A Randomized Sparsity Preserving Projection and its Exponential Approach for Image Recognition

To preserve the sparse representation of the original high-dimensional data, sparsity preserving projection (SPP) is proposed which involves solving a series of
$\ell _{1}$
minimization problems and a large-scale generalized eigenvalue problem. Compared to many other dimensionality reduction techniques, SPP does not contain any model parameters. Besides, it may have some discriminative abilities even though no class labels are provided. However, SPP tends to show signs of fatigue when encountering with the large-scale high-dimensional data sets since the process of solving the
$\ell _{1}$
minimization problems is quite time-consuming. To manage this problem, this paper proposes a randomized SPP (RSPP) via random sketching where the
$\ell _{1}$
minimization problems can be solved efficiently. Furthermore, a preprocessed approach and an exponential approach with a bit of extra efforts are presented to handle the small-sample-size problem which may be occurred in RSPP. Finally, some experiments on image recognition for four authentic data sets are implemented and the corresponding results demonstrate the superiority of the proposed methods.