Convex Subspace Clustering by Adaptive Block Diagonal Representation
This work addresses subspace clustering, a key task in machine learning for data analysis, by combining the benefits of direct and indirect methods, though it appears incremental as it builds on existing approaches like Convex BiClustering.
The paper tackles the problem of subspace clustering by proposing Adaptive Block Diagonal Representation (ABDR), which explicitly learns a block diagonal coefficient matrix while maintaining convexity, and experimental results show it outperforms state-of-the-art methods on synthetic and real benchmarks.
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 paper, we follow the direct line to propose Adaptive Block Diagonal Representation (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).