36,639 research outputs found
Truncated Nuclear Norm Minimization for Image Restoration Based On Iterative Support Detection
Recovering a large matrix from limited measurements is a challenging task
arising in many real applications, such as image inpainting, compressive
sensing and medical imaging, and this kind of problems are mostly formulated as
low-rank matrix approximation problems. Due to the rank operator being
non-convex and discontinuous, most of the recent theoretical studies use the
nuclear norm as a convex relaxation and the low-rank matrix recovery problem is
solved through minimization of the nuclear norm regularized problem. However, a
major limitation of nuclear norm minimization is that all the singular values
are simultaneously minimized and the rank may not be well approximated
\cite{hu2012fast}. Correspondingly, in this paper, we propose a new multi-stage
algorithm, which makes use of the concept of Truncated Nuclear Norm
Regularization (TNNR) proposed in \citep{hu2012fast} and Iterative Support
Detection (ISD) proposed in \citep{wang2010sparse} to overcome the above
limitation. Besides matrix completion problems considered in
\citep{hu2012fast}, the proposed method can be also extended to the general
low-rank matrix recovery problems. Extensive experiments well validate the
superiority of our new algorithms over other state-of-the-art methods
An Alternating Direction Algorithm for Matrix Completion with Nonnegative Factors
This paper introduces an algorithm for the nonnegative matrix
factorization-and-completion problem, which aims to find nonnegative low-rank
matrices X and Y so that the product XY approximates a nonnegative data matrix
M whose elements are partially known (to a certain accuracy). This problem
aggregates two existing problems: (i) nonnegative matrix factorization where
all entries of M are given, and (ii) low-rank matrix completion where
nonnegativity is not required. By taking the advantages of both nonnegativity
and low-rankness, one can generally obtain superior results than those of just
using one of the two properties. We propose to solve the non-convex constrained
least-squares problem using an algorithm based on the classic alternating
direction augmented Lagrangian method. Preliminary convergence properties of
the algorithm and numerical simulation results are presented. Compared to a
recent algorithm for nonnegative matrix factorization, the proposed algorithm
produces factorizations of similar quality using only about half of the matrix
entries. On tasks of recovering incomplete grayscale and hyperspectral images,
the proposed algorithm yields overall better qualities than those produced by
two recent matrix-completion algorithms that do not exploit nonnegativity
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