Here we study the extreme visual recovery problem, in which over 90 of pixel values in a given image are missing. Existing low rank-based algorithms are only effective for recovering data with at most 90 missing values. Thus, we exploit visual data's smoothness property to help solve this challenging extreme visual recovery problem. Based on the Discrete Cosine Transformation (DCT), we propose a novel DCT norm that involves all pixels and produces smooth estimations in any view. Our theoretical analysis shows that the total variation (TV) norm, which only achieves local smoothness, is a special case of the proposed DCT norm. We also develop a new visual recovery algorithm by minimizing the DCT and nuclear norms to achieve a more visually pleasing estimation. Experimental results on a benchmark image dataset demonstrate that the proposed approach is superior to state-of-the-art methods in terms of peak signal-to-noise ratio and structural similarity.
An ADMM-like optimization scheme @cite_12 can be adopted to solve Fcn.. However, the traditional TV norm only can guarantee an estimation presenting a locally smooth visualization, when in reality a natural image should be smoothed at every scale. In the next section, we propose a multi-scale DCT norm in the frequency domain.
@cite_12
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Abstract In this paper, we consider the problem of multi-class image classification when the classes behaviour has a low rank structure. That is, classes can be embedded into a low dimensional space. Traditional multi-class classification algorithms usually use nuclear norm to approximate the rank of the weight matrix. Considering the limited ability of the nuclear norm for the accurate approximation, we propose a new scalable large scale multi-class classification algorithm by using the recently proposed truncated nuclear norm as a better surrogate of the rank operator of matrices along with multinomial logisitic loss. To solve the non-convex and non-smooth optimization problem, we further develop an efficient iterative procedure. In each iteration, by lifting the non-smooth convex subproblem into an infinite dimensional l 1 norm regularized problem, a simple and efficient accelerated coordinate descent algorithm is applied to find the optimal solution. We conduct a series of evaluations on several public large scale image datasets, where the experimental results show the encouraging improvement of classification accuracy of the proposed algorithm in comparison with the state-of-the-art multi-class classification algorithms. Rain streaks removal is an important issue of the outdoor vision system and has been recently investigated extensively. In this paper, we propose a novel tensor based video rain streaks removal approach by fully considering the discriminatively intrinsic characteristics of rain streaks and clean videos, which needs neither rain detection nor time-consuming dictionary learning stage. In specific, on the one hand, rain streaks are sparse and smooth along the raindrops direction, and on the other hand, the clean videos possess smoothness along the rain-perpendicular direction and global and local correlation along time direction. We use the l1 norm to enhance the sparsity of the underlying rain, two unidirectional Total Variation (TV) regularizers to guarantee the different discriminative smoothness, and a tensor nuclear norm and a time directional difference operator to characterize the exclusive correlation of the clean video along time. Alternation direction method of multipliers (ADMM) is employed to solve the proposed concise tensor based convex model. Experiments implemented on synthetic and real data substantiate the effectiveness and efficiency of the proposed method. Under comprehensive quantitative performance measures, our approach outperforms other state-of-the-art methods. We describe a novel approach to optimizing matrix problems involving nuclear norm regularization and apply it to the matrix completion problem. We combine methods from nonsmooth and smooth optimization. At each step we use the proximal gradient to select an active subspace. We then find a smooth, convex relaxation of the smaller subspace problems and solve these using second order methods. We apply our methods to matrix completion problems including Netflix dataset, and show that they are more than 6 times faster than state-of-the-art nuclear norm solvers. Also, this is the first paper to scale nuclear norm solvers to the Yahoo-Music dataset, and the first time in the literature that the efficiency of nuclear norm solvers can be compared and even compete with non-convex solvers like Alternating Least Squares (ALS). We consider the minimization of a smooth loss function regularized by the trace norm of the matrix variable. Such formulation finds applications in many machine learning tasks including multi-task learning, matrix classification, and matrix completion. The standard semidefinite programming formulation for this problem is computationally expensive. In addition, due to the non-smooth nature of the trace norm, the optimal first-order black-box method for solving such class of problems converges as O(1 √k), where k is the iteration counter. In this paper, we exploit the special structure of the trace norm, based on which we propose an extended gradient algorithm that converges as O(1 k). We further propose an accelerated gradient algorithm, which achieves the optimal convergence rate of O(1 k2) for smooth problems. Experiments on multi-task learning problems demonstrate the efficiency of the proposed algorithms.