Abnormal miRNA functions are widely involved in many diseases recorded in the database of experimentally supported human miRNA-disease associations (HMDD). Some of the associations are complicated: There can be up to five heterogeneous association types of miRNA with the same disease, including genetics type, epigenetics type, circulating miRNAs type, miRNA tissue expression type and miRNA-target interaction type. When one type of association is known for an miRNA-disease pair, it is important to predict any other types of the association for a better understanding of the disease mechanism. It is even more important to reveal associations for currently unassociated miRNAs and diseases. Methods have been recently proposed to make predictions on the association types of miRNA-disease pairs through restricted Boltzman machines, label propagation theories and tensor completion algorithms. None of them has exploited the non-linear characteristics in the miRNA-disease association network to improve the performance. We propose to use attributed multi-layer heterogeneous network embedding to learn the latent representations of miRNAs and diseases from each association type and then to predict the existence of the association type for all the miRNA-disease pairs. The performance of our method is compared with two newest methods via 10-fold cross-validation on the database HMDD v3.2 to demonstrate the superior prediction achieved by our method under different settings. Moreover, our real predictions made beyond the HMDD database can be all validated by NCBI literatures, confirming that our method is capable of accurately predicting new associations of miRNAs with diseases and their association types as well.