Questions tagged [persistent-homology]
The persistent-homology tag has no usage guidance.
30
questions
0
votes
0
answers
45
views
Two-parameter “$\varepsilon$-$\delta$ filtration” given a function between metric spaces
Let $X,Y$ be metric space and $f : X \to Y$ a (not necessarily continuous) function. I'm interested in the two-parameter filtration $(X_{\varepsilon, \delta})_{{\varepsilon, \delta} > 0}$ where $X_{...
- 121
2
votes
0
answers
52
views
Computational complexity of exact computation of the doubling dimension
Given a finite metric space $X$, the doubling constant of $X$ is the smallest integer $k$ such that any ball of arbitrary radius $r$ can be covered by at most $k$ balls of radius $r/2$. The doubling ...
4
votes
1
answer
347
views
Upper bounds on the Gromov–Hausdorff distance using persistent homology
In persistent homology theory, stability theorems are important to show that the topological signatures extracted are stable under small changes. A key result is the following bound on the bottleneck ...
- 183
2
votes
0
answers
92
views
Description of a point cloud being "undersampled" wrt persistent homology, confidence level?
I am completely new to topological data analysis, so I apologise if this is a well-known area of persistent homology, as well as for any imprecise language.
Suppose we know completely the topological ...
- 21
6
votes
1
answer
639
views
Can you do geometry with persistent homology?
Setup
In practice, persistent homology of data $X$ is often used to infer the homology of the underlying (Riemannian) manifold $M$ that the data is sampled from.
However most filtrations (Vietoris, ...
- 139
5
votes
1
answer
184
views
Persistent homology of $\mathbb{F}_p$-points of elliptic curves
I'm currently teaching a short summer course on cryptography to high school students. Today, I taught them about elliptic curves. After spending some time playing around with their graphs over $\...
- 573
3
votes
0
answers
153
views
Sheaf theory in TDA
I was wondering wether anyone had any examples as to why it more useful to consider a sheaf theory approach to TDA problems.
I am familiar with some of the benefits of using cellular cosheaves to ...
- 333
0
votes
0
answers
116
views
Can you explain to me how to decompose this persistence module and why?
I am learning topological data analysis on my own. I am currently basically watching This course. But there this thing in the course note that I didn't understand.
So for this persistence module:
$$
\...
- 1
2
votes
0
answers
130
views
Is Morse theory local?
I am currently trying to use Morse theory in my work. Say I have $X$ smooth compact manifold and $f : X \to \mathbb{R}$ a smooth function. The classical result I know is : when $f$ has non-degenerate ...
- 93
2
votes
1
answer
140
views
Do there exist smaller simplicial models of barycentric subdivisions?
Let $S$ be a simplicial complex and let $Bary(S)$ denote its barycentric subdivision.
Of course, the geometric realizations of $S$ and $Bary(S)$ are homeomorphic.
However, one issue that arises in ...
4
votes
0
answers
229
views
Being a product - from homology to topology
The famous Kunneth formula expresses the homology of a product manifold as the tensor product of the two algebras.
Now suppose we know that a manifold $X$ has a decomposition $H_*(X) \simeq A \otimes ...
- 1,944
4
votes
0
answers
358
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 ...
- 1,944
1
vote
0
answers
97
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 ...
- 261
3
votes
0
answers
124
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 ...
- 2,490
5
votes
2
answers
336
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 ...
- 1,219
2
votes
1
answer
128
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$-...
- 1,219
7
votes
0
answers
176
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 ...
- 877
2
votes
0
answers
181
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 ...
- 612
6
votes
1
answer
329
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 ...
- 612
40
votes
5
answers
3k
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 ...
- 3,071
2
votes
1
answer
180
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 ...
- 417
35
votes
1
answer
3k
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).
...
- 1,219
10
votes
1
answer
665
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{...
- 1,219
0
votes
1
answer
378
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 ...
- 35
14
votes
4
answers
737
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:...
- 1,396
7
votes
1
answer
417
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 ...
- 1,219
24
votes
2
answers
2k
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
1
answer
223
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 ...
- 543
24
votes
3
answers
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 ...
- 43.1k
7
votes
1
answer
812
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 ...
- 15.3k