Dimensionality reduction via kernel sparse representation

Dimensionality reduction (DR) methods based on sparse representation as one of the hottest research topics have achieved remarkable performance in many applications in recent years. However, it’s a challenge for existing sparse representation based methods to solve nonlinear problem due to the limitations of seeking sparse representation of data in the original space. Motivated by kernel tricks, we proposed a new framework called empirical kernel sparse representation (EKSR) to solve nonlinear problem. In this framework, nonlinear separable data are mapped into kernel space in which the nonlinear similarity can be captured, and then the data in kernel space is reconstructed by sparse representation to preserve the sparse structure, which is obtained by minimizing a ℓ1 regularization-related objective function. EKSR provides new insights into dimensionality reduction and extends two models: 1) empirical kernel sparsity preserving projection (EKSPP), which is a feature extraction method based on sparsity preserving projection (SPP); 2) empirical kernel sparsity score (EKSS), which is a feature selection method based on sparsity score (SS). Both of the two methods can choose neighborhood automatically as the natural discriminative power of sparse representation. Compared with several existing approaches, the proposed framework can reduce computational complexity and be more convenient in practice.