Weighted Sparse Subspace Representation: A Unified Framework for Subspace Clustering, Constrained Clustering, and Active Learning

In many challenging real-world applications involving the grouping of high-dimensional data, different clusters can be well approximated as lower dimensional subspaces. This is the case for example in gene sequencing (McWilliams and Montana, 2014), face clustering (Elhamifar and Vidal, 2013), motion segmentation (Rao et al., 2010), and text mining (Peng et al., 2018). The problem of simultaneously estimating the subspace corresponding to each cluster and partitioning a group of points into clusters according to these subspaces is called subspace clustering (Vidal, 2011). Spectral methods for subspace clustering have demonstrated excellent performance in numerous real-world applications (Liu et al., 2012; Lu et al., 2012; Elhamifar and Vidal, 2013; Li and Vidal, 2015; Huang et al., 2015). These methods construct an affinity matrix for spectral clustering by solving an optimisation problem that aims to approximate each point through a linear combination of other points from the same subspace.