Preserving the intrinsic structure of data is very important for unsupervised dimensionality reduction. For structure preserving, graph embedding technique is widely considered. However, most of the existing unsupervised graph embedding based methods cannot effectively preserve the intrinsic structure of data since these methods either use the constant graph or only explore the geometric structure based on the distance information or representation information. To solve this problem, a novel method, called locality preserving projection with symmetric graph embedding (LPP_SGE), is proposed. LPP_SGE introduces a novel adaptive graph learning model and can obtain the intrinsic graph and projection in a unified framework by fully exploring the representation information and distance information of the original data. Different from the existing works which generally introduce no less than two constraints to capture the representation information and distance information, LPP_SGE can simultaneously capture the above two kinds of structure information in one term. Moreover, LPP_SGE introduces an ‘l’ norm based projection constraint to select the most discriminative features from the complex data for dimensionality reduction, such that the robustness is enhanced. Experimental results on four databases and two kinds of noisy databases show that LPP_SGE performs better than many well-known methods.