Link prediction is a fundamental problem of data science, which usually calls for unfolding the mechanisms that govern the micro-dynamics of networks. In this regard, using features obtained from network embedding for predicting links has drawn widespread attention. Though edge features-based or node similarity-based methods have been proposed to solve the link prediction problem, many technical challenges still exist due to the unique structural properties of networks, especially when the networks are sparse. From the graph mining perspective, we first give empirical evidence of the inconsistency between heuristic and learned edge features. Then we propose a novel link prediction framework, AdaSim, by introducing an Adaptive Similarity function using features obtained from network embedding based on random walks. The node feature representations are obtained by optimizing a graph-based objective function. Instead of generating edge features using binary operators, we perform link prediction solely leveraging the node features of the network. We define a flexible similarity function with one tunable parameter, which serves as a penalty of the original similarity measure. The optimal value is learned through supervised learning thus is adaptive to data distribution. To evaluate the performance of our proposed algorithm, we conduct extensive experiments on eleven disparate networks of the real world. Experimental results show that AdaSim achieves better performance than state-of-the-art algorithms and is robust to different sparsities of the networks.