Suppose that A∈RN×N is symmetric positive
semidefinite with rank K≤N. Our goal is to decompose A into K
rank-one matrices ∑k=1KgkgkT where the modes {gk}k=1K
are required to be as sparse as possible. In contrast to eigen decomposition,
these sparse modes are not required to be orthogonal. Such a problem arises in
random field parametrization where A is the covariance function and is
intractable to solve in general. In this paper, we partition the indices from 1
to N into several patches and propose to quantify the sparseness of a vector
by the number of patches on which it is nonzero, which is called patch-wise
sparseness. Our aim is to find the decomposition which minimizes the total
patch-wise sparseness of the decomposed modes. We propose a
domain-decomposition type method, called intrinsic sparse mode decomposition
(ISMD), which follows the "local-modes-construction + patching-up" procedure.
The key step in the ISMD is to construct local pieces of the intrinsic sparse
modes by a joint diagonalization problem. Thereafter a pivoted Cholesky
decomposition is utilized to glue these local pieces together. Optimal sparse
decomposition, consistency with different domain decomposition and robustness
to small perturbation are proved under the so called regular-sparse assumption
(see Definition 1.2). We provide simulation results to show the efficiency and
robustness of the ISMD. We also compare the ISMD to other existing methods,
e.g., eigen decomposition, pivoted Cholesky decomposition and convex relaxation
of sparse principal component analysis [25] and [40]