Differentiable Expectation-Maximization for Set Representation Learning

Minyoung Kim

Keywords: Representation learning, Bayesian models, Mixture estimation, Optimal transport, Attention

Abstract: We tackle the set2vec problem, the task of extracting a vector representation from an input set comprised of a variable number of feature vectors. Although recent approaches based on self attention such as (Set)Transformers were very successful due to the capability of capturing complex interaction between set elements, the computational overhead is the well-known downside. The inducing-point attention and the latest optimal transport kernel embedding (OTKE) are promising remedies that attain comparable or better performance with reduced computational cost, by incorporating a fixed number of learnable queries in attention. In this paper we approach the set2vec problem from a completely different perspective. The elements of an input set are considered as i.i.d.~samples from a mixture distribution, and we define our set embedding feed-forward network as the maximum-a-posterior (MAP) estimate of the mixture which is approximately attained by a few Expectation-Maximization (EM) steps. The whole MAP-EM steps are differentiable operations with a fixed number of mixture parameters, allowing efficient auto-diff back-propagation for any given downstream task. Furthermore, the proposed mixture set data fitting framework allows unsupervised set representation learning naturally via marginal likelihood maximization aka the empirical Bayes. Interestingly, we also find that OTKE can be seen as a special case of our framework, specifically a single-step EM with extra balanced assignment constraints on the E-step. Compared to OTKE, our approach provides more flexible set embedding as well as prior-induced model regularization. We evaluate our approach on various tasks demonstrating improved performance over the state-of-the-arts.

One-sentence Summary: We propose a novel set embedding function, a feed-forward network defined as the (differentiable) maximum-a-posterior estimate of the mixture, approximately attained by a few Expectation-Maximization steps.