  We introduce the sparse operator compression to compress a self-adjoint
higher-order elliptic operator with rough coefficients and various boundary
conditions. The operator compression is achieved by using localized basis
functions, which are energy-minimizing functions on local patches. On a regular
mesh with mesh size $h$, the localized basis functions have supports of
diameter $O(h\log(1/h))$ and give optimal compression rate of the solution
operator. We show that by using localized basis functions with supports of
diameter $O(h\log(1/h))$, our method achieves the optimal compression rate of
the solution operator. From the perspective of the generalized finite element
method to solve elliptic equations, the localized basis functions have the
optimal convergence rate $O(h^k)$ for a $(2k)$th-order elliptic problem in the
energy norm. From the perspective of the sparse PCA, our results show that a
large set of Mat\'{e}rn covariance functions can be approximated by a rank-$n$
operator with a localized basis and with the optimal accuracy.
