Localized bases of eigensubspaces and operator compression

Given a complex local operator, such as the generator of a Markov chain on a large network, a differential operator, or a large sparse matrix that comes fromthe discretization of a differential operator, we would like to find its best finite dimensional approximation with a given dimension. The answer to this question is often given simply by the projection of the original operator to its eigensubspace of the given dimension that corresponds to the smallest or largest eigenvalues, depending on the setting. The representation of such subspaces, however, is far from being unique and our interest is to find the most localized bases for these subspaces. The reduced operator using these bases would have sparsity features similar to that of the original operator. We will discuss different ways of obtaining localized bases, and we will give an explicit characterization of the decay rate of these basis functions. We will also discuss efficient numerical algorithms for finding such basis functions and the reduced (or compressed) operator.