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 points over which the homology is computed.
So say I am look at some point cloud of observations $x_i, i \in \{1, ..., n\}$. Each observation is a vector, representing different features or measurements from the observation.
$$ x_i = \begin{bmatrix} x_{i,1} \\ x_{i,2} \\ x_{i,3} \\ \vdots \\ x_{i,m} \end{bmatrix} $$
Now if I compute the persistent homology of the set of points, I will obtain a barcode or persistence diagram, and the world is good.
BUT, here is my question. If I were to reorder the elements in the each vector, would this change the barcode? For example if I just did a simple reordering like
$$ \hat{x_i} = \begin{bmatrix} x_{i,3} \\ x_{i,2} \\ x_{i,1} \\ \vdots \\ x_{i,m} \end{bmatrix} $$
Here I just rearranged some of the elements of the original vector $x_i$ to produce $\hat{x_i}$. If I did this same reordering to all of the other observations in the dataset, would this change the barcode or persistent homology in any way?
It seems like most homology calculations are based upon some norm, and so reordering would not necessarily affect that norm. But I am not enough of an algebraic topologist to know whether there are other theoretical questions that can affect the stability of the persistent homology.
Thanks.
I did a bit of TDA. To my knowledge/understanding, no, it shouldn't matter. TDA measures how long various "holes" of different dimensions last as you move the parameter. All permuting the dimensions would do would be to rotate the shapes involved, and rotations preserve the topological structures.