# Questions tagged [topological-data-analysis]

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

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61 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 ...
29 views

### Gluing axiom for a sheaf of sets over closed sets.

I've trying to get to grips with a sheaf not over the opens but rather the closed sets of some topological space $X$. I wasn't able to find any good information or even terminology on it even though ...
32 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 ...
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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 ...
33 views

### Definition of the betti curve in persistent homology

I'm a little confused about the definition of the betti curve in persistent homology. If $t$ is the filtration parameter is $\beta(t)$ just the betti number of the associated complex we're taking ...
141 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 ...
21 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 ...
32 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 ...
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### 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 (...
168 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 ...
71 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? ...
58 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. ...
91 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 ...
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### 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 ...
348 views

### Prerequisites for Gunnar Carlsson's Topology and Data

I am planning on doing a project on topological data analysis in the near future and intend to use Gunnar Carlsson's paper "Topology and Data" as my introduction to the field. I am familiar with point-...
91 views

### Extra barcodes in persistent cohomology

In the following filtration, I get 6 persistent cohomology intervals ("barcodes"): $[0,1)$ $[0,2)$ $[0,\infty)$ $[1,\infty)$ $[2,\infty)$ $[3,\infty)$ However, according to "duality" with ...
223 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 ...
156 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 ...
148 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 ...
76 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 ...
140 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 $k$ ...
144 views

### Persistent Homology with Integer Coefficients

Does anyone know if persistent homology with integer coefficients are being used anywhere? From what I understand, Carlsson's persistent module theory (http://citeseerx.ist.psu.edu/viewdoc/download?...
169 views

### Intuitive meaning of Persistent Group / Persistent Module

This question relates to the topic of Persistent Homology, a branch of (applied) algebraic topology. In the paper (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.116.2471&rep=rep1&...
109 views

### Is there a reference for an explicit computation of persistent homology?

Does anyone know any source containing an example of persistent homology being explicitly, manually calculated? That is, as opposed to being calculated on by computer. I am just looking for a simple ...
In the paper by Zomorodian, page 6, it is stated without proof that $\mathrm{im}\ \eta_k^{i,p}\cong H_k^{i,p}$. May I know how do we prove it? Also, it states that $\eta_k^{i,p}: H_k^i\to H_k^{i+p}$ ...