Sparse Circular Coordinates via Principal $\mathbb{Z}$-Bundles

TL;DR

This paper introduces a novel method for nonlinear dimensionality reduction using principal bundles to derive sparse, circle-valued coordinates from persistent cohomology, capturing data topology efficiently.

Contribution

It applies principal bundle theory to create sparse, circle-valued coordinates from persistent cohomology, enabling topology-aware dimensionality reduction with fewer landmarks.

Findings

Coordinates effectively capture data topology.

Method produces sparse, interpretable embeddings.

Theoretical foundations support practical applications.

Abstract

We present in this paper an application of the theory of principal bundles to the problem of nonlinear dimensionality reduction in data analysis. More explicitly, we derive, from a 1-dimensional persistent cohomology computation, explicit formulas for circle-valued functions on data with nontrivial underlying topology. We show that the language of principal bundles leads to coordinates defined on an open neighborhood of the data, but computed using only a smaller subset of landmarks. It is in this sense that the coordinates are sparse. Several data examples are presented, as well as theoretical results underlying the construction.