  This paper explores methods for estimating or approximating the total
variation distance and the chi-squared divergence of probability measures
within topological sample spaces, using independent and identically distributed
samples. Our focus is on the practical scenario where the sample space is
homeomorphic to subsets of Euclidean space, with the specific homeomorphism
remaining unknown. Our proposed methods rely on the integral probability metric
with witness functions in universal reproducing kernel Hilbert spaces (RKHSs).
The estimators we develop consist of learnable parametric functions mapping the
sample space to Euclidean space, paired with universal kernels defined in
Euclidean space. This approach effectively overcomes the challenge of
constructing universal kernels directly on non-Euclidean spaces. Furthermore,
the estimators we devise demonstrate asymptotic consistency, and we provide a
detailed statistical analysis, shedding light on their practical
implementation.
