In the field of recommender systems, shallow autoencoders have recently gained significant attention. One of the most highly acclaimed shallow autoencoders is EASE, favored for its competitive recommendation accuracy and simultaneous simplicity. However, the poor scalability of EASE (both in time and especially in memory) severely restricts its use in production environments with vast item sets. In this paper, we propose a hyperefficient factorization technique for sparse approximate inversion of the data-Gram matrix used in EASE. The resulting autoencoder, SANSA, is an end-to-end sparse solution with prescribable density and almost arbitrarily low memory requirements (even for training). As such, SANSA allows us to effortlessly scale the concept of EASE to millions of items and beyond.