Deep neural networks can model images with rich latent representations, but they cannot naturally conceptualize structures of object categories in a human-perceptible way. This paper addresses the problem of learning object structures in an image modeling process without supervision. We propose an autoencoding formulation to discover landmarks as explicit structural representations. The encoding module outputs landmark coordinates, whose validity is ensured by constraints that reflect the necessary properties for landmarks. The decoding module takes the landmarks as a part of the learnable input representations in an end-to-end differentiable framework. Our discovered landmarks are semantically meaningful and more predictive of manually annotated landmarks than those discovered by previous methods. The coordinates of our landmarks are also complementary features to pretrained deep-neural-network representations in recognizing visual attributes. In addition, the proposed method naturally creates an unsupervised, perceptible interface to manipulate object shapes and decode images with controllable structures. The project webpage is at http://ytzhang.net/projects/lmdis-rep
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