Questions tagged [topological-data-analysis]

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

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Barcodes Decomposition of Persistent Homology

Does anyone know if the barcode decomposition of a simplex-wise filtration a multiset? More specifically, can we have multiple barcodes with the same birth time? When I read the paper by Gunnar ...
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Copy of $\mathbb{Z}$ in a homology group [closed]

What does it mean to have a copy of $\mathbb{Z}$ (or in general, a vector space/module) in a homology group?
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2D Ternary plot equivalent for >3 dimensions

Ternary plots can be good for visualising systems where there are three different values that always add up to a constant value (i.e. there are only two degrees of freedom). A classic example is three-...
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Persistent betti numbers and birth and death of classes

I'll copy and paste the background information in my other question: Given a filtration of complexes $\emptyset = K_0 \subseteq K_1 \subseteq \ldots \subseteq K_n = K$, we have induced homomorphisms $...
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General Applications of Persistent Homology [closed]

Are there any general applications of persistent homology (or topological data analysis in general) ? . All the applications I seem to find relate to very specific applications so I was wondering if ...
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How does persistent homology detects curvature?

I am currently studying the paper Persistent homology detects curvature by Peter Bubenic (https://doi.org/10.1088/1361-6420/ab4ac0). I am stuck at a very fundamental idea of this paper. It claims that ...
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What skills are necessary for a career in applied topology? [closed]

What skills are necessary for a career in applied topology? I'm currently finishing my Bachelors in pure Mathematics and I love math, but have always wanted to work on anti-aging research in some ...
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When can 2 different simplicial chains share the same boundary?

Suppose I have a filtration of a simplicial complex $K$: $$ \emptyset=K_0\subseteq K_1\subseteq K_2\subseteq,...,\subseteq K_n=K $$ Suppose $\sigma_j$ is a $d$-dimensional simplex that first appears ...
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Counting birth and death classes of persistent homology.

Suppose we have a filtration of simplicial complexes $\emptyset = K_0 \subseteq K_1 \subseteq \ldots \subseteq K_n = K$. Then, for $i \leq j$, we have induced homomorphisms $f^{i,j}_p \colon H_p(K_i) \...
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Extending simplicial maps between filtrations to homology groups.

Suppose I have 2 filtrations of simplicial complexes $K,G$: $\{K_{\alpha}\}_{\alpha\in\mathbb{R}},\{G_{\beta}\}_{\beta\in\mathbb{R}}$. Here, $K_{\alpha}$ is a subcomplex of $K$ and if $\alpha\leq\beta$...
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Computing Persistent Barcodes.

I'm currently reading the following paper on persistent homology: https://geometry.stanford.edu/papers/zc-cph-05/zc-cph-05.pdf. Given a filtration of a simplicial complex, $K$, $$\{0\}=K^0\subseteq K^...
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What is a clever or efficient way to compute this variant of the Wasserstein distance between persistence diagrams?

A two-dimensional persistence diagram in $[0,1]$ say is just a multiset of points of $\mathbb R^2$. Given two diagrams $P=\{p_1=(a_1,b_1),\ldots, p_n= (a_n,b_n)\}$ and $Q=\{q_1=(c_1,d_1),\ldots, q_n= (...
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Equivalent definitions of alpha-complex? (topology)

The $\alpha$-complex is a widely used combinatorial structure in topological data analysis. However, I consistently find two different definitions of the $\alpha$-complex of a point cloud: Through ...
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Why are giotto-tda and cripser giving different persistent diagrams?

When I find the persistence diagrams using cubical homology and using the natural grayscale filtration of the image, I get two different answers depending on the package I use. By inspection, it seems ...
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How much more efficiently can one compute persistent homology after restricting the point cloud's shape?

Topological data analysis employs topology to study discrete multidimensional data sets. One often treats these data as point clouds embedded in $\mathbb{R}^n$. And in practice, it may be hard to ...
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Introducing myself to discrete Morse theory

I plan to write my maths masters' dissertation on discrete Morse theory. I intend to write it from a theoretical point of view, relating it to classical Morse theory. I still have to decide exactly ...
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Persistent Homology of circular point data set

I was experimenting with simple data points like squares, rectangles, and polygons to forecast my 0D and 1D persistent homology. I'm having trouble predicting persistent homology in the case of a ...
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Why is the following a cubical complex

The german Wikipedia-page on cubical complexes has the following example for a cubical complex. I don't understand how the 45° rotated square on the right is the product of elementary intervals. As I ...
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Persistent Homology: Birth and death of cycles

So I'm trying to understand death and birth in persistent homology. Given a filtration of complexes $\emptyset = K_0 \subseteq K_1 \subseteq \ldots \subseteq K_n = K$, we have induced homomorphisms $f^...
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persistent homology: how sensitive is the persistent homology of a dataset to reorderings of the elements in vector.

I have been looking at some of the applications of topological data analysis and persistent homology lately. I had a question about how sensitive persistent homology was to reordering of the data or ...
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Difference between bottleneck and matching distances

Can anyone explain me what's the difference between matching and bottleneck distances? I found a definition here where one is used for diagrams and one for the betti number function induced by the ...
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Persistent homology base definition

I'm currently dealing with some problems with the definitions of persistence homology. I have two different definition right now, the first one comes from here, while the second from here. The first ...
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persistent homology group as vector subspace

A persistent homology group is defined as $i^\ast (H_k(X^i))$ where $i$ is the function $i^\ast:H_k(X^i)\to H_k(X^j)$ for any $i<j$. All my homology groups have coefficient in a field $K$ so they ...
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Calculating Betti numbers in GUDHI

I am currently trying to write a program, which creates a simplicial complex, plots the persistence diagram and outputs the Betti numbers. I completed the first two steps using GUDHI, but I am not ...
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How topological fingerprints are effectively used in a Machine Learning model

I was just perplexed about the practical usage of topological fingerprints coming out from persistence homology approaches. Once I've obtained persistence diagrams, how do I effectively use them to ...
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Persistent Homology Betti Numbers definition

shifting from standard simplicial homology to persistent homology, there is something that I don't understand. In simplicial homology one builds a chain complex of the form $$\dots \rightarrow C_n(K) \...
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Why do we "vectorize" persistence diagrams?

Recently I've been going through some papers and tutorials on using persistent homology in machine learning and pretty quickly, when all algebraic topology stuff ended, I've found that, in order to ...
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Computing Persistence Diagram in a Persistence Homology Framework

I was recently reading with interest the following paper:https://arxiv.org/pdf/2102.07835.pdf and, going to appendix to retrieve some general notions of TDA, I've been stuck for a while trying to ...
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Periodic Boundary Conditions for Persistent Homology

Is there a standard method/library for implementing persistent homology on points with periodic boundary conditions? For example, see here, where red lines indicate the desired location of periodicity....
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Geometrical considerations behind simplicial homology construction

recently I've jumped into Topological Data Analysis (TDA) and I'm trying to get some insights about what's behind it in terms of math, in particular regarding simplicial Homology. I'll briefly recap ...
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Is there a natural topology for the set of measurable sets (i.e. a given sigma algebra)?

Given a sigma algebra $\mathcal{F}$, is there a natural topology worth defining on it? More specifically, is there a topology you can put on $\mathcal{F}$ which ensures a measure of interest $\mu: \...
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Where does the elder rule appear in the structure theorem for persistent homology?

I'm reading Computational Topology (by Edelsbrunner & Harer). The authors describe (pg 180) generating the persistence diagram from a filtration of simplicial complexes. The approach is to define ...
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Computing the persistence homology of the sublevel sets of a function

I have a question somewhat in line with the one asked here. That is, I am interested in how the persistent homology for a sublevel set of a function ($\{x \: : \: f(x) \leq c\}$) is computed. For ...
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Question about use of set of integers in persistent homology paper

I am unclear what is meant here with the notation $\mathbb{Z}(...)$ and 'extends linearly over $\mathbb{Z}$'. I'm reading this paper 'embedded homology of hypergraphs and applications', and ...
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Persistence barcodes given a sequence of abstract homology groups

For a sequence of nested complexes $K_1\subset K_2 \subset K_3$, I have calculated the first homology groups at each level, $$ H_1(K_1)=\left<a,b\right>\cong \mathbb{Z}^2\\ H_1(K_2)=\left<a,b,...
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Equally Distributed Data Set Measurement

I will be creating my own dataset with scores ranging from 50.00 to 100.00. How will I say that the dataset I chose is equally distributed and unbiased ? Is there a formula to know this?
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Persistant homology - Point data sets from images

I have been reading about topological data analysis techniques and specifically about Persistent Homology. The examples I have seen so far use point clouds as the data sets. But what if we have, say, ...
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visualizing 1-parameter family of persistence modules by vineyard technique

In the paper "The structure and stability of persistence modules", Page 49, they use vineyard technique to visualize the 1-parameter family of persistence modules produced by three ...
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Distances between two complexes when using Persistence Homology

I am using Persistence Homology to look at two different facebook networks. I can generate a distance matrix between individuals and then create the usual barcodes and persistence diagrams according ...
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A question about death (persistent homology)

I've been referring to this set of notes on persistent homology, and am confused with the definition and intuition for the death of a homology class for the persistent homology of a filtration. Given ...
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Maximal number of generators of first homology in Vietoris-Rips complex

For a point cloud $P$ with $n$ vertices is there a nice formula for the maximum number of points in a persistence diagram of the Vietoris-Rips complex on this point cloud? Since in a VR complex a $1$-...
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Equivalence of the persistence landscape diagram and the barcode?

I am studying persistent homology for the first time. I was reading Peter Bubenik's paper "Statistical Topological Data Analysis using Persistence Landscapes" from 2015 introducing ...
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recent research work on computational homotopy/ persistent homotopy

Persistent homology has been broadly used in topological data analysis since we have some ways to calculate them efficiently. However, homotopy is very different to compute so it is hard to use for ...
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What is the difference between a clique and a simplex?

I have seen several descriptions of simplicial complexes and clique complexes as being a combination of simplexes and cliques. I have heard descriptions of cliques being a subnetwork or subgraph ...
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Problem understanding barcodes in persistent homology.

I am currently reading the following paper by Gunnar Carlsson: https://www.cambridge.org/core/services/aop-cambridge-core/content/view/BB0DA0F0EBD79809C563AF80B555A23C/S0962492914000051a.pdf/...
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Question regarding the Vietoris-Rips construction as persistent vector space.

I am currently teaching myself the basics of persistent homology by reading the following set of notes by Gunnar Carlsson. https://www.cambridge.org/core/services/aop-cambridge-core/content/view/...
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Theoretical epidemiology and general mathematical investigations

First of all, let me say that I'am a mathematician working on mathematical physics. My wife was working on epidemiology on her master's and discussing with her I found the theme very interesting. When ...
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In persistent homology, when two homological features merge, how to determine which one dies?

Consider a W shaped function with local minimums at $y=1$ and $y=2$ and local maximum at $y=3$. When we look at the persistence diagram induced by the lower level sets of this function, Two ...
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How to generate a distance matrix from the height function applied on the point cloud?

I am new to the idea of topology data analysis, this is a figure in the paper: Persistent Homology Transform for Modeling Shapes and Surfaces, and I am wondering about how the distance matrix is ...
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What's the connection between persistent homology and tensor networks?

Tensor networks are mathematical representations of quantum many-body systems. Persistent homology is a method for computing topological features. Are these two related? It has at least two ...
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