In this work, we present a novel method to learn a local cross-domain descriptor for 2D image and 3D point cloud matching. Our proposed method is a dual auto-encoder neural network that maps 2D and 3D input into a shared latent space representation. We show that such local cross-domain descriptors in the shared embedding are more discriminative than those obtained from individual training in 2D and 3D domains. To facilitate the training process, we built a new dataset by collecting $approx 1.4$ millions of 2D-3D correspondences with various lighting conditions and settings from publicly available RGB-D scenes. Our descriptor is evaluated in three main experiments: 2D-3D matching, cross-domain retrieval, and sparse-to-dense depth estimation. Experimental results confirm the robustness of our approach as well as its competitive performance not only in solving cross-domain tasks but also in being able to generalize to solve sole 2D and 3D tasks. Our dataset and code are released publicly at url{https://hkust-vgd.github.io/lcd}.
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