Recommendation algorithms that incorporate techniques from deep learning are becoming increasingly popular. Due to the structure of the data coming from recommendation domains (i.e., one-hot-encoded vectors of item preferences), these algorithms tend to have large input and output dimensionalities that dominate their overall size. This makes them difficult to train, due to the limited memory of graphical processing units, and difficult to deploy on mobile devices with limited hardware. To address these difficulties, we propose Bloom embeddings, a compression technique that can be applied to the input and output of neural network models dealing with sparse high-dimensional binary-coded instances. Bloom embeddings are computationally efficient, and do not seriously compromise the accuracy of the model up to 1/5 compression ratios. In some cases, they even improve over the original accuracy, with relative increases up to 12%. We evaluate Bloom embeddings on 7 data sets and compare it against 4 alternative methods, obtaining favorable results. We also discuss a number of further advantages of Bloom embeddings, such as 'on-the-fly' constant-time operation, zero or marginal space requirements, training time speedups, or the fact that they do not require any change to the core model architecture or training configuration.