Equivalence of the persistence landscape diagram and the barcode?I am studying persistent homology for the...
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 persistent landscapes. I am quite confused on the approach on finding the values of the persistence landscape function using a barcode/persistence diagram. I feel like I have a naive misunderstanding of this topic as I shall attempt to explain.
Suppose X is a finite set of points in Euclidean space. From my understanding, if we consider the (finite length) persistence vector space given by the simplicial complex homology for a fixed dimension l, with maps , for ,, we have
(Theorem 4.10 of this paper) for some multiset , where gives the persistence vector space of length n,
with non-zero vector spaces at values of the specified interval.
This multiset corresponds to the persistence diagram/barcode so that the k-th Betti number can be identified by finding the number of lines of the barcode that intersect the line x=k in
Now Bubenik defines the Betti number of the persistence vector space for an interval [a,b] by , and the persistence landscape functions , for by
Shouldn't then correspond to the number of lines on the barcode that contain the interval ], so that is the largest value of m that has at least k lines of the barcode intersecting ?
I am confused on how the triangle construction is equivalent to the persistence landscape function instead. Any help would be much appreciated!