# Questions tagged [topological-data-analysis]

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

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### Critical Simplices of a Discrete Gradient Vector Field

I just started learning about discrete Morse Theory and I got stuck on a corollary that in the book I'm reading is described as simply following from a lemma. Denote by $P$ an almost linear metric ...
1 vote
52 views

### Understanding when laplacian eigen-coordinates on a 2-torus to $\mathbb{R}^3$ is an immersion

So I am trying to understand when the laplacian eigenfunctions from a torus–when graphed against each other–form a smooth immersion. We take the laplacian operator $L=-\partial_{x_1}-\partial_{x_2}$ ...
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### Using persistent homology to measure the "isotropy" of a graph

Suppose we have a random graph where the positions of the vertices are significant. How can I measure the isotropy of this graph? I may not be using the correct terminology, but essentially, I'm ...
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### Application of stability theorem in persistent homology

I'm currently studying some basic concepts of persistent homology used in topological data analysis (TDA). The stability theorem (in the form presented in the book i'm reading) roughly states that ...
1 vote
43 views

### Vietoris-Rips complex with repeated vertices

I am trying to study Vietoris-Rips complexes that arise from a point data sample, in the context of topological data analysis. Each data point maps to a point in a metric space by some measurement ...
1 vote
18 views

### Smith normal form of morphisms between non-free $R$-modules

If $R$ is a ring and, further, a PID, a morphism of $f : M \to N$ of finitely generated, free $R$-modules has a Smith normal form. Does this also hold when $M$ and $N$ are finitely generated but not ...
1 vote
34 views

### Efficient algorithms to detect connected components

I am reading the book "Computational Homology" by Tomasz Kaczynski, Konstantin Mischaikow and Marian Mrozek and in several places it says something to the effect of "of course, from the ...
• 187
1 vote
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### Persistence Vector Spaces

I am currently reading Gunnar Carlsson's "Topological Pattern Recognition for Point Cloud Data", you can find it here: http://math.stanford.edu/~gunnar/actanumericathree.pdf I have a ...
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1 vote
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### Can you do geometry with persistent homology?

Setup In practice, persistent homology of data $X$ is often used to infer the homology of the underlying (Riemannian) manifold $M$ that the data is sampled from. However most filtrations (Vietoris, ...
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1 vote
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### Isoparametric function on hypersurfaces in Euclidean space

A isoparametric function $f$ on a Riemannian manifold is a function that satisfies the followings identities: $|\nabla f|=a(f)$ and $\Delta f=b(f)$. Now, I would like to deal with hypersurfaces in the ...
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### Short exact sequence of persistence modules

I am currently trying to work out a elementary proof of the following statement: Let $X$ be a simplicial complex with a filtration $\mathbb{X}: X=\bigcup_{n\in\mathbb{N}} X_n$, let $k\in\mathbb{N}$ ...
188 views

### Understanding nearest neighbors in high-dimensional data

Let's have a random sample of points in an euclidean $n$-space: assume a iid sample from a standard normal distribution. To each point $p$, I assign the number $N(p)$ defined as "how many times ...
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### Using topological data analysis to analyze a graph

My question may seem CS-related at first, but it's essentially mathematical, so, please, bear with me. I'm doing Neural Architecture Search (NAS) by varying the number of layers and neurons per layer ...
1 vote
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### 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 ...
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1 vote
126 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 ...
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### 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 ...
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### 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 ...
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### 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?
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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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1 vote