In this paper, we propose an arbitrarily-conditioned data imputation framework built upon variational autoencoders and normalizing flows. The proposed model is capable of mapping any partial data to a multi-modal latent variational distribution. Sampling from such a distribution leads to stochastic imputation. Preliminary evaluation on MNIST dataset shows promising stochastic imputation conditioned on partial images as input. Neural network based algorithms have been shown effective and promising for various downstream tasks including classification (Deng et al., 2009; Damianou and Lawrence, 2013) , retrieval (Carvalho et al., 2018) , prediction (He et al., 2018) , and more. In order to correctly learn how to perform these tasks, they usually rely strictly on access to fully-observed data. However, acquiring this type of data in real life requires tremendous human effort, limiting the applicability of this family of models. Having a framework designed to perform inference on partially-observed data will not only alleviate the aforementioned constraint, but also open possibilities to perform data imputation, in which the missing data is inferred. Data imputation, also referred to conditional generation, has been an active research area (Little and Rubin, 1986; Song et al., 2018; Zadeh et al., 2019) . The probabilistic nature of this task makes it difficult to adopt off-the-shelf deterministic models widely studied. In other words, conditioned on the same partially-observed data as input, multiple plausible fully-observed data should be able to be imputed. Variational autoencoders (VAEs) (Kingma and Welling, 2013) , as a popular probabilistic modelling approach, have been applied to the data imputation task recently. A variational autoencoder defines a generative process that jointly models the distribution p θ (x, z) of the observed variable x and latent variable z, governed by parameters θ. Instead of performing local inference, VAEs include an inference network parameterized by φ to output an approximate posterior distribution q φ (z|x). Both the generative model and the inference model are optimized with a unified evidence lower bound (ELBO) on marginal data likelihood: . Recent literature on utilizing VAEbased models mainly focus on the effectiveness of combination of various obversed parts (Ma et al., 2019; Ivanov et al., 2018) . Different from the related works described above, we propose to enrich the latent space of variational autoencoders to enable multi-modal posterior inference, and therefore probabilistic imputation. Specifically, we use a two-stage model, with first-stage focusing on learning a representation space based on fully-observed data, and second-stage focusing on aligning the representation space embedded from partially-observed data to the one in stage-one. Using flow-based transformations for constructing a rich latent distribution, the proposed model is capable of inferring multi-modal variational latent distributions.
We propose an arbitrarily-conditioned data imputation framework built upon variational autoencoders and normalizing flows
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