# Questions tagged [topological-data-analysis]

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

46 questions
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### detecting flares with persistent homology

Can persistent homology detect "flares" how does it do so, if it can. I know persistent homology can certainly find "loopy" structure, like noisy circles, but I'm not sure about "flares".
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### How is a simplicial map a cellular map?

I’m fairly new to studying topology and I’m currently reviewing cellular maps, can someone explain how a simplicial map between geometric complexes is also a cellular map?
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### morse reduction

I was reading the paper by Mischaikow and was confused by how the MorseReduce function would run, particularly what the output would be. Say I had a triangle with a simplex-wise filtration as given <...
186 views

### Probability of two random points being orthogonal in higher-dimensional unit sphere

I understand that most points will be close to surface due to volume concentration. Also I also understand the concentration of volume near the equator, relative to any specific point (North pole). ...
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### Where to start learning about topological data analysis?

I was wondering if anyone could help me out with finding a nice introductory introductory text for topological data analysis (I'm speaking as somebody who has two semesters of experience with topology,...
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### computing wasserstein distance vs. bottleneck distance between persistence diagrams

According to the software Hera, bottleneck distance is computed by finding an perfect matching with minimal cost using the Hopcroft-Karp algorithm. furthermore, the wasserstein distance between ...
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### bottle neck distance: distance to diagonal points

recall the bottleneck distance is defined as the minimum of max_x |x-f(x)| over all bijections f between points in persistence diagram A and persistence diagram B. Recall we include all diagonal ...
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### Computing natural pseudo distance.

Consider the functions $\varphi , \ \psi: \, D^2= \{ (x,y) \in \mathbb R^2 : x^2+y^2 \leq 1 \} \rightarrow \mathbb R$ defined by $$\varphi (x,y):= \frac{5(1-x^2-y^2)}{2}$$ and  \psi (x,y):= 1 + \...
212 views

### Elevator Talk on Topology

I am interested in what others do when trying to give an elevator talk on their research interests, particularly on trying to explain what topology is. I am particularly interested in giving an ...
169 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 ...
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### 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 ...
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### Equivalent definitions of persistent homology

Q1) In the paper ([Zomorodian, page 6 https://pdfs.semanticscholar.org/b7ae/132ef88bde28903ac8b14a29a76130f61ac2.pdf), it is stated without proof that $im\ \eta_k^{i,p}\cong H_k^{i,p}$. May I know how ...
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### Understanding the last step in computing persistent homology

I'm reading Zomorodian and Carlsson's paper on computing persistent homology. The object of their algorithm is, given a filtered complex, to find a representation of the kth persistent homology group ...
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### The definition of an equivalence relation in persistent homology

I am reading a paper by Carlsson on "Topological pattern recognition for point cloud data." I was having a little trouble understanding the formal definition of an ...
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### 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 $k$ ...
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### Explicit computation of Persistent Homology (reference)

Does anyone know any source (article/book/etc) where an example of explicit (and manual) computation of persistent homology is shown? So far most examples are calculated by computer. I am just ...
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### 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 ...
127 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 ...
786 views

### Applications of Persistent Homology

Can anyone describe to me in Layman's terms what kind of use does Topology have through Persistent Homology in Data Analysis...that's can you give me some real life examples in which which this has ...
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### 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&...
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### 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-...
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### Identify the appropriate magnitude of K when building a Vietoris-Rips complex for Topological Data Analysis

I am writing some code to generate a Vietoris-Rips complex given a set of data points. The VR complex is generated from 0-complexes, 1-complexes, 2-complex, etc., where $k+1$ represents the number of ...
438 views

### libraries to plot topological data analysis persistence diagrams, complexes, and barcodes

I am starting to do some Topological Data Analysis and am looking for a good plotting library to produce the diagrams of complexes, barcodes, and persistence diagrams. The most popular TDA libraries ...
132 views

### How is Tensor Decomposition (Factorization) related to Topological Data Analysis?

I have been researching modern exploratory data analysis techniques, and came across two promising approaches: Topological Data Analysis (TDA) and Tensor Decomposition/Factorization (TF). I am ...
265 views

### Is topological data analysis useful for low dimensional data?

I often read how Topological Data Analysis (TDA) is useful especially for highly dimensional data. But, what about (apparently) low dimensional ones? Example: consider measuring the resistance of a ...
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### Difference between Topological Data Analysis and Graph Technology

I'm trying to understand the difference between Oracle's graph technology, which apparently has an inherent understanding of topology, and Ayasdi's Topological Data Analysis technology. Are these two ...
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### Roadmap for learning Topological Data Analysis?

I'm a math major who has recently graduated and I will be starting full time work in 'data analysis'. Having finished with decent marks and still being incredibly interested in mathematics, I was ...
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### Question about the classification theorem for finitely generated graded $F[t]$-modules

I am in the beginnings on learning persistence homology, and as a start I'm studying Gunnar Carlsson's survey "Topology and Data". Theorem 2.10 states the following: "Suppose $M_{\star}$ is a ...
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### Beginner's Guide to Persistent Homology (focusing on Theoretical Mathematical Aspects)

I am interested in seeking out a reference for learning Persistence Homology (more of the theoretical mathematical aspects rather than the applications/ computer science aspects). Currently, the two ...
144 views

### matrix columns represented by binary search tree

I've been reading this paper on the persistence algorithm: https://people.mpi-inf.mpg.de/~mkerber/ck-phcwat-11.pdf Given a matrix M with columns $M_j$. it states on page 3 that we may add two ...
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### Are the results from persistent homology complete?

Do the homology groups for each Betti number as produced by persistent homology completely capture all the topological characteristics of the space? If not, what topological features are not captured? ...
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### 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?...
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### TDA- Persistence Diagram and Barcodes using image data (and TDA R package)

I am very very new to Topological Data Analysis (TDA) and I am trying to apply the method described in this paper. I understood everything but stuck on methodology section (3.2) where he described: ...
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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 ...
624 views

### TDA - Persistence diagrams and Barcodes

I am relatively new in the field of persistent homology and topological data analysis. I would like to use RIPSER, DIPHA or GUDHI to calculate barcodes which will give a persistence diagram. Here are ...
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### How does one construct a persistence diagram in Persistence Homology

I am learning a bit about persistence homology and was hoping that someone could point me to a systematic explanation of how to construct a persistence diagram, such as the one pictured below. I ...