Low rank subspace clustering (LRSC)

Abstract

a b s t r a c tWe consider the problem of fitting a union of subspaces to a collection of data points drawn from one or more subspaces and corrupted by noise and/or gross errors. We pose this problem as a non-convex optimization problem, where the goal is to decompose the corrupted data matrix as the sum of a clean and self-expressive dictionary plus a matrix of noise and/or gross errors. By self-expressive we mean a dictionary whose atoms can be expressed as linear combinations of themselves with low-rank coefficients. In the case of noisy data, our key contribution is to show that this non-convex matrix decomposition problem can be solved in closed form from the SVD of the noisy data matrix. The solution involves a novel polynomial thresholding operator on the singular values of the data matrix, which requires minimal shrinkage. For one subspace, a particular case of our framework leads to classical PCA, which requires no shrinkage. For multiple subspaces, the low-rank coefficients obtained by our framework can be used to construct a data affinity matrix from which the clustering of the data according to the subspaces can be obtained by spectral clustering. In the case of data corrupted by gross errors, we solve the problem using an alternating minimization approach, which combines our polynomial thresholding operator with the more traditional shrinkage-thresholding operator. Experiments on motion segmentation and face clustering show that our framework performs on par with state-of-the-art techniques at a reduced computational cost.

Determining a Flexible Low-Rank Graph Using Subspace Clustering

“…As of late, enlivened by the advances of SSC and LRR, numerous chart based subspace grouping calculations have been developed [13] [19][4] [14]. For instance, to protect the complex structure of information, Cai et.…”

“…fused Laplacain diagram requirements into the standard LRR model to safeguard the geometric data from single side [14] and twoside [4] viewpoint. When all is said in done, this kind of subspace bunching can be assembled into ghostly grouping based strategies, which have been exhibited to perform extremely well for some applications in PC vision [19]. On a fundamental level, this sort of subspace bunching calculations comprises of two stages.…”

Symmetric low-rank representation for subspace clustering

“…Subspace clustering is one of the fundamental topics in machine learning, computer vision, and pattern recognition, e.g., image representation [1,2], face clustering [2][3][4], and motion segmentation [5][6][7][8][9]. The importance of subspace clustering is evident in the vast amount of literature thereon, because it is a crucial step in inferring structure information of data from subspaces through data analysis [10][11][12].…”

“…Low-rank representation based techniques have been proposed to address these drawbacks [2,9,34]. Liu et al [2] proposed the low-rank representation (LRR) method to learn a lowrank representation of data by capturing the global structure of the data.…”

“…Although an inexact variation of the augmented Lagrange multiplier (ALM) method [35,36], which is used to solve the optimization problem in LRR, performs well, and generally converges adequately in many practical applications, its convergence property still lacks a theoretical guarantee. Vidala and Favarob [9] considered low-rank subspace clustering (LRSC) as a non-convex matrix decomposition problem, which can be solved in closed form using SVD of the noisy data matrix. Although LRSC can be carried out on data contaminated by noise with reduced computational cost, the clustering performance could be seriously degraded owing to the presence of such corrupted data.…”