Robust Subspace Clustering via Smoothed Rank Approximation

Matrix rank minimizing subject to affine constraints arises in many application areas, ranging from signal processing to machine learning. Nuclear norm is a convex relaxation for this problem which can recover the rank exactly under some restricted and theoretically interesting conditions. However, for many real-world applications, nuclear norm approximation to the rank function can only produce a result far from the optimum. To seek a solution of higher accuracy than the nuclear norm, in this paper, we propose a rank approximation based on Logarithm-Determinant. We consider using this rank approximation for subspace clustering application. Our framework can model different kinds of errors and noise. Effective optimization strategy is developed with theoretical guarantee to converge to a stationary point. The proposed method gives promising results on face clustering and motion segmentation tasks compared to the state-of-the-art subspace clustering algorithms.Comment: Journal, code is availabl

Top-N Recommender System via Matrix Completion

Top-N recommender systems have been investigated widely both in industry and academia. However, the recommendation quality is far from satisfactory. In this paper, we propose a simple yet promising algorithm. We fill the user-item matrix based on a low-rank assumption and simultaneously keep the original information. To do that, a nonconvex rank relaxation rather than the nuclear norm is adopted to provide a better rank approximation and an efficient optimization strategy is designed. A comprehensive set of experiments on real datasets demonstrates that our method pushes the accuracy of Top-N recommendation to a new level.Comment: AAAI 201

Twin Learning for Similarity and Clustering: A Unified Kernel Approach

Many similarity-based clustering methods work in two separate steps including similarity matrix computation and subsequent spectral clustering. However, similarity measurement is challenging because it is usually impacted by many factors, e.g., the choice of similarity metric, neighborhood size, scale of data, noise and outliers. Thus the learned similarity matrix is often not suitable, let alone optimal, for the subsequent clustering. In addition, nonlinear similarity often exists in many real world data which, however, has not been effectively considered by most existing methods. To tackle these two challenges, we propose a model to simultaneously learn cluster indicator matrix and similarity information in kernel spaces in a principled way. We show theoretical relationships to kernel k-means, k-means, and spectral clustering methods. Then, to address the practical issue of how to select the most suitable kernel for a particular clustering task, we further extend our model with a multiple kernel learning ability. With this joint model, we can automatically accomplish three subtasks of finding the best cluster indicator matrix, the most accurate similarity relations and the optimal combination of multiple kernels. By leveraging the interactions between these three subtasks in a joint framework, each subtask can be iteratively boosted by using the results of the others towards an overall optimal solution. Extensive experiments are performed to demonstrate the effectiveness of our method.Comment: Published in AAAI 201

LogDet Rank Minimization with Application to Subspace Clustering

Low-rank matrix is desired in many machine learning and computer vision problems. Most of the recent studies use the nuclear norm as a convex surrogate of the rank operator. However, all singular values are simply added together by the nuclear norm, and thus the rank may not be well approximated in practical problems. In this paper, we propose to use a log-determinant (LogDet) function as a smooth and closer, though non-convex, approximation to rank for obtaining a low-rank representation in subspace clustering. Augmented Lagrange multipliers strategy is applied to iteratively optimize the LogDet-based non-convex objective function on potentially large-scale data. By making use of the angular information of principal directions of the resultant low-rank representation, an affinity graph matrix is constructed for spectral clustering. Experimental results on motion segmentation and face clustering data demonstrate that the proposed method often outperforms state-of-the-art subspace clustering algorithms.Comment: 10 pages, 4 figure