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Questions tagged [topological-data-analysis]

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

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Critical Simplices of a Discrete Gradient Vector Field

I just started learning about discrete Morse Theory and I got stuck on a corollary that in the book I'm reading is described as simply following from a lemma. Denote by $P$ an almost linear metric ...
moschops's user avatar
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Understanding when laplacian eigen-coordinates on a 2-torus to $\mathbb{R}^3$ is an immersion

So I am trying to understand when the laplacian eigenfunctions from a torus–when graphed against each other–form a smooth immersion. We take the laplacian operator $L=-\partial_{x_1}-\partial_{x_2}$ ...
4u9ust's user avatar
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Using persistent homology to measure the "isotropy" of a graph

Suppose we have a random graph where the positions of the vertices are significant. How can I measure the isotropy of this graph? I may not be using the correct terminology, but essentially, I'm ...
sam wolfe's user avatar
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Application of stability theorem in persistent homology

I'm currently studying some basic concepts of persistent homology used in topological data analysis (TDA). The stability theorem (in the form presented in the book i'm reading) roughly states that ...
Michele Busti's user avatar
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1 answer
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Vietoris-Rips complex with repeated vertices

I am trying to study Vietoris-Rips complexes that arise from a point data sample, in the context of topological data analysis. Each data point maps to a point in a metric space by some measurement ...
Student005's user avatar
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Smith normal form of morphisms between non-free $R$-modules

If $R$ is a ring and, further, a PID, a morphism of $f : M \to N$ of finitely generated, free $R$-modules has a Smith normal form. Does this also hold when $M$ and $N$ are finitely generated but not ...
richokicked800goals's user avatar
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Efficient algorithms to detect connected components

I am reading the book "Computational Homology" by Tomasz Kaczynski, Konstantin Mischaikow and Marian Mrozek and in several places it says something to the effect of "of course, from the ...
12345's user avatar
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Persistence Vector Spaces

I am currently reading Gunnar Carlsson's "Topological Pattern Recognition for Point Cloud Data", you can find it here: http://math.stanford.edu/~gunnar/actanumericathree.pdf I have a ...
Red Phoenix's user avatar
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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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Isoparametric function on hypersurfaces in Euclidean space

A isoparametric function $f$ on a Riemannian manifold is a function that satisfies the followings identities: $|\nabla f|=a(f)$ and $\Delta f=b(f)$. Now, I would like to deal with hypersurfaces in the ...
Santos's user avatar
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Short exact sequence of persistence modules

I am currently trying to work out a elementary proof of the following statement: Let $X$ be a simplicial complex with a filtration $\mathbb{X}: X=\bigcup_{n\in\mathbb{N}} X_n$, let $k\in\mathbb{N}$ ...
littleD's user avatar
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Understanding nearest neighbors in high-dimensional data

Let's have a random sample of points in an euclidean $n$-space: assume a iid sample from a standard normal distribution. To each point $p$, I assign the number $N(p)$ defined as "how many times ...
Peter Franek's user avatar
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Using topological data analysis to analyze a graph

My question may seem CS-related at first, but it's essentially mathematical, so, please, bear with me. I'm doing Neural Architecture Search (NAS) by varying the number of layers and neurons per layer ...
user314159's user avatar
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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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Indecomposable Persistence Module

A persitence module is a functor $F:\mathbb{N} \rightarrow \mathbb{A}$ where $\mathbb{N}$ is the category of natural numbers with a partial order and $\mathbb{A}$ is some abelian category. There is a ...
amd1234's user avatar
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Simplicial Complex as a topological space

I am new to topological data analysis. I just do not get the idea that a simplicial complex can be considered as a topological space when in the first place, it does not satisfy the definition of a ...
Ruth's user avatar
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Books on topological data analysis and persistent homology

I am a physicist/data analyst and I am trying to get into topological data analysis. Needless to say, I severely lack background. My math education in this direction terminated at analytical geometry ...
MsTais's user avatar
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Reeb Graph of Topological Space and Lemma about Induced map Between Homologies

I am a physicist following a course in Topological Data Analysis (MasterMath), and I need to prove the following lemma about the graphs. I know I am supposed to write what I have tried, but honestly I ...
George Smyridis's user avatar
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Is there a different barcode for the same filtration of complex?

The example at the beginning of the video https://youtu.be/qGkIuJmXhts, (Filtration of the example), I have a question about the barcode of a 1-dim persistent barcode. The video's author gave two ...
Billal's user avatar
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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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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-...
big-o's user avatar
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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 $...
Phil's user avatar
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1 answer
130 views

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 ...
Ilgaz's user avatar
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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 ...
Jhon Doe's user avatar
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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 $\alpha\leq\beta$...
Jhon Doe's user avatar
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2 votes
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140 views

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^...
Jhon Doe's user avatar
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205 views

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= (...
Daron's user avatar
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1 vote
1 answer
257 views

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 ...
Saud Molaib's user avatar
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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 ...
Matthew Niemiro's user avatar
2 votes
1 answer
122 views

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 ...
MathMole's user avatar
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1 answer
135 views

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 ...
Ashley Chraya's user avatar
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0 answers
129 views

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 ...
Moritz Groß's user avatar
2 votes
1 answer
476 views

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^...
Phil's user avatar
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0 votes
1 answer
51 views

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 ...
krishnab's user avatar
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1 vote
1 answer
126 views

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 ...
Barbamento's user avatar
1 vote
0 answers
327 views

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 ...
Anika's user avatar
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2 votes
1 answer
70 views

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 ...
James Arten's user avatar
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1 vote
1 answer
124 views

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) \...
James Arten's user avatar
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4 votes
1 answer
584 views

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 ...
Pavel Snopov's user avatar
1 vote
1 answer
114 views

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 ...
James Arten's user avatar
  • 1,953
0 votes
0 answers
94 views

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....
Ayodan's user avatar
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2 votes
0 answers
38 views

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 ...
James Arten's user avatar
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2 votes
1 answer
170 views

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: \...
Nick Bishop's user avatar
2 votes
1 answer
287 views

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 ...
D. Jude's user avatar
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4 votes
1 answer
657 views

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 ...
Nick Bishop's user avatar
0 votes
2 answers
48 views

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 ...
Chris's user avatar
  • 85
1 vote
1 answer
96 views

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,...
sougonde's user avatar
  • 113
0 votes
1 answer
159 views

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?
alyssaeliyah's user avatar
1 vote
1 answer
108 views

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, ...
ozera ozera's user avatar
1 vote
1 answer
102 views

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 ...
cbyh's user avatar
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