Permutation Invariant Likelihoods and Equivariant Transformations

In this work, we fill a substantial void in machine learning and statistical methodology by developing extensive generative density estimation techniques for exchangeable non-iid data. We do so through the use of permutation invariant likelihoods and permutation equivariant transformations of variables. These methods exploit the intradependencies within sets in ways that are independent of ordering (for likelihoods) or order preserving (for transformations). The proposed techniques are able to directly model exchangeable data (such as sets) without the need to account for permutations or assume independence of elements. We consider applications to point clouds and provide several interesting experiments on both synthetic and real-world datasets.