Without using any category, topology of sheaf theory, this is what I believe is in this paper:
(1) the prior hypothesis is that data points in R^n are a sample from a uniform distribution in a Riemann space.
(2) Try to define a Riemann metric such that the number of sample points in any ball B is propotional to the volume of B.
(3) Since (2) doesn't define a global Riemann metric, they define a fuzzy membership relation. I suppose the role of the fuzzy tool is that local distance information is weighted according to the variance of the local distance
estimations.
Without using any category, topology of sheaf theory, this is what I believe is in this paper:
(1) the prior hypothesis is that data points in R^n are a sample from a uniform distribution in a Riemann space.
(2) Try to define a Riemann metric such that the number of sample points in any ball B is propotional to the volume of B.
(3) Since (2) doesn't define a global Riemann metric, they define a fuzzy membership relation. I suppose the role of the fuzzy tool is that local distance information is weighted according to the variance of the local distance estimations.
Disclaimer, I could be completely wrong.