# Tensioned Multi-View Ordered Kernel Subspace Clustering
## 3 The Proposed TMOKSC Algorithm  
### Multi-View Ordered Kernel Subspace Clustering Based on Tensor  
In the multimodal sparse subspace clustering model, the similarity kernel function of samples often appears in the form of pairwise inner products. The proposed TMOKSC algorithm considers the preservation of local sequential information coinciding with the fusion of global information, as shown in Figure 1. Its goal is to capture the overall structure of the clustering by matrix operations. Multi-view subspace clustering algorithms often assume that heterogeneous data are located in the union of the same linear subspaces. It often roughly learns association graphs from the original data containing noise and missing views. In extreme cases, it even damages the clustering structure, which is originally inseparable in a low-dimensional linear space. By analyzing the context of data across views, we can observe the sparsity of the original data and the sequential relationship among the inherence data. Suppose \(\left\|.\right\|_{p}\)_and_\(\left\|.\right\|_{q}\) represent the specific norm of the noises matrix of thevthview \(E^{(v)}\) and Similaritygraphmatrixofthevthview \(Z^{(v)}\). We respectively explore the inherent consistent information to represent noise by integrating the sparse regularization term \(\sum_{v=1}^{V}\left\|Z^{(v)}\right\|_{q}\), meanwhile, we adopt the consistent regularization domain \(\Omega\) to modify the sequential complementary information. So its objective function is generally expressed as:  
\[\underset{\{Z^{(v)},E^{(v)}\}_{v=1}^{V}=1}{min}\sum_{v=1}^{V}\left\|E^{(v)} \right\|_{p}+\alpha\sum_{v=1}^{V}\left\|Z^{(v)}\right\|_{q}+\Omega s.t.X^{(v) }=X^{(v)}Z^{(v)}+E^{(v)},\forall v, \tag{1}\]  
Figure 1: Tensioned Multi-view Ordered Kernel Subspace Clustering. If we replace the noise term according to the norm of the difference between the sample and the reconstructed sample. Then the formula turns to:  
\[\underset{\{Z^{(p)}\}_{p=1}^{V}}{\underset{v=1}{min}}\sum_{v=1}^{V}\left\|X^{(v) }-X^{(v)}Z^{(v)}\right\|_{p}+\alpha\sum_{v=1}^{V}\left\|Z^{(v)}\right\|_{q}+\Omega, \tag{2}\]  
At present, the dot product \(\left\langle x_{i},x_{j}\right\rangle\) in the kernel function mainly reflects the included angle relationship of the samples. This rough similarity function may cause an inaccurate result. The spatial distance dissimilarity measure is timely added to reflect the distribution of the samples in the feature space. From the perspective of sequence characteristics, the selection of kernel functions is detrimental to the performance of kernel learning. So, a good selection of kernel functions is necessary for the success of the kernel subspace clustering method. In the sample set X, the dot product \(\left\langle x_{i},x_{j}\right\rangle\) is often used to initially describe the similarity of two vectors \(x_{i},x_{j}\). In addition to reflecting the global relationship of the vector field, the sequential kernel function \(\phi_{ord}\big{(}x_{i},x_{j}\big{)}\) is proposed. Taking into account the sequential relationship among vector samples, the function is represented as follows:  
\[\phi_{ord}\big{(}x_{i},x_{j}\big{)}=\big{\langle}x_{i},x_{j}\big{\rangle}+ \lambda\big{|}order(x_{i})-order\big{(}x_{j}\big{)}\big{|}, \tag{3}\]  
Here, \(order()\) means the physical position of the sample. In the construction of this kernel function, the sequential function \(\big{|}order(x_{i})-order\big{(}x_{j}\big{)}\big{|}\) is defined in the spatiotemporal domain to reflect a new ordered feature. The sequential constraint kernel function can obey its following properties:  
1. Non-negativity: \[\phi_{ord}\big{(}x_{i},x_{j}\big{)}\geq 0,\] (4)
2. Symmetry: \[\phi_{ord}\big{(}x_{i},x_{j}\big{)}=\phi_{ord}\big{(}x_{j},x_{i}\big{)},\] (5)  
This kernel matrix K induced by this kernel function \(\phi_{ord}\big{(}x_{i},x_{j}\big{)}\) also satisfies non-negativity and symmetry. Thus, the representation of order norm in the spatial domain is added to the feature space.