then, through a conceptual “lowering” process implied by the chain of mappings, this point is effectively represented on a lower-dimensional sphere S^d. Finally, a stereographic projection maps this point from S^d to the target low-dimensional Euclidean space R^{d+1}. Since every step is a conformal mapping, their composition is also conformal, guaranteeing that all angles between data points are preserved in the final low-dimensional representation.

Conversely, the dimensionality increase algorithm (Section 3) follows a symmetric but opposite chain: from the original space to a sphere, then to a higher-dimensional sphere, and finally via stereographic projection to a higher-dimensional Euclidean space.

A significant theoretical contribution is that these methods make no assumptions about the intrinsic dimension or manifold structure of the data. They are applicable to any set of points, even those lying on non-manifold or disconnected structures. The transformation preserves local shapes due to angle preservation but distorts global sizes and shapes increasingly with distance from a central point of the transformation, as stereographic projection is not isometric.

The paper provides rigorous mathematical definitions and proofs for the properties of the generalized stereographic projection, establishing it as a conformal homeomorphism. The algorithms are presented as explicit compositional formulas, highlighting their deterministic and low-computational complexity nature.

In summary, this work offers a purely geometric approach to dimension alteration that prioritizes the exact preservation of angular relationships. While it may not directly address feature extraction or noise reduction goals common in machine learning, it provides a powerful mathematical tool for understanding and manipulating the geometric structure of data across dimensions, with potential applications in fields where relative directional information is paramount. The trade-off is the inevitable global distortion of distances, a characteristic inherited from the underlying stereographic projection.