Leveraging the framework of Optimal Transport, we introduce a new family of generative autoencoders with a learnable prior, called Symmetric Wasserstein Autoencoders (SWAEs). We propose to symmetrically match the joint distributions of the observed data and the latent representation induced by the encoder and the decoder. The resulting algorithm jointly optimizes the modelling losses in both the data and the latent spaces with the loss in the data space leading to the denoising effect. With the symmetric treatment of the data and the latent representation, the algorithm implicitly preserves the local structure of the data in the latent space. To further improve the quality of the latent representation, we incorporate a reconstruction loss into the objective, which significantly benefits both the generation and reconstruction. We empirically show the superior performance of SWAEs over the state-of-the-art generative autoencoders in terms of classification, reconstruction, and generation.
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