  Multiple modalities often co-occur when describing natural phenomena.
Learning a joint representation of these modalities should yield deeper and
more useful representations. Previous generative approaches to multi-modal
input either do not learn a joint distribution or require additional
computation to handle missing data. Here, we introduce a multimodal variational
autoencoder (MVAE) that uses a product-of-experts inference network and a
sub-sampled training paradigm to solve the multi-modal inference problem.
Notably, our model shares parameters to efficiently learn under any combination
of missing modalities. We apply the MVAE on four datasets and match
state-of-the-art performance using many fewer parameters. In addition, we show
that the MVAE is directly applicable to weakly-supervised learning, and is
robust to incomplete supervision. We then consider two case studies, one of
learning image transformations---edge detection, colorization,
segmentation---as a set of modalities, followed by one of machine translation
between two languages. We find appealing results across this range of tasks.
