An effective unsupervised hashing algorithm leads to compact binary codes preserving the neighborhood structure of data as much as possible. One of the most established schemes for unsupervised hashing is to reduce the dimensionality of data and then find a rigid (neighborhood-preserving) transformation that reduces the quantization error. Although employing rigid transformations is effective, we may not reduce quantization loss to the ultimate limits. As well, reducing dimensionality and quantization loss in two separate steps seems to be sub-optimal. Motivated by these shortcomings, we propose to employ both rigid and non-rigid transformations to reduce quantization error and dimensionality simultaneously. We relax the orthogonality constraint on the projection in a PCA-formulation and regularize this by a quantization term. We show that both the non-rigid projection matrix and rotation matrix contribute towards minimizing quantization loss but in different ways. A scalable nested coordinate descent approach is proposed to optimize this mixed-integer optimization problem. We evaluate the proposed method on five public benchmark datasets providing almost half a million images. Comparative results indicate that the proposed method mostly outperforms state-of-art linear methods and competes with end-to-end deep solutions.

一种有效的无监督散列算法会产生紧凑的二进制代码，尽可能多地保留数据的邻域结构。最成熟的无监督散列方案之一是降低数据的维数，然后找到一种减少量化误差的刚性（保持邻域）变换。尽管采用刚性变换是有效的，但我们可能无法将量化损失降低到极限。同样，在两个单独的步骤中减少维度和量化损失似乎是次优的。受这些缺点的启发，我们建议同时采用刚性和非刚性变换来减少量化误差和维度。我们在 PCA 公式中放松对投影的正交性约束，并通过量化项对其进行正则化。我们表明非刚性投影矩阵和旋转矩阵都有助于最小化量化损失，但方式不同。提出了一种可扩展的嵌套坐标下降方法来优化这个混合整数优化问题。我们在提供近 50 万张图像的五个公共基准数据集上评估了所提出的方法。比较结果表明，所提出的方法主要优于最先进的线性方法，并与端到端深度解决方案竞争。