## 3 Proposed Method  
### Multi-Attack Reference Hashing  
Currently, most image hashing method take the original image as the reference. However, the image hashes arising from the original image may not be the hash centroid of the distorted copies. As shown in Figure 1(a), we applied 15 classes of attacks on five original images and represent their hashes in 2-dimensional space for both the original images and their distorted copies. From Figure 1(a), we can observe five clusters in the hashing space. From Figure 1(b) by zooming into one hash cluster, we note an observation that the hashes of the original image actually may not be the centroid of its cluster. For \(l\) original images in the dataset, we apply \(V\) type content preserving attacks with different types of parameter settings to generate simulated distorted copies. Let us denote the feature matrix of attacked instances in set \(\Psi_{v}\) as \(\mathbf{X}^{v}\in\mathbb{R}^{m\times t}\). Here, \(v=1,2,...,V\), \(m\) is the dimensionality of data feature  
Figure 1: Block diagram of the proposed algorithm. and \(t\) is the number of instances for attack \(v\). Finally, we get the feature matrices for the total \(n\) instance as \(\mathbf{X}=\{\mathbf{X}^{1},...,\mathbf{X}^{V}\}\), and here \(n=tV\). Note that the feature matrices are normalized to zero-centered. By considering the total reconstruction errors of all the training objects, we have the following minimization problem in a matrix form, which jointly exploits the information from various content preserving multi-attack data. \[J_{1}(\mathbf{\tilde{U}},\mathbf{\tilde{X}})=\alpha(||\mathbf{\tilde{X}}- \mathbf{\tilde{U}}\mathbf{X}||_{F}^{2}+\beta||\mathbf{\tilde{U}}||_{F}^{2}), \tag{1}\]  
where \(\mathbf{\tilde{X}}\) is the shared latent multi-attack feature representation. The matrix \(\mathbf{\tilde{U}}\) can be viewed as the basis matrix, which maps the input multi-attack features onto the corresponding latent features. Parameter \(\alpha\), \(\beta\) is a nonnegative weighting vector to balance the significance. From the information-theoretic point of view, the variance over all data is measured, and taken as a regularization term:  
\[J_{2}(\mathbf{\tilde{U}})=\gamma||\mathbf{\tilde{U}}\mathbf{X}||_{F}^{2}, \tag{2}\]  
where \(\gamma\) is a nonnegative constant parameter. The image reference for authentication is actually an infinite clustering problem. The reference is usually generated based on the cluster centroid image. Therefore, we also consider keeping the cluster structures. We formulate this objective function as:  
\[J_{3}(\mathbf{C},\mathbf{G})=\lambda||\mathbf{\tilde{X}}-\mathbf{C}\mathbf{G} ||_{F}^{2}, \tag{3}\]  
where \(\mathbf{C}\in\mathbb{R}^{k\times l}\) and \(\mathbf{G}\in\{0,1\}^{l\times n}\) are the clustering centroid and indicator. Finally, the formulation can be written as:  
\[\underset{\mathbf{\tilde{U}},\mathbf{\tilde{X}},\mathbf{C},\mathbf{G}}{min} \alpha||\mathbf{\tilde{X}}-\mathbf{\tilde{U}}\mathbf{X}||_{F}^{2}+\beta|| \mathbf{\tilde{U}}||_{F}^{2}-\gamma||\mathbf{\tilde{U}}\mathbf{X}||_{F}^{2}+ \lambda||\mathbf{\tilde{X}}-\mathbf{C}\mathbf{G}||_{F}^{2}. \tag{4}\]  
Our objective function simultaneously learns the feature representations \(\mathbf{\tilde{X}}\) and finds the mapping matrix \(\mathbf{\tilde{U}}\), the cluster centroid \(\mathbf{C}\) and indicator \(\mathbf{G}\). The iterative optimization algorithm is as follows.