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Network embedding is an effective technique to learn the low-dimensional representations of nodes in networks. Real-world networks are usually with multiplex or having multi-view representations from different relations. Recently, there has been increasing interest in network embedding on multiplex data. However, most existing multiplex approaches assume that the data is complete in all views. But in real applications, it is often the case that each view suffers from the missing of some data and therefore results in partial multiplex data. In this paper, we present a novel Deep Partial Multiplex Network Embedding approach to deal with incomplete data. In particular, the network embeddings are learned by simultaneously minimizing the deep reconstruction loss with the autoencoder neural network, enforcing the data consistency across views via common latent subspace learning, and preserving the data topological structure within the same network through graph Laplacian. We further prove the orthogonal invariant property of the learned embeddings and connect our approach with the binary embedding techniques. Experiments on four multiplex benchmarks demonstrate the superior performance of the proposed approach over several state-of-the-art methods on node classification, link prediction and clustering tasks.