Questions tagged [topological-data-analysis]

Questions about persistent homology, computational topology, discrete morse theory, and applied algebraic topology in general.

23 questions with no upvoted or accepted answers
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9
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0answers
187 views

Topological Features and Graph Spectra

I was just thinking recently about if there are any possible meaningful connections between tools such as persistent homology used for things like topological data analysis and tools used in spectral ...
6
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70 views

TDA and knot theory

I'm new to topological data analysis, and I learned some basics of it including persistent homology and mapper. In this paper, authors suggest a method to detect circle $S^{1}$, which is 1-dimensional ...
4
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0answers
91 views

Are the results from persistent homology complete?

Do the homology groups for each Betti number as produced by persistent homology completely capture all the topological characteristics of the space? If not, what topological features are not captured? ...
4
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0answers
131 views

Reduction algorithm for Persistent Homology

The reduction algorithm (pg. 5 of http://geometry.stanford.edu/papers/zc-cph-05/zc-cph-05.pdf) enables us to compute homology for modules over a PID. I am curious why the reduction algorithm cannot ...
4
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137 views

How is Persistent Homology different from just calculating ordinary homology at each filtration

I apologize if this question is not well-phrased, as I am not very familiar with the subject. Let say we have a filtration $K^0\subseteq K^1\subseteq K^2\subseteq K^3$. The $p$-persistent $k$ ...
4
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0answers
116 views

Beginner's Guide to Persistent Homology (focusing on Theoretical Mathematical Aspects)

I am interested in seeking out a reference for learning Persistence Homology (more of the theoretical mathematical aspects rather than the applications/ computer science aspects). Currently, the two ...
3
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0answers
123 views

Complexity of computing Persistent Homology vs Homology

I am curious whether it is significantly harder (computational-wise) to compute persistent homology as compared to computing homology. Or is it the same time complexity. I am aware that there is a ...
3
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0answers
137 views

How is Tensor Decomposition (Factorization) related to Topological Data Analysis?

I have been researching modern exploratory data analysis techniques, and came across two promising approaches: Topological Data Analysis (TDA) and Tensor Decomposition/Factorization (TF). I am ...
2
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0answers
121 views

computing wasserstein distance vs. bottleneck distance between persistence diagrams

According to the software Hera, bottleneck distance is computed by finding an perfect matching with minimal cost using the Hopcroft-Karp algorithm. furthermore, the wasserstein distance between ...
1
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0answers
40 views

What are the 3-dimensional subspaces (or quotient spaces) to which the projections are made in the given figures? (Topological Data Analysis)

EDIT: I was told by my supervisor to implement the algorithm first and then look back over the question because "biologists' papers do not always contain the information that is necessary to reproduce ...
1
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0answers
26 views

Computing Homology

I need A little bit more clarification when computing the homology of a chain complex. So the problem is: Compute the simplicial homology of the graph with vertices $$V=\left\{ 1, 2, 3, 4 \right\}$$ ...
1
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0answers
150 views

TDA- Persistence Diagram and Barcodes using image data (and TDA R package)

I am very very new to Topological Data Analysis (TDA) and I am trying to apply the method described in this paper. I understood everything but stuck on methodology section (3.2) where he described: ...
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44 views

Understanding the last step in computing persistent homology

I'm reading Zomorodian and Carlsson's paper on computing persistent homology. The object of their algorithm is, given a filtered complex, to find a representation of the kth persistent homology group ...
1
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71 views

Computing natural pseudo distance.

Consider the functions $\varphi , \ \psi: \, D^2= \{ (x,y) \in \mathbb R^2 : x^2+y^2 \leq 1 \} \rightarrow \mathbb R $ defined by $$ \varphi (x,y):= \frac{5(1-x^2-y^2)}{2} $$ and $$ \psi (x,y):= 1 + \...
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55 views

Persistence Homology on a grid Distance measure

I am just beginning to learn about topological data analysis and understand the basics. With respect to constructing a persistence diagram, I understand level sets etc. My question is regarding how ...
1
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0answers
70 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 ...
0
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27 views

n-graded vector space

I am reading Gunnar Carlsson and Afra Zomorodian's "The theory of multidimensional persistence" and I'm stuck on what an n-graded vector space V is. Their definition seems to be that V can be written ...
0
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1answer
46 views

the structure of persistence modules

I am reading the book The Structure and Stability of Persistence Modules. In chapter 2 it's told that under some conditions, a persistence module can be expressed as a direct sum of interval modules, ...
0
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24 views

morse reduction

I was reading the paper by Mischaikow and was confused by how the MorseReduce function would run, particularly what the output would be. Say I had a triangle with a simplex-wise filtration as given <...
0
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0answers
30 views

Acyclic Chain Complex

I’m a little confused, for a chain to be acyclic, all Betti numbers must be zero. For a Betti number $\beta$ $\beta_i=\dim(Z_i)-\dim(B_i)$ where $Z_i=\ker(\partial_i)$ and $B_i$=im$(\partial_i{+}_1))$...
0
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1answer
52 views

distance between two points at infinity in persistence diagrams

Say we have an essential class (barcode $[a,\infty)$, meaning its represents a feature that never gets killed) represented in a persistence diagram on $\bar{\mathbb{R}}^2$ as x= $(a,\infty)$. Where $\...
0
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1answer
165 views

matrix columns represented by binary search tree

I've been reading this paper on the persistence algorithm: https://people.mpi-inf.mpg.de/~mkerber/ck-phcwat-11.pdf Given a matrix M with columns $M_j$. it states on page 3 that we may add two ...
0
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71 views

Question about the classification theorem for finitely generated graded $F[t]$-modules

I am in the beginnings on learning persistence homology, and as a start I'm studying Gunnar Carlsson's survey "Topology and Data". Theorem 2.10 states the following: "Suppose $M_{\star}$ is a ...