Mixed-Projection Conic Optimization: A New Paradigm for Modeling Rank Constraints

Many central problems throughout optimization, machine learning, and statistics are equivalent to optimizing a low-rank matrix over a convex set. However, although rank constraints offer unparalleled modeling flexibility, no generic code currently solves these problems to certifiable optimality at even moderate sizes. Instead, low-rank optimization problems are solved via convex relaxations or heuristics that do not enjoy optimality guarantees. In “Mixed-Projection Conic Optimization: A New Paradigm for Modeling Rank Constraints,” Bertsimas, Cory-Wright, and Pauphilet propose a new approach for modeling and optimizing over rank constraints. They generalize mixed-integer optimization by replacing binary variables z that satisfy z2 =z with orthogonal projection matrices Y that satisfy Y2 = Y. This approach offers the following contributions: First, it supplies certificates of (near) optimality for low-rank problems. Second, it demonstrates that some of the best ideas in mixed-integer optimization, such as decomposition methods, cutting planes, relaxations, and random rounding schemes, admit straightforward extensions to mixed-projection optimization.

Anomaly Detection With Kernel Preserving Embedding

Similarity representation plays a central role in increasingly popular anomaly detection techniques, which have been successfully applied in various realistic scenes. Until now, many low-rank representation techniques have been introduced to measure the similarity relations of data; yet, they only concern to minimize reconstruction errors, without involving the structural information of data. Besides, the traditional low-rank representation methods often take nuclear norm as their low-rank constraints, easily yielding a suboptimal solution. To address the problems above, in this article, we propose a novel anomaly detection method, which exploits kernel preserving embedding, as well as the double nuclear norm, to explore the similarity relations of data. Based on the similarity relations, a kind of probability transition matrix is derived, and a tailored random walk is further adopted to reveal anomalies. The proposed method can not only preserve the manifold structural properties of the data, but also alleviate the suboptimal problem. To validate the superiority of our method, extensive experiments with eight popular anomaly detection algorithms were conducted on 12 widely used datasets. The experimental results show that our detection method outperformed the state-of-the-art anomaly detection algorithms in most cases.

MR Brain Image Segmentation Using a Fuzzy Weighted Multiview Possibility Clustering Algorithm with Low-Rank Constraints

To expand the multiview clustering abilities of traditional PCM in increasingly complex MR brain image segmentation tasks, a fuzzy weighted multiview possibility clustering algorithm with low-rank constraints (LR-FW-MVPCM) is proposed. The LR-FW-MVPCM can effectively mine both the internal consistency and diversity of multiview data, which are two principles for constructing a multiview clustering algorithm. First, a kernel norm is introduced as a low-rank constraint of the fuzzy membership matrix among multiple perspectives. Second, to ensure the clustering accuracy of the algorithm, the view fuzzy weighted mechanism is introduced to the framework of possibility c-means clustering, and the weights of each view are adaptively allocated during the iterative optimization process. The segmentation results of different brain tissues based on the proposed algorithm and three other algorithms illustrate that the LR-FW-MVPCM algorithm can segment MR brain images much more effectively and ensure better segmentation performance.