Questions tagged [topological-data-analysis]
Questions about persistent homology, computational topology, discrete morse theory, and applied algebraic topology in general.
142 questions
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Doubt about effect of Standard Scaler on persistent homology
I’m currently studying persistent homology, and I’m quite new to the topic — so please forgive me if this is a naive question.
I’m trying to analyze 2ドルD$ data where each axis represents quantities ...
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Birth definition in persistent homology theory
I have some confusion about the definition of birth time, and I hope someone can clarify a few things. I am reading "Computational Topology: An Introduction" by Edelsbrunner and Harer. They ...
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In what sense are persistence modules actually modules
A persistence module can be defined as a functor from a totally ordered poset to Vect. In what sense can this be viewed as a module? I read a comment that they can be thought of as modules over a ...
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how to show that generic persistence modules are dense for the interleaving distance
In a paper of TDA, the authors defined a kind of persistence modules called 'generic'. They are defined as follows : Consider a persistence module $\mathbf{V}=(V_t,i_{s,t})_{s<t}$. If $t\in\mathbb ...
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Understanding the persistent homology groups
I have been trying to understand persistent homology groups from several sources now and still not getting the point. It says that for a filtered complex $X_0\subseteq X_1\subseteq \ldots X_n =X,ドル ...
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Is $E_I^+(N)$ a subspace of $N_b$ in the context of persistence modules?
I'm currently working through a proposition from the paper by Polterovich, Rosen, Samvelyan, and Zhang, which states:
Let $ p : (M, \varphi) \twoheadrightarrow (N, \phi) $ be a surjective morphism of ...
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isomorphism invariance of the spectrum of persistence modules
Is this proof of the isomorphism invariance of the spectrum of persistence modules correct?
I developed the following proof that the spectrum of a persistence module is an isomorphism invariant. I ...
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How to Visualize the Action of $t$ in a Persistence Module Over $\mathbf{k}[t]$?
I got stuck on the proof of structure theorem.
In persistent homology, a persistence module $M$ is often described as a module over the polynomial ring $\mathbf{k}[t],ドル where $t$ acts as a shift ...
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Proof of the Structure Theorem for Persistent Homology
Proof of the Structure Theorem for Persistent Homology
I am trying to understand the Structure Theorem for Persistent Homology, which states that any finitely generated persistence module over a field ...
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Direct Sum of Persistence Modules
I've been following Chapter 6 of https://people.maths.ox.ac.uk/nanda/cat/TDANotes.pdf and I wanted to come up with my own example for their definition of direct sums. I'm struggling to fund other ...
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Persistent Vector Space barcode decomposition
I have a question about Topological Data Analysis (TDA), specifically related to material available at https://tgda.osu.edu/math-4570-applied-algebraic-topology/.
From the notes, my understanding is ...
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Question on Gunnar Carlsson's 2010 work "Zigzag Persistence"
I am currently learning persistence theory, and have found Gunnar Carlsson and Vin de Silva's 2010 work "Zigzag Persistence", which is an extension of traditional persistent homology theory ...
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Reducing boundary matrix to Smith normal form in computational topological data analysis
I have been reading Topological data analysis: concepts, computation, and applications in chemical engineering and I am struggling with the idea of reducing boundary matrices to the Smith normal form (...
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Different dimension of homology over different rings
Can the dimension of the $n^\text{th}$ homology of a simplicial complex vary over different commutative rings with unity?
Is it possible that $\dim(H_n (K,\mathbb{F}_2))\ne \dim(H_n (K,\mathbb{Z}))$?
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Can I get a spherical coordinate from a real cocycle?
The Setting
I am currently working on a project in Topological Data Analysis (TDA), where I aim to construct a density-robust spherical coordinate associated with a dataset $X,ドル sampled from a ...