On Unifying Multi-view Self-Representations for Clustering by Tensor Multi-rank Minimization

Abstract

In this paper, we address the multi-view subspace clustering problem. Our method utilizes the circulant algebra for tensor, which is constructed by stacking the subspace representation matrices of different views and then rotating, to capture the low rank tensor subspace so that the refinement of the view-specific subspaces can be achieved, as well as the high order correlations underlying multi-view data can be explored. By introducing a recently proposed tensor factorization, namely tensor-Singular Value Decomposition (t-SVD) (Kilmer et al. in SIAM J Matrix Anal Appl 34(1):148–172, 2013), we can impose a new type of low-rank tensor constraint on the rotated tensor to ensure the consensus among multiple views. Different from traditional unfolding based tensor norm, this low-rank tensor constraint has optimality properties similar to that of matrix rank derived from SVD, so the complementary information can be explored and propagated among all the views more thoroughly and effectively. The established model, called t-SVD based Multi-view Subspace Clustering (t-SVD-MSC), falls into the applicable scope of augmented Lagrangian method, and its minimization problem can be efficiently solved with theoretical convergence guarantee and relatively low computational complexity. Extensive experimental testing on eight challenging image datasets shows that the proposed method has achieved highly competent objective performance compared to several state-of-the-art multi-view clustering methods.

Appendix

Proof of the Theorem 2:

Proof

In Fourier domain, the optimization problem of Eq. (29) can be reformulated as

$$\begin{aligned} \varvec{{\mathcal {G}}}_{f}&= {{\mathrm{argmin}}}_{\varvec{{\mathcal {G}}}_{f}} ~ \tau ||\mathrm {bdiag}(\varvec{{\mathcal {G}}}_{f})||_{*} + \frac{1}{2n_{3}}||\varvec{{\mathcal {G}}}_{f} - \varvec{{\mathcal {F}}}_{f}||_{F}^{2} \end{aligned}$$

(39)

$$\begin{aligned}&={{\mathrm{argmin}}}_{\varvec{{\mathcal {G}}}_{f}} ~ \sum _{j=1}^{n_{3}}\tau '||\varvec{{\mathcal {G}}}_{f}^{(j)}||_{*} + \frac{1}{2}||\varvec{{\mathcal {G}}}_{f}^{(j)} - \varvec{{\mathcal {F}}}_{f}^{(j)}||_{F}^{2}, \end{aligned}$$

(40)

where $\tau '=n_{3}\tau$. Then Eq. (40) can be separated into $n_{3}$ independent subproblems,

$$\begin{aligned} \varvec{{\mathcal {G}}}_{f}^{(j)} = {{\mathrm{argmin}}}_{\varvec{{\mathcal {G}}}_{f}^{(j)}} ~ \tau '||\varvec{{\mathcal {G}}}_{f}^{(j)}||_{*} + \frac{1}{2}||\varvec{{\mathcal {G}}}_{f}^{(j)} - \varvec{{\mathcal {F}}}_{f}^{(j)}||_{F}^{2}, \end{aligned}$$

(41)

where $j = 1,2,\ldots ,n_{3}$. Note that Eq. (41) is the F-norm based nuclear norm low rank matrix approximation problem represented in Fourier domain. According to the result on subgradients of unitarily invariant norms, Eq. (41) can also be solved by a soft-thresholding operation (Cai et al. 2010),

$$\begin{aligned} \varvec{{\mathcal {G}}}_{f}^{(j)} = D_{\tau '}(\varvec{{\mathcal {F}}}_{f}^{(j)})= \varvec{{\mathcal {U}}}_{f}^{(j)}\varvec{{\mathcal {S}}}_{f,\tau '}^{(j)}\varvec{{\mathcal {V}}}_{f}^{(j)^{\mathrm {T}}}, \end{aligned}$$

(42)

here, $\varvec{{\mathcal {G}}}_{f}^{(j)} = \varvec{{\mathcal {U}}}_{f}^{(j)}\varvec{{\mathcal {S}}}_{f}^{(j)}\varvec{{\mathcal {V}}}_{f}^{(j)^{\mathrm {T}}}$, ${\mathcal {D}}_{\tau '}(\cdot )$ is the SVT operation with with threshold $\tau '$ (see Sect. 3), and ${\mathcal {S}}_{f,\tau '}^{(j)}= \mathrm {diag}\{(\mathcal{{\mathcal {S}}}_{f}^{(j)}(i,i)-\tau ' )_{+}\}$. Then, we can get

$$\begin{aligned} \varvec{{\mathcal {G}}}_{f} = \mathrm {bdfold}\left\{ \mathrm {bdiag}(\varvec{\mathcal{U}}_{f})\mathrm {bdiag}(\varvec{\mathcal{S}}_{f,\tau '}) \mathrm {bdiag}(\varvec{\mathcal{V}}_{f})^{\mathrm {T}}\right\} \end{aligned}$$

(43)

and

$$\begin{aligned} \varvec{{\mathcal {G}}} = \varvec{{\mathcal {U}}}*\varvec{\mathcal {{\tilde{S}}}}*\varvec{{\mathcal {V}}^{\mathrm {T}}} \end{aligned}$$

(44)

where $\varvec{\mathcal {{\tilde{S}}}} = \mathrm {ifft}(\varvec{\mathcal{S}}_{f,\tau '},[~], 3)$. Suppose that $\varvec{\mathcal{J}}$ is an $n_{1} \times n_{2} \times n_{3}$ f-diagonal tensor whose diagonal element in the Fourier domain is $\varvec{\mathcal{J}}_{f}(i,i,j) = (1 - \frac{\tau '}{\varvec{\mathcal{S}}_{f}^{(j)}(i,i)})_{+}$, then we have that $\varvec{\mathcal{S}}_{f,\tau '}(i,i,:) = \varvec{\mathcal{S}}_{f}(i,i,:)\varvec{\mathcal{J}}_{f}(i,i,:)$ in the Fourier domain, as well as $\varvec{\mathcal{{\tilde{S}}}}(i,i,:) = \varvec{\mathcal{S}}(i,i,:) \circ \varvec{\mathcal{J}}(i,i,:)$ in the original domain. Because both $\varvec{\mathcal{S}}$ and $\varvec{\mathcal{J}}$ are f-diagonal, $\varvec{\mathcal{{\tilde{S}}}}$ can be formulated as $\varvec{\mathcal{{\tilde{S}}}} = \varvec{\mathcal{S}} * \varvec{\mathcal{J}}$. Therefore, a convolution based tubal-shrinkage operator in the original domain is equivalent to the tensor SVT in the Fourier domain. $\square$