51,096 research outputs found
A sparse decomposition of low rank symmetric positive semi-definite matrices
Suppose that is symmetric positive
semidefinite with rank . Our goal is to decompose into
rank-one matrices where the modes
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 is the covariance function and is
intractable to solve in general. In this paper, we partition the indices from 1
to 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]
Flux rope proxies and fan-spine structures in active region NOAA 11897
Employing the high-resolution observations from the Solar Dynamics
Observatory (SDO) and the Interface Region Imaging Spectrograph (IRIS), we
statistically investigate flux rope proxies in NOAA AR 11897 from 14-Nov-2013
to 19-Nov-2013 and display two fan-spine structures in this AR. For the first
time, we detect flux rope proxies of NOAA 11897 for total 30 times in 4
different locations. These flux rope proxies were either tracked in both lower
and higher temperature wavelengths or only detected in hot channels. Specially,
none of these flux rope proxies was observed to erupt, but just faded away
gradually. In addition to these flux rope proxies, we firstly detect a
secondary fan-spine structure. It was covered by dome-shaped magnetic fields
which belong to a larger fan-spine topology. These new observations imply that
considerable amounts of flux ropes can exist in an AR and the complexity of AR
magnetic configuration is far beyond our imagination.Comment: 8 pages, 8 figures, Accepted for publication in A&
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