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.
Nuclear norm minimization-based matrix completion @cite_24 is formulated as where @math is the optimal estimation of @math . The truncated nuclear norm minimization problem @cite_16 is defined as: where @math is the sum of @math minimum singular values. It is widely known that the TV norm @cite_4 is an efficient smoothness regularization that accumulates all the gradients of a given image @math : where @math and @math denote the vertical and horizontal positions of @math , respectively. Since the TV norm accumulates gradient modules of the entire image, minimizing the TV norm can result in a smooth estimation:
@cite_24 @cite_16 @cite_4
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Motivated by some applications in signal processing and machine learning, we consider two convex optimization problems where, given a cone @math K, a norm @math ?·? and a smooth convex function @math f, we want either (1) to minimize the norm over the intersection of the cone and a level set of @math f, or (2) to minimize over the cone the sum of @math f and a multiple of the norm. We focus on the case where (a) the dimension of the problem is too large to allow for interior point algorithms, (b) @math ?·? is "too complicated" to allow for computationally cheap Bregman projections required in the first-order proximal gradient algorithms. On the other hand, we assume that it is relatively easy to minimize linear forms over the intersection of @math K and the unit @math ?·?-ball. Motivating examples are given by the nuclear norm with @math K being the entire space of matrices, or the positive semidefinite cone in the space of symmetric matrices, and the Total Variation norm on the space of 2D images. We discuss versions of the Conditional Gradient algorithm capable to handle our problems of interest, provide the related theoretical efficiency estimates and outline some applications. Recovering a low-rank tensor from incomplete information is a recurring problem in signal processing and machine learning. The most popular convex relaxation of this problem minimizes the sum of the nuclear norms of the unfoldings of the tensor. We show that this approach can be substantially suboptimal: reliably recovering a @math -way tensor of length @math and Tucker rank @math from Gaussian measurements requires @math observations. In contrast, a certain (intractable) nonconvex formulation needs only @math observations. We introduce a very simple, new convex relaxation, which partially bridges this gap. Our new formulation succeeds with @math observations. While these results pertain to Gaussian measurements, simulations strongly suggest that the new norm also outperforms the sum of nuclear norms for tensor completion from a random subset of entries. Our lower bound for the sum-of-nuclear-norms model follows from a new result on recovering signals with multiple sparse structures (e.g. sparse, low rank), which perhaps surprisingly demonstrates the significant suboptimality of the commonly used recovery approach via minimizing the sum of individual sparsity inducing norms (e.g. @math , nuclear norm). Our new formulation for low-rank tensor recovery however opens the possibility in reducing the sample complexity by exploiting several structures jointly. We present a new method for solving total variation (TV) minimization problems in image restoration. The main idea is to remove some of the singularity caused by the nondifferentiability of the quantity @math in the definition of the TV-norm before we apply a linearization technique such as Newton's method. This is accomplished by introducing an additional variable for the flux quantity appearing in the gradient of the objective function, which can be interpreted as the normal vector to the level sets of the image u. Our method can be viewed as a primal-dual method as proposed by Conn and Overton [ A Primal-Dual Interior Point Method for Minimizing a Sum of Euclidean Norms, preprint, 1994] and Andersen [Ph. D. thesis, Odense University, Denmark, 1995] for the minimization of a sum of Euclidean norms. In addition to possessing local quadratic convergence, experimental results show that the new method seems to be globally convergent. Recovering a large matrix from a small subset of its entries is a challenging problem arising in many real applications, such as image inpainting and recommender systems. Many existing approaches formulate this problem as a general low-rank matrix approximation problem. Since the rank operator is nonconvex and discontinuous, most of the recent theoretical studies use the nuclear norm as a convex relaxation. One major limitation of the existing approaches based on nuclear norm minimization is that all the singular values are simultaneously minimized, and thus the rank may not be well approximated in practice. In this paper, we propose to achieve a better approximation to the rank of matrix by truncated nuclear norm, which is given by the nuclear norm subtracted by the sum of the largest few singular values. In addition, we develop a novel matrix completion algorithm by minimizing the Truncated Nuclear Norm. We further develop three efficient iterative procedures, TNNR-ADMM, TNNR-APGL, and TNNR-ADMMAP, to solve the optimization problem. TNNR-ADMM utilizes the alternating direction method of multipliers (ADMM), while TNNR-AGPL applies the accelerated proximal gradient line search method (APGL) for the final optimization. For TNNR-ADMMAP, we make use of an adaptive penalty according to a novel update rule for ADMM to achieve a faster convergence rate. Our empirical study shows encouraging results of the proposed algorithms in comparison to the state-of-the-art matrix completion algorithms on both synthetic and real visual datasets.