Questions tagged [persistent-homology]
The persistent-homology tag has no usage guidance.
19
questions
3
votes
0answers
189 views
Persistent homotopy groups
Everybody in algebraic topology loves homology and cohomology, but sometimes we like homotopy groups also, since they detect different things (think about spheres) .
An interesting and recent ...
1
vote
0answers
57 views
Direct representation of simplical complexes in a HoTT implementation
Persistent homology can be used to transform a point-cloud into a simplical complex.
Do such simplical complexes have a first-class representation:
Conceptually, within HoTT?
Concretely, within some ...
3
votes
0answers
90 views
Motivation for persistent homology with respect to eigenfunctions of distance kernel operator in a recent preprint
I have a question about a recent preprint https://arxiv.org/pdf/1912.02225.pdf by Maria, Oudot, and Solomon. As far as I understand, in Section 8 they prove that persistent homology (persistence ...
4
votes
2answers
218 views
Is there an upper bound on the number of points in point cloud for which we compute the persistent homology?
I am interested in the topic of persistent homology (topological data analysis). According to what I read, there is some roadblock in the analysis of "big data" using persistent homology as it is ...
2
votes
1answer
104 views
Persistent homology stability results (query about Lipschitz functions)
One of the beneficial properties of persistent homology is its stability results (so called robustness to noise).
Usually the referenced paper is this paper
titled "Lipschitz functions have $L_p$-...
6
votes
0answers
158 views
Coarsifying persistence modules
The context
Let $I=[0,∞)$ and consider the category of persistence modules $(V,π)$ indexed over $I$ satisfying:
For all $t$ in $I$ but a closed discrete set of points $T$, there exists a ...
2
votes
0answers
120 views
Discrete Morse theory, choice of Morse function, and removing noise
If I have a simplicial complex, and a discrete Morse function defined on the simplices, I can use persistent homology to produce a barcode which helps me distinguish "persistent" shape from noise. To ...
6
votes
1answer
263 views
Approximate homology of a large simplicial complex
I can use software to calculate the Betti numbers $\beta_0,\beta_1,\beta_2,\dots$ of a finite simplicial complex.
This is prohibitive for large complexes, built on say > 100,000 nodes.
Is there some ...
39
votes
5answers
2k views
Reference on Persistent Homology
I will be teaching a course on algebraic topology for MSc students and this semester, unlike previous ones where I used to begin with the fundamental group, I would like to start with ideas of ...
2
votes
1answer
118 views
On the entries of a matrix representation for a boundary operator of a persistence module
In equation 6 of Computing Persistent Homology (page 8), the authors put forward the following identity:
$$\deg \hat{e_i}+\deg M_k (i,j)=\deg e_j$$
Where $\hat{e_i}$ and $e_j$ are elements of ...
32
votes
1answer
2k views
Why is persistent cohomology so much faster than persistent homology
I refer to this paper: de Silva, Vin; Morozov, Dmitriy; Vejdemo-Johansson, Mikael. Dualities in persistent (co)homology. Inverse Problems 27 (2011), no. 12, 124003, 17 pp. (Journal link, arXiv link).
...
9
votes
1answer
476 views
Persistent homology over the integers
Is it likely that in the future, there will be interest in computing persistent homology over the integers (or other PIDs)?
Currently, persistent homology is usually done over a field (like $\mathbb{...
0
votes
1answer
329 views
Clarification of “death event” in persistent homology
Before I ask my question let me clarify some notation:
$f^{i,j}_r$, where $i < j$, refers to the inclusion map $f: H_r(X_i) \hookrightarrow H_r(X_j)$. $X_i$ and $X_j$ are subcomplexes of a filtered ...
12
votes
3answers
540 views
Category of data sets, motivated by persistent homology?
Is there a useful or agreed-upon category of data sets? In particular, I'm thinking about a point cloud and wondering what an acceptable morphism between point clouds "should" be.
Edit/Clarification:...
6
votes
1answer
293 views
Correspondence between persistence module and graded module over $R[t]$
In the paper "Computing Persistent Homology" by Zomorodian and Carlsson, it is stated as Theorem 3.1 that:
The correspondence $\alpha$ defines an equivalence of categories between the category of ...
23
votes
2answers
1k views
Research directions in persistent homology
I am interested in what are the possible directions for new research in persistent homology (more of the mathematical theoretical aspects rather than the computer algorithm aspects).
So far from ...
7
votes
1answer
201 views
Computing homology of subvarieties of Euclidean spaces by persistent homology
Let $M$ be a submanifold of the Euclidean space $\mathbb{R}^n$. Let $G$ be a finite group acting on $M$ freely. I want to compute the homology (or even the cohomology ring) of $M/G$.
Suppose the ...
22
votes
3answers
2k views
Persistent homology of Gaussian fields in Euclidean space
If you generate points in $\mathbb R^n$ via a process that respects a Gaussian normal distribution, then compute the persistent homology / barcodes, to my eye something fairly regular seems to be ...
7
votes
1answer
720 views
Persistent homology of Markovian dynamical systems
Consider a dynamical system $(T,X)$ that admits a Markov partition $\mathcal{M}$ (e.g., an Anosov map), and consider the corresponding 0-1 transition matrix $A$. It is commonplace to study information ...
