Block diagonal representation learning for robust subspace clustering

Convex Subspace Clustering by Adaptive Block Diagonal Representation

Subspace clustering is a class of extensively studied clustering methods where the spectral-type approaches are its important subclass. Its key first step is to desire learning a representation coefficient matrix with block diagonal structure. To realize this step, many methods were successively proposed by imposing different structure priors on the coefficient matrix. These impositions can be roughly divided into two categories, i.e., indirect and direct. The former introduces the priors such as sparsity and low rankness to indirectly or implicitly learn the block diagonal structure. However, the desired block diagonalty cannot necessarily be guaranteed for noisy data. While the latter directly or explicitly imposes the block diagonal structure prior such as block diagonal representation (BDR) to ensure so-desired block diagonalty even if the data is noisy but at the expense of losing the convexity that the former's objective possesses. For compensating their respective shortcomings, in this article, we follow the direct line to propose adaptive BDR (ABDR) which explicitly pursues block diagonalty without sacrificing the convexity of the indirect one. Specifically, inspired by Convex BiClustering, ABDR coercively fuses both columns and rows of the coefficient matrix via a specially designed convex regularizer, thus naturally enjoying their merits and adaptively obtaining the number of blocks. Finally, experimental results on synthetic and real benchmarks demonstrate the superiority of ABDR to the state-of-the-arts (SOTAs).

Latent Low-Rank Representation With Weighted Distance Penalty for Clustering

Latent low-rank representation (LatLRR) is a critical self-representation technique that improves low-rank representation (LRR) by using observed and unobserved samples. It can simultaneously learn the low-dimensional structure embedded in the data space and capture the salient features. However, LatLRR ignores the local geometry structure and can be affected by the noise and redundancy in the original data space. To solve the above problems, we propose a latent LRR with weighted distance penalty (LLRRWD) for clustering in this article. First, a weighted distance is proposed to enhance the original Euclidean distance by enlarging the distance among the unconnected samples, which can enhance the discriminitation of the distance among the samples. By leveraging on the weighted distance, a weighted distance penalty is introduced to the LatLRR model to enable the method to preserve both the local geometric information and global information, improving discrimination of the learned affinity matrix. Moreover, a weight matrix is imposed on the sparse error norm to reduce the effect of noise and redundancy. Experimental results based on several benchmark databases show the effectiveness of our method in clustering.

Block Diagonal Least Squares Regression for Subspace Clustering

Least squares regression (LSR) is an effective method that has been widely used for subspace clustering. Under the conditions of independent subspaces and noise-free data, coefficient matrices can satisfy enforced block diagonal (EBD) structures and achieve good clustering results. More importantly, LSR produces closed solutions that are easier to solve. However, solutions with block diagonal properties that have been solved using LSR are sensitive to noise or corruption as they are fragile and easily destroyed. Moreover, when using actual datasets, these structures cannot always guarantee satisfactory clustering results. Considering that block diagonal representation has excellent clustering performance, the idea of block diagonal constraints has been introduced into LSR and a new subspace clustering method, which is named block diagonal least squares regression (BDLSR), has been proposed. By using a block diagonal regularizer, BDLSR can effectively reinforce the fragile block diagonal structures of the obtained matrices and improve the clustering performance. Our experiments using several real datasets illustrated that BDLSR produced a higher clustering performance compared to other algorithms.

A Robust Tensor-Based Submodule Clustering for Imaging Data Using l12 Regularization and Simultaneous Noise Recovery via Sparse and Low Rank Decomposition Approach

The massive generation of data, which includes images and videos, has made data management, analysis, information extraction difficult in recent years. To gather relevant information, this large amount of data needs to be grouped. Real-life data may be noise corrupted during data collection or transmission, and the majority of them are unlabeled, allowing for the use of robust unsupervised clustering techniques. Traditional clustering techniques, which vectorize the images, are unable to keep the geometrical structure of the images. Hence, a robust tensor-based submodule clustering method based on l12 regularization with improved clustering capability is formulated. The l12 induced tensor nuclear norm (TNN), integrated into the proposed method, offers better low rankness while retaining the self-expressiveness property of submodules. Unlike existing methods, the proposed method employs a simultaneous noise removal technique by twisting the lateral image slices of the input data tensor into frontal slices and eliminates the noise content in each image, using the principles of the sparse and low rank decomposition technique. Experiments are carried out over three datasets with varying amounts of sparse, Gaussian and salt and pepper noise. The experimental results demonstrate the superior performance of the proposed method over the existing state-of-the-art methods.