Suppose certain data points are overly contaminated, then the existing principal component analysis (PCA) methods are frequently incapable of filtering out and eliminating the excessively polluted ones, which potentially lead to the functional degeneration of the corresponding models. To tackle the issue, we propose a general framework namely robust weight learning with adaptive neighbors (RWL-AN), via which adaptive weight vector is automatically obtained with both robustness and sparse neighbors. More significantly, the degree of the sparsity is steerable such that only exact k well-fitting samples with least reconstruction errors are activated during the optimization, while the residual samples, i.e., the extreme noised ones are eliminated for the global robustness. Additionally, the framework is further applied to PCA problem to demonstrate the superiority and effectiveness of the proposed RWL-AN model.