Questions tagged [topological-data-analysis]
Questions about persistent homology, computational topology, discrete morse theory, and applied algebraic topology in general.
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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 ...
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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, ...
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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 ...
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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 $k\in\mathbb{N}$ ...
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Counter-example involving 2-out-of-3 property for simplicial collapses
Let $K$ be a simplicial complex. An elementary simplicial collapse is a formal operation on $K$ involving removing a free face (i.e. a simplex with a unique cofacet) and its unique cofacet. If $L \...
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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 ...
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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 ...
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Machine learning with algebraic information
My inquiry pertains to the exploration and understanding of academic literature, a meta-question by nature. Specifically, I have come across a multitude of machine learning research papers that ...
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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:
$$
\...
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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 ...
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How to calculate number of working in Sales
For the first sample, choosing random 1000 people in labor market, getting 50 people who work in Sales.
For the second sample, choosing random 1000 people in labor market, getting X total Sales people ...
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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 ...
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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 ...
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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 ...
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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 ...
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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-...
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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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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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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
$$
Suppose $\sigma_j$ is a $d$-dimensional simplex that first appears ...
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Extending simplicial maps between filtrations to homology groups.
Suppose I have 2 filtrations of simplicial complexes $K,G$: $\{K_{\alpha}\}_{\alpha\in\mathbb{R}},\{G_{\beta}\}_{\beta\in\mathbb{R}}$. Here, $K_{\alpha}$ is a subcomplex of $K$ and if $\alpha\leq\beta$...
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Computing Persistent Barcodes.
I'm currently reading the following paper on persistent homology: https://geometry.stanford.edu/papers/zc-cph-05/zc-cph-05.pdf.
Given a filtration of a simplicial complex, $K$,
$$\{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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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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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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