Truncated Nuclear Norm Minimization Based Group Sparse Representation for Image Restoration

Efficient Implementation of Truncated Reweighting Low-Rank Matrix Approximation

“…Note that the descent property established in Lemma 1 is independent of the convexity of f and g, so it is applicable to (3) even with the concavity of TRNM regularization. Literally, it seems that Algorithm 1 requires two loops for optimization, which may yield more overall iterations as similar to the twostep strategy for TNNM [18]. However, with the fact that the sufficient descent property of Θ δ can be satisfied given µ k large enough and β k small enough, we present Proposition 1 that the condition in step 1.3 can be easily satisfied given θ > 1 and η < 1.…”

“…To validate the effectiveness of the proposed TRNM constraint and MURP scheme, we apply them to two typical data mining applications: matrix completion and subspace clustering. For matrix completion, similar to TNNM [18], [21], [22], TRNM is applied directly to the input matrix for structure recovery. For subspace clustering, like the TSCLR [30], we propose a TRNM constrained SC method to recover the "low-rank + temporal" structure of the data matrix.…”

“…The main problem is that the nuclear norm is still a suboptimal relaxation to the rank minimization problem. Therefore, lots of attempts for improvements have been investigated [4], [12], [13], [14], [15], [16], [17], [18]. An intuitive scheme is to enforce low-rank property by using nonconvex constraints for closer approximation, e.g., Schatten p [12], Logarithm [12], [15], and Rational [15], etc.…”

“…Although WNNM is more flexible compared with NNM by penalizing larger singular values less than the smaller ones such as to preserve the major data components, it does not fully take into consideration a priori rank information for the encountered problem. To address this limitation, the truncated nuclear norm minimization (TNNM) [18], [19], i.e., X r = q i=r+1 σ i (X), is proposed for achieving better control of the target rank to be a given parameter r. However, it is also questionable due to the missing consideration of different rank contributions, which leads to suboptimal soft thresholding.…”

“…Additionally, for the TNNM problem, some extra manipulations are further required due to the discrete nature of truncation step. To avoid the directly minimizing of TNNM, a common way is to alternatively optimize it by a two-step scheme [18]. The first approximates TNNM with a difference of two convex surrogates by introducing some auxiliary variables.…”