Abstract
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 using a log-determinant (LogDet) function as a smooth and closer, though nonconvex, approximation to rank for obtaining a low-rank representation in subspace clustering. Augmented Lagrange multipliers strategy is applied to iteratively optimize the LogDet-based nonconvex 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.
Highlights
Matrix rank minimizing [1] is ubiquitous in machine learning, computer vision, control, signal processing, and system identification
Subspace clustering is an intrinsically difficult problem, since we need to simultaneously cluster all data points into multiple groups and find a low-dimensional subspace fitting each group of points
We propose using a log-determinant (LogDet) function for rank approximation and study its minimization in subspace clustering
Summary
Matrix rank minimizing [1] is ubiquitous in machine learning, computer vision, control, signal processing, and system identification. Low-rank representation based subspace clustering [2,3,4] and matrix completion [5, 6] methods have achieved great success recently. Spectral clustering-based methods have achieved promising performance, where the key is to learn a good affinity matrix of data points. Some spectral clustering-based methods, such as sparse representation (SSC) [10] and low-rank representation (LRR) [3], have been proposed to obtain state-of-theart results in subspace clustering. By employing a rather simple formulation based on the LogDet function, we obtain a low-rank representation for subspace clustering. When used for subspace clustering, our simple formulation shows favorable performance compared to other state-of-theart methods, we do not explicitly account for outliers in our model This demonstrates the robustness of our approach.
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