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Feature allocation models are popular models used in different applications such as unsupervised learning or network modeling. In particular, the Indian buffet process is a flexible and simple one-parameter feature allocation model where the number of features grows unboundedly with the number of objects. The Indian buffet process, like most feature allocation models, satisfies a symmetry property of exchangeability: the distribution is invariant under permutation of the objects. While this property is desirable in some cases, it has some strong implications. Importantly, the number of objects sharing a particular feature grows linearly with the number of objects. In this article, we describe a class of non-exchangeable feature allocation models where the number of objects sharing a given feature grows sublinearly, where the rate can be controlled by a tuning parameter. We derive the asymptotic properties of the model, and show that such model provides a better fit and better predictive performances on various datasets.