  Topological data analysis computes and analyses topological features of the
point clouds by constructing and studying a simplicial representation of the
underlying topological structure. The enthusiasm that followed the initial
successes of topological data analysis was curbed by the computational cost of
constructing such simplicial representations. The lazy witness complex is a
computationally feasible approximation of the underlying topological structure
of a point cloud. It is built in reference to a subset of points, called
landmarks, rather than considering all the points as in the \v{C}ech and
Vietoris-Rips complexes. The choice and the number of landmarks dictate the
effectiveness and efficiency of the approximation. We adopt the notion of
$\epsilon$-cover to define $\epsilon$-net. We prove that $\epsilon$-net, as a
choice of landmarks, is an $\epsilon$-approximate representation of the point
cloud and the induced lazy witness complex is a $3$-approximation of the
induced Vietoris-Rips complex. Furthermore, we propose three algorithms to
construct $\epsilon$-net landmarks. We establish the relationship of these
algorithms with the existing landmark selection algorithms. We empirically
validate our theoretical claims. We empirically and comparatively evaluate the
effectiveness, efficiency, and stability of the proposed algorithms on
synthetic and real datasets.
