# Persistent Cohomology and Circular Coordinates

@article{Silva2011PersistentCA, title={Persistent Cohomology and Circular Coordinates}, author={Vin de Silva and Dmitriy Morozov and Mikael Vejdemo-Johansson}, journal={Discrete \& Computational Geometry}, year={2011}, volume={45}, pages={737-759} }

Nonlinear dimensionality reduction (NLDR) algorithms such as Isomap, LLE, and Laplacian Eigenmaps address the problem of representing high-dimensional nonlinear data in terms of low-dimensional coordinates which represent the intrinsic structure of the data. This paradigm incorporates the assumption that real-valued coordinates provide a rich enough class of functions to represent the data faithfully and efficiently. On the other hand, there are simple structures which challenge this assumption…

## 55 Citations

Persistent Cohomology for Data With Multicomponent Heterogeneous Information

- Computer ScienceSIAM J. Math. Data Sci.
- 2020

It is found that the proposed framework outperforms or at least matches the state-of-the-art methods in the protein-ligand binding affinity prediction from massive biomolecular datasets without resorting to any deep learning formulation.

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

- Mathematics, Computer Science
- 2018

It is shown 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, so that the coordinates are sparse.

Sparse Circular Coordinates via Principal ℤ-Bundles

- Mathematics, Computer ScienceArXiv
- 2018

It is shown 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, so that the coordinates are sparse.

Approximation algorithms for Vietoris-Rips and Čech filtrations

- Computer Science, Mathematics
- 2017

A lower bound result is provided: a point cloud is constructed that requires super-polynomial complexity for a high-quality approximation of the persistence, and it is shown that polynomial complexity is achievable for rough approximation, but impossible for sufficiently fine approximations.

Evaluating State Space Discovery by Persistent Cohomology in the Spatial Representation System

- MathematicsFrontiers in Computational Neuroscience
- 2021

The results reveal how dataset parameters affect the success of topological discovery and suggest principles for applying persistent cohomology, as well as persistent homology, to experimental neural recordings.

Evaluating state space discovery by persistent cohomology in the spatial representation system

- Mathematics
- 2020

This work comprehensively and rigorously assess its performance in simulated neural recordings of the brain’s spatial representation system, and identifies regimes under which mixtures of populations form product topologies that can be detected.

25 HIGH-DIMENSIONAL TOPOLOGICAL DATA ANALYSIS

- Mathematics
- 2016

simplicial complex: Given a set X, an abstract simplicial complex C with vertex set X is a set of finite subsets of X, the simplices, such that the elements of X belong to C and if σ ∈ C and τ ⊂ σ,…

Computing persistent homology with various coefficient fields in a single pass

- MathematicsJ. Appl. Comput. Topol.
- 2019

An algorithm to compute the persistent homology of a filtered complex with various coefficient fields in a single matrix reduction allows us to infer the prime divisors of the torsion coefficients of the integral homology groups of the topological space at any scale, hence furnishing a more informative description of topology than persistence in asingle coefficient field.

Twisty Takens: A Geometric Characterization of Good Observations on Dense Trajectories

- MathematicsJ. Appl. Comput. Topol.
- 2019

This work state conditions on observation functions defined on compact Riemannian manifolds that lead to successful reconstructions for particular dynamics, and construct families of time series whose sliding window embeddings trace tori, Klein bottles, spheres, and projective planes.

Persistence for Circle Valued Maps

- MathematicsArXiv
- 2011

A finite collection of computable invariants which answer the basic questions on persistence and in addition encode the topology of the source space and its relevant subspaces with these invariants.