Abstract We investigate the `1 -constrained representation which demonstrates robustness to quantization errors, utilizing the tool of deep learning. Based on the Alternating Direction Method of Multipliers (ADMM), we formulate the original convex minimization problem as a feed-forward neural network, named Deep `1 Encoder, by introducing the novel Bounded Linear Unit (BLU) neuron and modeling the Lagrange multipliers as network biases. Such a structural prior acts as an effective network regularization, and facilitates the model initialization. We then investigate the effective use of the proposed model in the application of hashing, by coupling the proposed encoders under a supervised pairwise loss, to develop a Deep Siamese `1 Network, which can be optimized from end to end. Extensive experiments demonstrate the impressive performances of the proposed model. We also provide an in-depth analysis of its behaviors against the competitors.