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.

本文提出了一种使用 LogDet 函数来近似低秩表示和相似矩阵构造的方法，并利用 Augmented Lagrange Multipliers 进行优化，该方法在运动分割和人脸聚类数据上的实验结果表明其优于当前先进的子空间聚类算法。

基于对数行列式秩最小化的子空间聚类应用

LogDet Rank Minimization with Application to Subspace Clustering

This paper proposes scalable and fast algorithms for solving the Robust PCA
problem, namely recovering a low-rank matrix with an unknown fraction of its
entries being arbitrarily corrupted. This problem arises in many applications,
such as image processing, web data ranking, and bioinformatic data analysis. It
was recently shown that under surprisingly broad conditions, the Robust PCA
problem can be exactly solved via convex optimization that minimizes a
combination of the nuclear norm and the $\ell^1$-norm . In this paper, we apply
the method of augmented Lagrange multipliers (ALM) to solve this convex
program. As the objective function is non-smooth, we show how to extend the
classical analysis of ALM to such new objective functions and prove the
optimality of the proposed algorithms and characterize their convergence rate.
Empirically, the proposed new algorithms can be more than five times faster
than the previous state-of-the-art algorithms for Robust PCA, such as the
accelerated proximal gradient (APG) algorithm. Moreover, the new algorithms
achieve higher precision, yet being less storage/memory demanding. We also show
that the ALM technique can be used to solve the (related but somewhat simpler)
matrix completion problem and obtain rather promising results too. We further
prove the necessary and sufficient condition for the inexact ALM to converge
globally. Matlab code of all algorithms discussed are available at
this http URL

本文提出了一些可扩展且快速的算法来解决鲁棒 PCA 问题，即恢复具有未知一部分被任意损坏的低秩矩阵，通过优化核范数和  l1 范数的组合实现凸优化进行解决该问题，并利用增广拉格朗日乘数法来解决此凸问题。提出的新算法比先前的最先进的 Robust PCA 算法快五倍以上，达到更高精度，且存储 / 内存需求更少，并证明了不精确增广拉格朗日乘数法全局收敛必要和充分条件。