LogDet Rank Minimization with Application to Subspace Clustering

TL;DR

This paper introduces a novel non-convex LogDet function to better approximate matrix rank for subspace clustering, leading to improved clustering performance over existing methods.

Contribution

It proposes a LogDet-based low-rank approximation method optimized via augmented Lagrange multipliers, enhancing subspace clustering accuracy.

Findings

Outperforms state-of-the-art algorithms in motion segmentation

Achieves superior face clustering results

Effective on large-scale data sets

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 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.