Questions tagged [topological-data-analysis]

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

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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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12 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 ...
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79 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 ...
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18 views

Characterisation of the persistence landscape of the sum of two persistence module.

Let us consider two persistence module $M,N$ tame enough. Let's define $D(M),D(N)$ their persistence diagrams and $\lambda_M,\lambda_N$ their persistence landscape. We know that $$D(M\bigoplus N)=D(M)\...
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Feasibility of inventing a new RSA inspired cryptography system using dimensionality reduction techniques inspired by PCA/T-SNE?

In RSA, we take advantage of the fact that even if someone knows the public key, finding the private key is hard. Encryption is fast. Decryption is slow without the private key. Is it possible to ...
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28 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 ...
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1answer
89 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 ...
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1answer
32 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 $1$-...
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45 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 ...
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38 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 ...
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78 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 ...
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56 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/...
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54 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/...
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67 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 ...
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39 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 ...
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40 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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38 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 ...
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105 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 ...
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183 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 ...
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27 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 ...
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43 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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37 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 (...
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199 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 ...
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1answer
90 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? ...
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66 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. ...
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1answer
169 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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81 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 \...
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43 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 ...
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1answer
113 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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114 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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198 views

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
43 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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1answer
613 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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81 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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153 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. Please ...
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51 views

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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26 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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35 views

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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31 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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135 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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111 views

Topological Data Analysis

Beyond the basic math symbols you can type holding down shift on your keyboard, I don't know math symbols, or even how to meaningfully search for them. I came across Topological Data Analysis https://...
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300 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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73 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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2answers
81 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).
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130 views

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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256 views

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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1answer
276 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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58 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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83 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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75 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 ...