  This study proposes multivariate kernel density estimation by stagewise
minimization algorithm based on $U$-divergence and a simple dictionary. The
dictionary consists of an appropriate scalar bandwidth matrix and a part of the
original data. The resulting estimator brings us data-adaptive weighting
parameters and bandwidth matrices, and realizes a sparse representation of
kernel density estimation. We develop the non-asymptotic error bound of
estimator obtained via the proposed stagewise minimization algorithm. It is
confirmed from simulation studies that the proposed estimator performs
competitive to or sometime better than other well-known density estimators.
