Equivalent Continuous Formulation of General Hashing Problem.

Hashing-based approximate nearest neighbors search has attracted broad research interest, due to its low computational cost and fast retrieval speed. The hashing technique maps the data points into binary codes and, meanwhile, preserves the similarity in the original space. Generally, we need to solve a discrete optimization problem to learn the binary codes and hash functions, which is NP-hard. In the literature, most hashing methods choose to solve a relaxed problem by discarding the discrete constraints. However, such a relaxation scheme will cause large quantization error, which makes the learned binary codes less effective. In this paper, we present an equivalent continuous formulation of the discrete hashing problem. Specifically, we show that the discrete hashing problem can be transformed into a continuous optimization problem without any relaxations, while the transformed continuous optimization problem has the same optimal solutions and the same optimal value as the original discrete hashing problem. After transformation, the continuous optimization methods can be applied. We devise the algorithms based on the idea of DC (difference of convex functions) programming to solve this problem. The proposed continuous hashing scheme can be easily applied to the existing hashing models, including both supervised and unsupervised hashing. We evaluate the proposed method on several benchmarks and the results show the superiority of the proposed method compared with the state-of-the-art hashing methods.