  In constructing the $\mathcal{H}^2$ representation of dense matrices defined
by the Laplace kernel, the interpolative decomposition of certain off-diagonal
submatrices that dominates the computation can be dramatically accelerated
using the concept of a proxy surface. We refer to the computation of such
interpolative decompositions as the proxy surface method. We present an error
bound for the proxy surface method in the 3D case and thus provide theoretical
guidance for the discretization of the proxy surface in the method.
