A Quick Review
How to take average of persistence diagrams is a central problem in TDA. The problem is first addressed by Mileyko et al. in Probability measures on the space of persistence diagrams. Let \(\mathcal{D}_p\) be the space of persistence diagrams with finite \(p\)-total persistence, e...
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Frechet means for distributions of persistence diagrams
Consider the measure \(\rho = 1/m \sum_{i=1}^m \delta_{Z_i}\) on the space of persistence diagrams. The population Frechet mean is defined by
\[
Y = \arg\min_Z \int_{\mathcal{D}} d(X,Z)^2 {\rm d}\rho(X)
\]
Let \(X_1,\cdots, X_n\) be iid...
This is a useful technique I learned from Convergence rates for persistence diagram estimation in topological data analysis when one wants to prove convergence results for some estimator under Hausdorff distance. Typical settings are
A metric space \((X,\rho)\);
A probability measure \(\mu\) ...
When I want to save a bunch of tex equations/symbols as svg files I find there is no efficient software to fulfill this goal. But I find the interesting pipeline which is implemented on a Linux system. The following is what I did on my CentOS7 server.
Install Latex
Login in as root. Then instal...
Motivation
Let \(A\) stand for the true symmetric matrix and \(E\) represent the perturbation. Let \(\lambda_i\)'s be the eigenvalues of \(A\), sorted in descending order and denote \(\delta=\lambda_1-\lambda_2\) to be the eigengap. Let \(u_i\) be the eigenvector of \(A\) and \(v_i\) be the eige...
Problem Statement
Let \(A\) be an \(n\times n\) Hermitian matrix, and suppose we have the following spectral decomposition for \(A\)
\[
A=\sum_{i=1}^n\lambda_iu_iu_i^*
\]
where \(\lambda_i\)'s are eigenvalues of \(A\) (we do not need to sort eigenvalues here), and \(u_i\)'s are corresponding...
Amit Singer's paper From graph to manifold Laplacian: The convergence rate presents a very clear proof of the convergence rate of graph Laplacian that everyone can follow. Techniques involved in this proof have become standard in following works in manifold learning. We review the proof in this p...
If \(\mathcal{M}\) is an Euclidean submanifold, many intrinsic quantities can be expressed using information from the ambient Euclidean space. Here are two examples where the Weingarten map plays an important role: first is the Riemannian hessian of a smooth map, and second is the Ricci curvature...
In noiseless case we have a clean result for the convergence rate about tangent space estimation via local PCA ( see this post ). However, for noisy data the analysis is much more involved, because noise will cause points in the local neighborhood to exit, and points from the outside to enter. Th...
