Nonlinear mixture model motivated subspace clustering
Provides theoretical foundations for subspace clustering, enabling better post-processing of self-representation matrices, but the practical impact is incremental as it builds on existing models.
The paper derives the linear union-of-subspaces model for subspace clustering from a nonlinear mixture model used in blind source separation, establishing theoretical relationships that enable estimation of bounds on subspace dimension. Validation on six benchmark datasets using five algorithms confirms the theoretical results.
We derive the linear union-of-subspaces (UoS) model for subspace clustering (SC) from the nonlinear mixture model (NMM) used in blind source separation (BSS) to represent a D-dimensional observation vector as an unknown multivariate nonlinear mapping of C latent variables. Assuming the mapping is differentiable up to an unknown order K, we approximate NMM by a K-th order Taylor expansion, yielding a model equivalent to the linear UoS framework underlying SC. This establishes that: (i) the smoothness order K corresponds to the unknown subspace dimension d; (ii) KC equals the number of anchors; and (iii) the sparsity of the representation vector equals K (i.e., d). These relationships enable estimation of bounds on subspace dimension, and that is validated on six benchmark datasets using five established SC algorithms. Established theoretical results are important for post-processing of self-representation matrices estimated by SC algorithms.