Questions tagged [topological-data-analysis]

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

Filter by
Sorted by
Tagged with
0
votes
0answers
7 views

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 ...
2
votes
1answer
40 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 ...
0
votes
1answer
25 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 ...
0
votes
0answers
21 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 ...
2
votes
1answer
49 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^...
0
votes
1answer
29 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 ...
0
votes
0answers
14 views

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 ...
0
votes
0answers
22 views

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 ...
1
vote
1answer
50 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 ...
1
vote
0answers
57 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 ...
2
votes
1answer
44 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 ...
1
vote
1answer
53 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) \...
4
votes
1answer
146 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 ...
1
vote
1answer
29 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 ...
0
votes
0answers
26 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....
2
votes
0answers
32 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 ...
2
votes
1answer
87 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: \...
2
votes
1answer
85 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 ...
3
votes
1answer
87 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 ...
0
votes
0answers
42 views

Incremental Algorithm - How to compute betti numbers

Currently, I'm trying to learn a bit more about persistence homology and computational topology. While doing so, I came across a paper entitled "An incremental algorithm for Betti numbers of ...
0
votes
2answers
46 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 ...
1
vote
1answer
49 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,...
0
votes
0answers
52 views

Undergraduate books that you need to study before entering TDA?

I am a first year master's student of computer science and I wanna enter to the field of topological data analysis. I had elementary courses in analysis, linear algebra, abstract algebra, graph theory,...
0
votes
0answers
47 views

Can Takens' Embedding Theorem be used on noisy particle dynamics?

I am trying to use topological data analysis (TDA) to analyze the dynamics of some multi-particle simulations (position-time series) of swimming particles with memory. Since TDA was developed for high-...
0
votes
1answer
30 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?
1
vote
1answer
49 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, ...
0
votes
0answers
23 views

Mapping low-dimensional data onto low-dimensional manifold in high-dimensional space

I have 2D data that I am using for testing my clustering algorithm. Now, I would like to show that it also works well for high-dimensional data, which has the same properties as the 2D data, if the ...
1
vote
1answer
85 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 ...
1
vote
1answer
65 views

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 ...
2
votes
1answer
127 views

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 ...
1
vote
1answer
39 views

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 $1$-...
1
vote
1answer
70 views

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 ...
2
votes
0answers
54 views

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 ...
1
vote
1answer
198 views

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 ...
1
vote
1answer
81 views

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/...
1
vote
0answers
70 views

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/...
2
votes
0answers
73 views

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 ...
0
votes
1answer
49 views

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 ...
0
votes
1answer
48 views

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 ...
5
votes
1answer
253 views

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 ...
1
vote
0answers
34 views

What does `sparse' mean in the context of topological data analysis?

I am trying to read this article: https://arxiv.org/abs/1506.03797 but have not been able to find a definition of `sparse' except in the context of sparse matrices. If it is used heuristically, I am ...
1
vote
1answer
57 views

Intuitive notion of functoriality in topological data analysis

For school, we have to give a presentation about topological data analysis and I am in charge of motivating why topological data analysis is cool and useful. Most of what I say is based on "Topology ...
2
votes
0answers
42 views

Are there examples persistent homology being used to study non-linear data?

You can compute the persistent homology of any point cloud embedded in a metric space. In the real-world applications of persistent homology I've come across so far, the data points all have (...
10
votes
1answer
231 views

Good Stopping Criteria for Persistent Homology

I've recently coded up a suite of algorithms for computing the persistent homology for various data sets (small data sets roughly around 30 data points). A question has come to my mind about how to ...
0
votes
1answer
133 views

Persistent homology has to be free, right?

I've been convinced that the homology groups you get when computing the persistent homology of a data cloud have to be free. But now I'm second guessing myself. Can we quickly say why this has to be? ...
0
votes
1answer
77 views

Topological Invariance in Data Structures

I need to do a PhD in Pure Mathematics and I am thinking of Topological Data Analysis. I want to use persistent homology and quiver representation to obtain topological features in data structures. ...
0
votes
1answer
322 views

How is the persistent homology of sublevel (or superlevel) sets are calculated on the computer

With Rips complexes, the calculation of persistent homology is simply linear algebra. However, it takes a lot of computational time since even a few hundred data points yield lots of simplexes in the ...
4
votes
1answer
103 views

Decomposition of quiver representation with Jordan cell map

I am reading Persistence Theory: From Quiver Representations to Data Analysis by Steve Y. Oudot and have the following question on Gabriel's theorem. Consider the quiver $$\bullet \longrightarrow \...
1
vote
0answers
46 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 ...
3
votes
1answer
137 views

Prerequisites for discrete Morse theory

Does one need to know Morse theory to learn Discrete Morse theory? How much of Milnor's Morse theory is essential? Does one also need a background in differential topology for discrete Morse theory?