Extreme learning machine is characterized by less training parameters, fast training speed, and strong generalization ability. It has been applied to obtain feature representations from the complex data in the tasks of data clustering or classification. In this paper, a graph embedding-based denoising extreme learning machine autoencoder (GDELM-AE) is proposed for capturing the structure of the inputs. Specifically, in GDELM-AE, a graph embedding framework that contains an intrinsic graph and a penalty graph constructed by local Fisher discrimination analysis is integrated into the autoencoder. So, it can exploit both local structure and global structure information in extreme learning machine (ELM) spaces. Further, we propose a stacked graph embedded denoising (SGD)-ELM by stacking several GDELM-AEs. The experimental results on several benchmarks validate that GDELM-AE can obtain efficient and robust feature representation of original data; moreover, the stacked GDELM-AE can obtain high-level and noise-robust representations. The comparative results with the state-of-the-art algorithms indicate that the proposed algorithm can obtain better accuracy as well as faster training speed.