  In topology inference from data, current approaches face two major problems.
One concerns the selection of a correct parameter to build an appropriate
complex on top of the data points; the other involves with the typical `large'
size of this complex. We address these two issues in the context of inferring
homology from sample points of a smooth manifold of known dimension sitting in
an Euclidean space $\mathbb{R}^k$. We show that, for a sample size of $n$
points, we can identify a set of $O(n^2)$ points (as opposed to $O(n^{\lceil
\frac{k}{2}\rceil})$ Voronoi vertices) approximating a subset of the medial
axis that suffices to compute a distance sandwiched between the well known
local feature size and the local weak feature size (in fact, the approximating
set can be further reduced in size to $O(n)$). This distance, called the lean
feature size, helps pruning the input set at least to the level of local
feature size while making the data locally uniform. The local uniformity in
turn helps in building a complex for homology inference on top of the
sparsified data without requiring any user-supplied distance threshold. Unlike
most topology inference results, ours does not require that the input is dense
relative to a {\em global} feature such as {\em reach} or {\em weak feature
size}; instead it can be adaptive with respect to the local feature size. We
present some empirical evidence in support of our theoretical claims.
