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.

中文翻译：

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一般散列问题的等效连续公式


基于哈希的近似最近邻搜索由于其计算成本低和检索速度快而引起了广泛的研究兴趣。哈希技术将数据点映射为二进制代码，同时保留原始空间的相似性。一般来说，我们需要解决一个离散优化问题来学习二进制代码和哈希函数，这是NP-hard的。在文献中，大多数散列方法选择通过丢弃离散约束来解决宽松的问题。然而，这种松弛方案会导致较大的量化误差，从而使得学习到的二进制码的效果较差。在本文中，我们提出了离散哈希问题的等效连续公式。具体来说，我们证明离散哈希问题可以在没有任何松弛的情况下转化为连续优化问题，而转化后的连续优化问题与原始离散哈希问题具有相同的最优解和相同的最优值。改造后，可以应用持续优化的方法。我们设计了基于DC（凸函数差分）编程思想的算法来解决这个问题。所提出的连续哈希方案可以很容易地应用于现有的哈希模型，包括监督和无监督哈希。我们在几个基准上评估所提出的方法，结果表明所提出的方法与最先进的哈希方法相比具有优越性。