# Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations

arXiv: https://arxiv.org/abs/1612.02861 · HTML: https://mlsanity.com/paper/1612.02861/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Dec 2016.

## Built on by (most cited first, 4 of 4)

- [FibeRed: Fiberwise Dimensionality Reduction of Topologically Complex Data with Vector Bundles](https://mlsanity.com/paper/2206.06513/) (2022)
- [O(k)-Equivariant Dimensionality Reduction on Stiefel Manifolds](https://mlsanity.com/paper/2309.10775/) (2023)
- [Learning Tangent Bundles and Characteristic Classes with Autoencoder Atlases](https://mlsanity.com/paper/2602.22873/) (2026)
- [Selecting Interpretable Circular Coordinates from Data](https://mlsanity.com/paper/2607.08230/) (2026)

## Abstract (arXiv)

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.