We are born with the ability to learn concepts by comparing diverse observations. This helps us to understand the new world in a compositional manner and facilitates extrapolation, as objects naturally consist of multiple concepts. In this work, we argue that the cognitive mechanism of comparison, fundamental to human learning, is also vital for machines to recover true concepts underlying the data. This offers correctness guarantees for the field of concept learning, which, despite its impressive empirical successes, still lacks general theoretical support. Specifically, we aim to develop a theoretical framework for the identifiability of concepts with multiple classes of observations. We show that with sufficient diversity across classes, hidden concepts can be identified without assuming specific concept types, functional relations, or parametric generative models. Interestingly, even when conditions are not globally satisfied, we can still provide alternative guarantees for as many concepts as possible based on local comparisons, thereby extending the applicability of our theory to more flexible scenarios. Moreover, the hidden structure between classes and concepts can also be identified nonparametrically. We validate our theoretical results in both synthetic and real-world settings.

We begin by congratulating Yixin Wang and David Blei for their thought-provoking article that opens up a new research frontier in the field of causal inference. The authors directly tackle the challenging question of how to infer causal effects of many treatments in the presence of unmeasured confounding. We expect their article to have a major impact by further advancing our understanding of this important methodological problem. This commentary has two goals. We then briefly consider three possible ways to address some of the limitations of the deconfounder method. We first discuss several advantages offered by the deconfounder method. We then examine the assumptions required by the method and discuss its limitations.