We study the problem of finding structured low-rank matrices using nuclear norm regularization where the structure is encoded by a linear map. In contrast to most known approaches for linearly structured rank minimization, we do not (a) use the full SVD; nor (b) resort to augmented Lagrangian techniques; nor (c) solve linear systems per iteration. Instead, we formulate the problem differently so that it is amenable to a generalized conditional gradient method, which results in a practical improvement with low per iteration computational cost. Numerical results show that our approach significantly outperforms state-of-the-art competitors in terms of running time, while effectively recovering low rank solutions in stochastic system realization and spectral compressed sensing problems.

In this paper we consider general rank minimization problems with rank appearing in either objective function or constraint. We first show that a class of matrix optimization problems can be solved as lower dimensional vector optimization problems. As a consequence, we establish that a class of rank minimization problems have closed form solutions. Using this result, we then propose penalty decomposition methods for general rank minimization problems. The convergence results of the PD methods have been shown in the longer version of the paper [19]. Finally, we test the performance of our methods by applying them to matrix completion and nearest low-rank correlation matrix problems. The computational results demonstrate that our methods generally outperform the existing methods in terms of solution quality and/or speed.

We study the problem of finding structured low-rank matrices using nuclear norm regularization where the structure is encoded by a linear map. In contrast to most known approaches for linearly structured rank minimization, we do not (a) use the full SVD; nor (b) resort to augmented Lagrangian techniques; nor (c) solve linear systems per iteration. Instead, we formulate the problem differently so that it is amenable to a generalized conditional gradient method, which results in a practical improvement with low per iteration computational cost. Numerical results show that our approach significantly outperforms state-of-the-art competitors in terms of running time, while effectively recovering low rank solutions in stochastic system realization and spectral compressed sensing problems.

In this paper we consider general rank minimization problems with rank appearing in either objective function or constraint. We first show that a class of matrix optimization problems can be solved as lower dimensional vector optimization problems. As a consequence, we establish that a class of rank minimization problems have closed form solutions. Using this result, we then propose penalty decomposition methods for general rank minimization problems. The convergence results of the PD methods have been shown in the longer version of the paper.

In this paper we consider general rank minimization problems with rank appearing in either objective function or constraint. We first show that a class of matrix optimization problems can be solved as lower dimensional vector optimization problems. As a consequence, we establish that a class of rank minimization problems have closed form solutions. Using this result, we then propose penalty decomposition methods for general rank minimization problems. The convergence results of the PD methods have been shown in the longer version of the paper.

We first propose a novel nonconvex rank surrogate on the general rank minimization problem and apply this to the corrupted image completion problem. Then, we propose that nonconvex rank surrogates can be introduced into two well-known sparse and low-rank models: Robust Principal Component Analysis (RPCA) and Low-Rank Representation (LRR). For optimization, we use alternating direction method of multipliers (ADMM) and propose a trick, which is called the dual momentum. We add the difference of the dual variable between the current and the last iteration with a weight. This trick can avoid the local minimum problem and make the algorithm converge to a solution with smaller recovery error in the nonconvex optimization problem. Also, it can boost the convergence when the variable updates too slowly. We also give a severe proof and verify that the proposed algorithms are convergent. Then, several experiments are conducted, including image completion, denoising, and spectral clustering with outlier detection. These experiments show that the proposed methods are effective in image and signal processing applications, and have the best performance compared with state-of-the-art methods.

Semi-definite rank minimization problems model a wide range of applications in both signal processing and machine learning fields. This class of problem is NP-hard in general. In this paper, we propose a proximal Alternating Direction Method (ADM) for the well-known semi-definite rank regularized minimization problem. Specifically, we first reformulate this NP-hard problem as an equivalent biconvex MPEC (Mathematical Program with Equilibrium Constraints), and then solve it using proximal ADM, which involves solving a sequence of structured convex semi-definite subproblems to find a desirable solution to the original rank regularized optimization problem. Moreover, based on the Kurdyka-Lojasiewicz inequality, we prove that the proposed method always converges to a KKT stationary point under mild conditions. We apply the proposed method to the widely studied and popular sensor network localization problem. Our extensive experiments demonstrate that the proposed algorithm outperforms state-of-the-art low-rank semi-definite minimization algorithms in terms of solution quality.

We study the problem of finding structured low-rank matrices using nuclear norm regularization where the structure is encoded by a linear map. In contrast to most known approaches for linearly structured rank minimization, we do not (a) use the full SVD; nor (b) resort to augmented Lagrangian techniques; nor (c) solve linear systems per iteration. Instead, we formulate the problem differently so that it is amenable to a generalized conditional gradient method, which results in a practical improvement with low per iteration computational cost. Numerical results show that our approach significantly outperforms state-of-the-art competitors in terms of running time, while effectively recovering low rank solutions in stochastic system realization and spectral compressed sensing problems.

This paper concerns model reduction of dynamical systems using the nuclear norm of the Hankel matrix to make a trade-off between model fit and model complexity. This results in a convex optimization problem where this trade-off is determined by one crucial design parameter. The main contribution is a methodology to approximately calculate all solutions up to a certain tolerance to the model reduction problem as a function of the design parameter. This is called the regularization path in sparse estimation and is a very important tool in order to find the appropriate balance between fit and complexity. We extend this to the more complicated nuclear norm case. The key idea is to determine when to exactly calculate the optimal solution using an upper bound based on the so-called duality gap. Hence, by solving a fixed number of optimization problems the whole regularization path up to a given tolerance can be efficiently computed. We illustrate this approach on some numerical examples.

Rank minimization has attracted a lot of attention due to its robustness in data recovery. To overcome the computational difficulty, rank is often replaced with nuclear norm. For several rank minimization problems, such a replacement has been theoretically proven to be valid, i.e., the solution to nuclear norm minimization problem is also the solution to rank minimization problem. Although it is easy to believe that such a replacement may not always be valid, no concrete example has ever been found. We argue that such a validity checking cannot be done by numerical computation and show, by analyzing the noiseless latent low rank representation (LatLRR) model, that even for very simple rank minimization problems the validity may still break down. As a by-product, we find that the solution to the nuclear norm minimization formulation of LatLRR is non-unique. Hence the results of LatLRR reported in the literature may be questionable.