Analyzing the Weighted Nuclear Norm Minimization and Nuclear Norm Minimization based on Group Sparse Representation

Abstract

Rank minimization methods have attracted considerable interest in various areas, such as computer vision and machine learning. The most representative work is nuclear norm minimization (NNM), which can recover the matrix rank exactly under some restricted and theoretical guarantee conditions. However, for many real applications, NNM is not able to approximate the matrix rank accurately, since it often tends to over-shrink the rank components. To rectify the weakness of NNM, recent advances have shown that weighted nuclear norm minimization (WNNM) can achieve a better matrix rank approximation than NNM, which heuristically set the weight being inverse to the singular values. However, it still lacks a sound mathematical explanation on why WNNM is more feasible than NNM. In this paper, we propose a scheme to analyze WNNM and NNM from the perspective of the group sparse representation. Specifically, we design an adaptive dictionary to bridge the gap between the group sparse representation and the rank minimization models. Based on this scheme, we provide a mathematical derivation to explain why WNNM is more feasible than NNM. Moreover, due to the heuristical set of the weight, WNNM sometimes pops out error in the operation of SVD, and thus we present an adaptive weight setting scheme to avoid this error. We then employ the proposed scheme on two low-level vision tasks including image denoising and image inpainting. Experimental results demonstrate that WNNM is more feasible than NNM and the proposed scheme outperforms many current state-of-the-art methods.

Non-Rigid Structure from Motion: Prior-Free Factorization Method Revisited

“…The motivation for such formulation is quite clear, however, the proposed optimization (Eq:11) is generally nonconvex, and is more difficult to solve than the nuclear norm minimization. Fortunately, recent results [34,25,12] in compressed sensing have shown that we can achieve an effective optimal solution to Eq: (11) in the case when 0 ≤ Θ 1 ≤ Θ 2 ≤ .... ≤ Θ K §3.1.…”

“…This section provides the mathematical derivation to the optimization proposed in Eq: (11). Our solution use the following theorems and proofs as stated and used in [34,12,7].…”

“…The readers are encouraged to refer to [34,12] work for detailed derivations to the lemma's leading to the proof of the theorems. In conclusion, if the weights satisfies nondescending order, not necessarily with the same value, the weighted nuclear norm minimization problem is still convex and optimal solution can be obtained using a softthresholding operator with different weights [34,12].…”