Learning robust locality preserving projection via p-order minimization

Locality preserving projection (LPP) is an effective dimensionality reduction method based on manifold learning, which is defined over the graph weighted squared l₂-norm distances in the projected subspace. Since squared l₂-norm distance is prone to outliers, it is desirable to develop a robust LPP method. In this paper, motivated by existing studies that improve the robustness of statistical learning models via l₂-norm or not-squared l₂-norm formulations, we propose a robust LPP (rLPP) formulation to minimize the p-th order of the ¿2-norm distances, which can better tolerate large outlying data samples because it suppress the introduced biased more than the l₂-norm or not squared l₂-norm minimizations. However, solving the formulated objective is very challenging because it not only non-smooth but also non-convex. As an important theoretical contribution of this work, we systematically derive an efficient iterative algorithm to solve the general p-th order l₂-norm minimization problem, which, to the best of our knowledge, is solved for the first time in literature. Extensive empirical evaluations on the proposed rLPP method have been performed, in which our new method outperforms the related state-of-the-art methods in a variety of experimental settings and demonstrate its effectiveness in seeking better subspaces on both noiseless and noisy data.