Abstract
This paper proposes a proximal iteratively reweighted algorithm to recover a low-rank matrix based on the weighted fixed point method. The weighted singular value thresholding problem gains a closed form solution because of the special properties of nonconvex surrogate functions. Besides, this study also has shown that the proximal iteratively reweighted algorithm lessens the objective function value monotonically, and any limit point is a stationary point theoretically.
Highlights
1 Introduction The low-rank matrix recovery problem has been a research hotpot recently [1, 2], and it has a range of applications in many fields such as signal or image processing [3, 4], subspace segmentation [5], collaborative filtering [6], and system identification [7]
In theory, this study has proved that the proximal iteratively reweighted algorithm decreases the objective function value monotonically, and any limit point is a stationary point
Proof Since Xk+1Wk + ∇f (Xk)+1 is the globally optimal solution of problem (3.3), and the zero matrix is contained in the subgradient with respect to X
Summary
The low-rank matrix recovery problem has been a research hotpot recently [1, 2], and it has a range of applications in many fields such as signal or image processing [3, 4], subspace segmentation [5], collaborative filtering [6], and system identification [7]. The tightest convex relaxation of problem (1.1) is the following nuclear norm minimization problem: min X s.t. A(X) = b,. The exact recovery of the low-rank matrix requires more measurements via nuclear norm minimization. Different from previous studies, based on the weighted fixed point method, this paper puts forward a proximal iteratively reweighted algorithm to recover a low-rank matrix. In order to introduce the following lemma, the definitions of Lipschitz continuous of a function and the norm · F are given, namely a function is Lipschitz continuous with constant L if, for any x, y, |f (x) – f (y)| ≤ L x – y ; and the · F of a matrix X is defined as. Lemma 3.1 ([17]) Let f : Rm×n → R be a continuously differentiable function with Lipschitz continuous gradient and the Lipschitz constant L(f ).
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