  Data-driven neighborhood definitions and graph constructions are often used
in machine learning and signal processing applications. k-nearest
neighbor~(kNN) and $\epsilon$-neighborhood methods are among the most common
methods used for neighborhood selection, due to their computational simplicity.
However, the choice of parameters associated with these methods, such as k and
$\epsilon$, is still ad hoc. We make two main contributions in this paper.
First, we present an alternative view of neighborhood selection, where we show
that neighborhood construction is equivalent to a sparse signal approximation
problem. Second, we propose an algorithm, non-negative kernel regression~(NNK),
for obtaining neighborhoods that lead to better sparse representation. NNK
draws similarities to the orthogonal matching pursuit approach to signal
representation and possesses desirable geometric and theoretical properties.
Experiments demonstrate (i) the robustness of the NNK algorithm for
neighborhood and graph construction, (ii) its ability to adapt the number of
neighbors to the data properties, and (iii) its superior performance in local
neighborhood and graph-based machine learning tasks.
