Additional Feedback: Fixing the proof of Theorem 1 is not trivial. One can assume that the entire observed data \mathcal{D} is in r-tubular neighborhood of M, but that is out of the point of this paper because we only want the boundary to be in the tubular neighborhood of M. Another possible fix would be as follows: with probability 1, \mathcal{D} {0} will be a subset of {x:p_{X Y}(x 0) 0} and \mathcal{D} {1} will be a subset of {x:p_{X Y}(x 1) 0}. Then the edge {x_i,x_j} is in (kappa sqrt{rho_i rho_j})-tubular neighborhood of \mathfrak{D}, and hence r (kappa sqrt{rho_i rho_j})-tubular neighborhood of M. And hence the entire edge set E, and eventually the labeled Cech complex, is in r (kappa sup_i rho_i)-tubular neighborhood of M. The term sup_i rho_i goes to 0 as n goes to infty, so with proper range of kappa so that r (kappa sup_i rho_i) is bounded by (3-sqrt{8})tau in Assumption 1(b) in line 118 and also that \partial C is r (kappa sup_i rho_i) - dense in M, then the labeled Cech complex will deformation retracts to M, and hence this will appear in the persistent homology as well. This is very rough sketch and if the authors want to use this proof, they should fill the gap by themselves, although I can help to check more details if the authors want in the authors' response phase. And hence I increased my score to 6. Although, I would also like to mention that there is a slight mismatch between the framework of the paper and Theorem 1.