Most feature selection methods first compute a similarity matrix by assigning a fixed value to pairs of objects in the whole data or to pairs of objects in a class or by computing the similarity between two objects from the original data. The similarity matrix is fixed as a constant in the subsequent feature selection process. However, the similarities computed from the original data may be unreliable, because they are affected by noise features. Moreover, the local structure within classes cannot be recovered if the similarities between the pairs of objects in a class are equal. In this paper, we propose a novel local adaptive projection (LAP) framework. Instead of computing fixed similarities before performing feature selection, LAP simultaneously learns an adaptive similarity matrix and a projection matrix with an iterative method. In each iteration, is computed from the projected distance with the learned and W is computed with the learned . Therefore, LAP can learn better projection matrix by weakening the effect of noise features with the adaptive similarity matrix. A supervised feature selection with LAP (SLAP) method and an unsupervised feature selection with LAP (ULAP) method are proposed. Experimental results on eight data sets show the superiority of SLAP compared with seven supervised feature selection methods and the superiority of ULAP compared with five unsupervised feature selection methods.