Questions tagged [topological-data-analysis]

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

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27 views

n-graded vector space

I am reading Gunnar Carlsson and Afra Zomorodian's "The theory of multidimensional persistence" and I'm stuck on what an n-graded vector space V is. Their definition seems to be that V can be written ...
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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 ...
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2answers
5k views

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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1answer
70 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?
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46 views

the structure of persistence modules

I am reading the book The Structure and Stability of Persistence Modules. In chapter 2 it's told that under some conditions, a persistence module can be expressed as a direct sum of interval modules, ...
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TDA and knot theory

I'm new to topological data analysis, and I learned some basics of it including persistent homology and mapper. In this paper, authors suggest a method to detect circle $S^{1}$, which is 1-dimensional ...
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1answer
28 views

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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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 <...
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1answer
225 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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1answer
66 views

absolute persistent cohomology bar codes

Can anyone explain the persistent absolute cohomology bar codes? how are the indices defined in absolute persistent cohomology? For example, corresponds to the filtration $X_1 \subset ... \subset ...
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1answer
77 views

relative persistent homology of a filtration of a 2-sphere cell complex

I am reading this paper and am trying to understand the relative persistent homology of the 2-sphere cell complex filtration shown above. I am not familar with how to compute relative homology. ...
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1answer
22 views

Find an example of K, A, B, and I to show $\beta_i\neq \beta_i(A)+\beta_i(B)-\beta_i(A\cap B)$

I’ve been trying to find a simplicial complex $K$, where $A$ and $B$ are subcomplexes, and $K=A\cup B$ to show that the Betti number $\beta_i(K)\neq \beta_i(A)+\beta_i(B)-\beta_i(A\cap B)$ but ...
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Acyclic Chain Complex

I’m a little confused, for a chain to be acyclic, all Betti numbers must be zero. For a Betti number $\beta$ $\beta_i=\dim(Z_i)-\dim(B_i)$ where $Z_i=\ker(\partial_i)$ and $B_i$=im$(\partial_i{+}_1))$...
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0-chain Boundary

Can anyone explain how adding two vertices in a connected graph to create a $0$-Chain is the boundary of some 1-dimensional chain? I know that the definition of boundary is the collection of $n+1$ ...
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26 views

Computing Homology

I need A little bit more clarification when computing the homology of a chain complex. So the problem is: Compute the simplicial homology of the graph with vertices $$V=\left\{ 1, 2, 3, 4 \right\}$$ ...
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7answers
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Why does topology rarely come up outside of topology?

I am currently taking topology and it seems like a completely different branch of math than anything else I have encountered previously. I find it a little strange that things are not defined more ...
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52 views

distance between two points at infinity in persistence diagrams

Say we have an essential class (barcode $[a,\infty)$, meaning its represents a feature that never gets killed) represented in a persistence diagram on $\bar{\mathbb{R}}^2$ as x= $(a,\infty)$. Where $\...
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120 views

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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1answer
52 views

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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71 views

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 + \...
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217 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 ...
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187 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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70 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 ...
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1answer
61 views

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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44 views

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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1answer
124 views

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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137 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 $k$ ...
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1answer
88 views

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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123 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 ...
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131 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 ...
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2answers
868 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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1answer
156 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&...
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282 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-...
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51 views

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 ...
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2answers
477 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 ...
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137 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 ...
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1answer
273 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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2answers
968 views

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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2answers
1k views

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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71 views

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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116 views

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 ...
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1answer
165 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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1answer
125 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?...
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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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55 views

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 ...
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1answer
704 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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1answer
1k views

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 ...
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1answer
926 views

Self study Persistent Homology

I am a graduate student in mathematics interested in Persistent Homology. Can anyone recommend any good books / resources to self study Persistent Homology? I am taking a course in Algebraic ...
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1answer
66 views

Original source of the persistence algorithm?

Who was the first person/what was the first paper to invent/mention the persistence algorithm? (see page 6 of these lecture notes).