Iterative Reconstrained Low-rank Representation via Weighted Nonconvex Regularizer

OAPA Benefiting from the joint consideration of geometric structures and low-rank constraint, graph low-rank representation (GLRR) method has led to the state-of-the-art results in many applications. However, it faces the limitations that the structure of errors should be known a prior, the isolated construction of graph Laplacian matrix, and the over shrinkage of the leading rank components. To improve GLRR in these regards, this paper proposes a new LRR model, namely iterative reconstrained LRR via weighted nonconvex regularization (IRWNR), using three distinguished properties on the concerned representation matrix. The first characterizes various distributions of the errors into an adaptively learned weight factor for more flexibility of noise suppression. The second generates an accurate graph matrix from weighted observations for less afflicted by noisy features. The third employs a parameterized Rational function to reveal the importance of different rank components for better approximation to the intrinsic subspace structure. Following a deep exploration of automatic thresholding, parallel update, and partial SVD operation, we derive a computationally efficient low-rank representation algorithm using an iterative reconstrained framework and accelerated proximal gradient method. Comprehensive experiments are conducted on synthetic data, image clustering, and background subtraction to achieve several quantitative benchmarks as clustering accuracy, normalized mutual information, and execution time. Results demonstrate the robustness and efficiency of IRWNR compared with other state-of-the-art models

Weighted Schatten pp-Norm Minimization for Image Denoising and Background Subtraction

Low rank matrix approximation (LRMA), which aims to recover the underlying low rank matrix from its degraded observation, has a wide range of applications in computer vision. The latest LRMA methods resort to using the nuclear norm minimization (NNM) as a convex relaxation of the nonconvex rank minimization. However, NNM tends to over-shrink the rank components and treats the different rank components equally, limiting its flexibility in practical applications. We propose a more flexible model, namely the Weighted Schatten $p$-Norm Minimization (WSNM), to generalize the NNM to the Schatten $p$-norm minimization with weights assigned to different singular values. The proposed WSNM not only gives better approximation to the original low-rank assumption, but also considers the importance of different rank components. We analyze the solution of WSNM and prove that, under certain weights permutation, WSNM can be equivalently transformed into independent non-convex $l_p$-norm subproblems, whose global optimum can be efficiently solved by generalized iterated shrinkage algorithm. We apply WSNM to typical low-level vision problems, e.g., image denoising and background subtraction. Extensive experimental results show, both qualitatively and quantitatively, that the proposed WSNM can more effectively remove noise, and model complex and dynamic scenes compared with state-of-the-art methods.Comment: 13 pages, 11 figure

Generalized Nonconvex Nonsmooth Low-Rank Minimization

As surrogate functions of $L_0$-norm, many nonconvex penalty functions have been proposed to enhance the sparse vector recovery. It is easy to extend these nonconvex penalty functions on singular values of a matrix to enhance low-rank matrix recovery. However, different from convex optimization, solving the nonconvex low-rank minimization problem is much more challenging than the nonconvex sparse minimization problem. We observe that all the existing nonconvex penalty functions are concave and monotonically increasing on $[0,\infty)$. Thus their gradients are decreasing functions. Based on this property, we propose an Iteratively Reweighted Nuclear Norm (IRNN) algorithm to solve the nonconvex nonsmooth low-rank minimization problem. IRNN iteratively solves a Weighted Singular Value Thresholding (WSVT) problem. By setting the weight vector as the gradient of the concave penalty function, the WSVT problem has a closed form solution. In theory, we prove that IRNN decreases the objective function value monotonically, and any limit point is a stationary point. Extensive experiments on both synthetic data and real images demonstrate that IRNN enhances the low-rank matrix recovery compared with state-of-the-art convex algorithms.Comment: IEEE International Conference on Computer Vision and Pattern Recognition, 201