A nonconvex formulation for low rank subspace clustering: algorithms and convergence analysis

2018 ◽  
Vol 70 (2) ◽  
pp. 395-418 ◽  
Author(s):  
Hao Jiang ◽  
Daniel P. Robinson ◽  
René Vidal ◽  
Chong You
Keyword(s):  
Convergence Analysis ◽  
Clustering Algorithms ◽  
Subspace Clustering ◽  
Low Rank

scholarly journals Low-rank sparse subspace clustering with a clean dictionary

2021 ◽  
Vol 15 ◽  
pp. 174830262199962
Author(s):  
Cong-Zhe You ◽  
Zhen-Qiu Shu ◽  
Hong-Hui Fan
Keyword(s):  
Optimization Problem ◽  
Clustering Algorithms ◽  
Subspace Clustering ◽  
Low Rank ◽  
Data Matrix ◽  
Image Clustering ◽  
Convex Optimization Problem ◽  
Data Points ◽  
Low Rank Representation ◽  
Sparse Subspace Clustering

Low-Rank Representation (LRR) and Sparse Subspace Clustering (SSC) are considered as the hot topics of subspace clustering algorithms. SSC induces the sparsity through minimizing the l1-norm of the data matrix while LRR promotes a low-rank structure through minimizing the nuclear norm. In this paper, considering the problem of fitting a union of subspace to a collection of data points drawn from one more subspaces and corrupted by noise, 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. We propose a new algorithm, named Low-Rank and Sparse Subspace Clustering with a Clean dictionary (LRS2C2), by combining SSC and LRR, as the representation is often both sparse and low-rank. The effectiveness of the proposed algorithm is demonstrated through experiments on motion segmentation and image clustering.

scholarly journals LogDet Rank Minimization with Application to Subspace Clustering

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Zhao Kang ◽  
Chong Peng ◽  
Jie Cheng ◽  
Qiang Cheng
Keyword(s):  
Large Scale ◽  
Clustering Algorithms ◽  
Subspace Clustering ◽  
Singular Values ◽  
Nuclear Norm ◽  
Low Rank ◽  
Face Clustering ◽  
Rank Matrix ◽  
Low Rank Representation ◽  
Clustering Data

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.

scholarly journals Low-rank sparse subspace clustering with a clean dictionary

2021 ◽  
Vol 15 ◽  
pp. 174830262098369
Author(s):  
Cong-Zhe You ◽  
Zhen-Qiu Shu ◽  
Hong-Hui Fan
Keyword(s):  
Optimization Problem ◽  
Clustering Algorithms ◽  
Subspace Clustering ◽  
Low Rank ◽  
Data Matrix ◽  
Image Clustering ◽  
Convex Optimization Problem ◽  
Data Points ◽  
Low Rank Representation ◽  
Sparse Subspace Clustering

Low-Rank Representation (LRR) and Sparse Subspace Clustering (SSC) are considered as the hot topics of subspace clustering algorithms. SSC induces the sparsity through minimizing the [Formula: see text]-norm of the data matrix while LRR promotes a low-rank structure through minimizing the nuclear norm. In this paper, considering the problem of fitting a union of subspace to a collection of data points drawn from one more subspaces and corrupted by noise, 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. We propose a new algorithm, named Low-Rank and Sparse Subspace Clustering with a Clean dictionary (LRS2C2), by combining SSC and LRR, as the representation is often both sparse and low-rank. The effectiveness of the proposed algorithm is demonstrated through experiments on motion segmentation and image clustering.

Low-Rank And Sparse Tensor Representation For Multi-View Subspace Clustering

2021 ◽  
Author(s):  
Shuqin Wang ◽  
Yongyong Chen ◽  
Yigang Ce ◽  
Linna Zhang ◽  
Viacheslav Voronin
Keyword(s):  
Subspace Clustering ◽  
Low Rank ◽  
Tensor Representation

Low-Rank Tensor Graph Learning for Multi-view Subspace Clustering

2021 ◽  
pp. 1-1
Author(s):  
Yongyong Chen ◽  
Xiaolin Xiao ◽  
Chong Peng ◽  
Guangming Lu ◽  
Yicong Zhou
Keyword(s):  
Subspace Clustering ◽  
Low Rank ◽  
Graph Learning ◽  
Rank Tensor

scholarly journals Subspace clustering using a symmetric low-rank representation

2017 ◽  
Vol 127 ◽  
pp. 46-57 ◽  
Author(s):  
Jie Chen ◽  
Hua Mao ◽  
Yongsheng Sang ◽  
Zhang Yi
Keyword(s):  
Subspace Clustering ◽  
Low Rank ◽  
Low Rank Representation

Multiview Common Subspace Clustering via Coupled Low Rank Representation

2021 ◽  
Vol 12 (4) ◽  
pp. 1-25
Author(s):  
Stanley Ebhohimhen Abhadiomhen ◽  
Zhiyang Wang ◽  
Xiangjun Shen ◽  
Jianping Fan
Keyword(s):  
Common Knowledge ◽  
Subspace Clustering ◽  
Laplacian Matrix ◽  
Low Rank ◽  
Connected Components ◽  
Similarity Matrix ◽  
Benchmark Datasets ◽  
Low Rank Representation ◽  
Low Dimensional ◽  
Shared Structure

Multi-view subspace clustering (MVSC) finds a shared structure in latent low-dimensional subspaces of multi-view data to enhance clustering performance. Nonetheless, we observe that most existing MVSC methods neglect the diversity in multi-view data by considering only the common knowledge to find a shared structure either directly or by merging different similarity matrices learned for each view. In the presence of noise, this predefined shared structure becomes a biased representation of the different views. Thus, in this article, we propose a MVSC method based on coupled low-rank representation to address the above limitation. Our method first obtains a low-rank representation for each view, constrained to be a linear combination of the view-specific representation and the shared representation by simultaneously encouraging the sparsity of view-specific one. Then, it uses the k -block diagonal regularizer to learn a manifold recovery matrix for each view through respective low-rank matrices to recover more manifold structures from them. In this way, the proposed method can find an ideal similarity matrix by approximating clustering projection matrices obtained from the recovery structures. Hence, this similarity matrix denotes our clustering structure with exactly k connected components by applying a rank constraint on the similarity matrix’s relaxed Laplacian matrix to avoid spectral post-processing of the low-dimensional embedding matrix. The core of our idea is such that we introduce dynamic approximation into the low-rank representation to allow the clustering structure and the shared representation to guide each other to learn cleaner low-rank matrices that would lead to a better clustering structure. Therefore, our approach is notably different from existing methods in which the local manifold structure of data is captured in advance. Extensive experiments on six benchmark datasets show that our method outperforms 10 similar state-of-the-art compared methods in six evaluation metrics.

Subspace Clustering of High Dimensional Data Using Differential Evolution

2019 ◽  
pp. 47-74 ◽  
Author(s):  
Parul Agarwal ◽  
Shikha Mehta
Keyword(s):  
Differential Evolution ◽  
Distance Measure ◽  
Dimensional Space ◽  
Clustering Algorithms ◽  
High Dimensional Data ◽  
Subspace Clustering ◽  
High Dimensional ◽  
Dbscan Clustering ◽  
Evolution Algorithms ◽  
Self Adaptive

Subspace clustering approaches cluster high dimensional data in different subspaces. It means grouping the data with different relevant subsets of dimensions. This technique has become very effective as a distance measure becomes ineffective in a high dimensional space. This chapter presents a novel evolutionary approach to a bottom up subspace clustering SUBSPACE_DE which is scalable to high dimensional data. SUBSPACE_DE uses a self-adaptive DBSCAN algorithm to perform clustering in data instances of each attribute and maximal subspaces. Self-adaptive DBSCAN clustering algorithms accept input from differential evolution algorithms. The proposed SUBSPACE_DE algorithm is tested on 14 datasets, both real and synthetic. It is compared with 11 existing subspace clustering algorithms. Evaluation metrics such as F1_Measure and accuracy are used. Performance analysis of the proposed algorithms is considerably better on a success rate ratio ranking in both accuracy and F1_Measure. SUBSPACE_DE also has potential scalability on high dimensional datasets.

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