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Questions tagged [persistent-homology]

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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_{...
user1892304's user avatar
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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 ...
pyridoxal_trigeminus's user avatar
4 votes
1 answer
399 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 ...
Vahid Shams's user avatar
2 votes
0 answers
98 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 ...
Jake Lai's user avatar
6 votes
1 answer
666 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, ...
Alex's user avatar
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1 answer
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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 $\...
Stephen McKean's user avatar
3 votes
0 answers
160 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 ...
amd1234's user avatar
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117 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: $$ \...
egrr's user avatar
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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 ...
Taraellum's user avatar
2 votes
1 answer
145 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 ...
pyridoxal_trigeminus's user avatar
4 votes
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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 ...
Andrea Marino's user avatar
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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 ...
Andrea Marino's user avatar
1 vote
0 answers
98 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 ...
NietzscheanAI's user avatar
3 votes
0 answers
125 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 ...
Yellow Pig's user avatar
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5 votes
2 answers
342 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 ...
yoyostein's user avatar
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2 votes
1 answer
130 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 $L_p$-...
yoyostein's user avatar
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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 ...
user148575's user avatar
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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 ...
apg's user avatar
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6 votes
1 answer
334 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 ...
apg's user avatar
  • 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 ...
user51223's user avatar
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2 votes
1 answer
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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 ...
Eben Kadile's user avatar
35 votes
1 answer
4k 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). ...
yoyostein's user avatar
  • 1,219
10 votes
1 answer
679 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{...
yoyostein's user avatar
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0 votes
1 answer
382 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 ...
gf.c's user avatar
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14 votes
4 answers
738 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:...
cheyne's user avatar
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7 votes
1 answer
421 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 ...
yoyostein's user avatar
  • 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
224 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 ...
Shi Q.'s user avatar
  • 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 ...
Ryan Budney's user avatar
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7 votes
1 answer
830 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 ...
Steve Huntsman's user avatar