Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations

TL;DR

This paper introduces a topology-based framework for creating coordinate representations of data in projective spaces using persistent cohomology, enabling dimensionality reduction and explicit map construction.

Contribution

It develops a novel method to derive coordinate maps to projective spaces from persistent cohomology classes, including a dimensionality reduction technique called Principal Projective Components.

Findings

Constructs maps to real and complex projective spaces from data topology.

Provides a method for dimensionality reduction in projective space.

Includes examples and theoretical results on the construction process.

Abstract

We present in this paper a framework which leverages the underlying topology of a data set, in order to produce appropriate coordinate representations. In particular, we show how to construct maps to real and complex projective spaces, given appropriate persistent cohomology classes. An initial map is obtained in two steps: First, the persistent cohomology of a sparse filtration is used to compute systems of transition functions for (real and complex) line bundles over neighborhoods of the data. Next, the transition functions are used to produce explicit classifying maps for the induced bundles. A framework for dimensionality reduction in projective space (Principal Projective Components) is also developed, aimed at decreasing the target dimension of the original map. Several examples are provided as well as theorems addressing choices in the construction.