Simultaneous local clustering and unsupervised feature selection via strong space constraint

Clustering is a fashion method applied in machine learning tasks. However, high dimensional data brings many obstacles for clustering approaches. To address such a problem, the unsupervised feature selection (UFS) method can be incorporated into clustering to reduce dimensionality. In general, most of the UFS methods adopt ℓ_2,1-norm for subspace sparsity learning. However, its sparsity highly relies on the setting of trade-off parameter, which may lead to instability of ranking results and the difficulty in obtaining the optimal solution of projection matrix. In this paper, we propose to directly learn an absolutely row-sparsity subspace via the ℓ_2,0-norm constraint, called Sparse constraint and Local learning for Unsupervised Feature Selection (SLUFS). It is an ideal sparse subspace constraint which can overcome the drawbacks of the ℓ_2,1-norm. However, optimizing the ℓ_2,0-norm constraint is an NP-hard problem, and at present, only some approximate solutions can be given, but the convergence can not be guaranteed. To tackle this challenge, we design a novel alternative iterative algorithm to directly optimize the ℓ_2,0-norm based model. Most importantly, our strategy can obtain a closed-form solution with strict convergence guarantee. Comprehensive experiments are conducted on several real-world datasets to evaluate the performance of SLUFS with comparison to several related state-of-the-art methods.