  Suppose that there is a ground set which consists of a large number of
vectors in a Hilbert space. Consider the problem of selecting a subset of the
ground set such that the projection of a vector of interest onto the subspace
spanned by the vectors in the chosen subset reaches the maximum norm. This
problem is generally NP-hard, and alternative approximation algorithms such as
forward regression and orthogonal matching pursuit have been proposed as
heuristic approaches. In this paper, we investigate bounds on the performance
of these algorithms by introducing the notions of elemental curvatures. More
specifically, we derive lower bounds, as functions of these elemental
curvatures, for performance of the aforementioned algorithms with respect to
that of the optimal solution under uniform and non-uniform matroid constraints,
respectively. We show that if the elements in the ground set are mutually
orthogonal, then these algorithms are optimal when the matroid is uniform and
they achieve at least $1/2$-approximations of the optimal solution when the
matroid is non-uniform.
