Abstract: Hashing methods and apparatus for large-scale search learn a similarity-preserving transformation from feature space to a lower-dimensional binary space. The resulting binary codes are more compact to store than feature vectors and can be rapidly searched. The hashing methods may be performed without introducing continuous relaxations. Apparatus as described herein comprises a deep neural network that may be trained in a discrete optimization framework without continuous relaxations. The methods and apparatus can be applied to efficiently search image collections without class labels.