Fiberwise dimensionality reduction of topologically complex data with vector bundles

  title={Fiberwise dimensionality reduction of topologically complex data with vector bundles},
  author={Luis Scoccola and Jose A. Perea},
Datasets with non-trivial large scale topology can be hard to embed in low-dimensional Euclidean space with existing dimensionality reduction algorithms. We propose to model topologically complex datasets using vector bundles, in such a way that the base space accounts for the large scale topology, while the fibers account for the local geometry. This allows one to reduce the dimensionality of the fibers, while preserving the large scale topology. We formalize this point of view, and, as an… 

Figures and Tables from this paper

TOAST: Topological Algorithm for Singularity Tracking

A topological framework is developed that identifies singularities of complex spaces, while also capturing singular structures and local geometric complexity in image data, and yields a Euclidicity score for assessing the ‘manifoldness’ of a point along multiple scales.



Nonlinear dimensionality reduction of data manifolds with essential loops

Improving Metric Dimensionality Reduction with Distributed Topology

DIPOLE is a dimensionalityreduction post-processing step that corrects an initial embedding by minimizing a loss functional with both a local, metric term and a global, topological term.

A global geometric framework for nonlinear dimensionality reduction.

An approach to solving dimensionality reduction problems that uses easily measured local metric information to learn the underlying global geometry of a data set and efficiently computes a globally optimal solution, and is guaranteed to converge asymptotically to the true structure.

Orientability and Diffusion Maps.

Nonlinear dimensionality reduction by locally linear embedding.

Locally linear embedding (LLE) is introduced, an unsupervised learning algorithm that computes low-dimensional, neighborhood-preserving embeddings of high-dimensional inputs that learns the global structure of nonlinear manifolds.

Principal Manifolds and Nonlinear Dimensionality Reduction via Tangent Space Alignment

We present a new algorithm for manifold learning and nonlinear dimensionality reduction. Based on a set of unorganized da-ta points sampled with noise from a parameterized manifold, the local

Homology-Preserving Dimensionality Reduction via Manifold Landmarking and Tearing

Inspired by recent work in topological data analysis, this work is on the quest for a dimensionality reduction technique that achieves the criterion of homology preservation, a generalized version of topology preservation.

Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations

A framework which leverages the underlying topology of a data set, in order to produce appropriate coordinate representations, and shows how to construct maps to real and complex projective spaces, given appropriate persistent cohomology classes.

A study on validating non-linear dimensionality reduction using persistent homology

Laplacian Eigenmaps for Dimensionality Reduction and Data Representation

This work proposes a geometrically motivated algorithm for representing the high-dimensional data that provides a computationally efficient approach to nonlinear dimensionality reduction that has locality-preserving properties and a natural connection to clustering.