PTL-LTM model for complex action recognition using local-weighted NMF and deep dual-manifold regularized NMF with sparsity constraint

Abstract

Complex action recognition possesses significant academic research value, potential commercial value and broad market application prospect. For improving its performance, a local-weighted nonnegative matrix factorization with rank regularization constraint (LWNMF_RC) is firstly presented, which removes complex background and then obtains motion salient regions. Secondly, a dual-manifold regularized nonnegative matrix factorization with sparsity constraint (DMNMF_SC) is proposed, which not only considers the short-term and middle-term temporal dependencies implied in data manifold, but also mines the geometric structure hidden in feature manifold. In addition, the introduction of sparsity constraint makes features possess better discriminativeness. Thirdly, a deep DMNMF_SC method is constructed, which acquires more hierarchical and discriminative features. Finally, a long-term temporal memory model with probability transfer learning (PTL-LTM) is proposed, which accurately memorizes the long-term temporal dependency among multiple simple action segments and, meanwhile, makes full use of the probability features of rich labeled simple actions and then applies the knowledge learned from simple actions for complex action recognition. Consequently, the performance is effectively improved.

Appendices

Appendix 1: Proof of Theorem 1

To prove Theorem 1, it is required to show the non-increasing property of the objective function in Eq. (3) under the update rules in Eqs. (15) and (16). Firstly, the objective function is proved to have non-increasing property under the update rule in Eq. (15). Then, it is demonstrated to have non-increasing property under the update rule in Eq. (16). The proof procedure will utilize the following auxiliary function, which is the same as that employed in the expectation maximization (EM) algorithm.

Definition 1

If the conditions G(x, x^(t)) ≥ F(x) and G(x, x) = F(x) are satisfied, then G(x, x^(t)) is an auxiliary function of F(x).

Lemma 1

If G(x, x^(t)) is an auxiliary function of F(x), then F(x) is non-increasing under the following update rule:

$$ x^{{\left( {t + 1} \right)}} = \mathop {\arg \hbox{min} }\limits_{x} G\left( {x,x^{\left( t \right)} } \right). $$

(51)

Proof

$F\left( {x^{{\left( {t + 1} \right)}} } \right) \le G\left( {x^{{\left( {t + 1} \right)}} ,x^{\left( t \right)} } \right) \le G\left( {x^{\left( t \right)} ,x^{\left( t \right)} } \right) = F\left( {x^{\left( t \right)} } \right).$

Now, it will be shown in the following that the update rule for W in Eq. (3) is exactly the update rule in Eq. (15) with a proper auxiliary function.

Considering any element w_jk in W, $F_{{w_{jk} }}$ is used to represent the part of Eq. (3), which is only related to w_jk. It is easy to check that:

$$ F^{\prime}_{{w_{jk} }} = \left( {\frac{{\partial O_{LWNMF\_RC} }}{{\partial \varvec{W}}}} \right)_{jk} = \left( { - 2\varvec{X}^{\text{T}} \varvec{XV} + 2\varvec{X}^{\text{T}} \varvec{XWV}^{\text{T}} \varvec{V} + \lambda_{1}\varvec{\psi}} \right)_{jk} $$

(52)

$$ F^{\prime\prime}_{{w_{jk} }} = 2\left( {\varvec{X}^{\text{T}} \varvec{X}} \right)_{jj} \left( {\varvec{V}^{\text{T}} \varvec{V}} \right)_{kk} + \lambda_{1} \left( {\frac{{\partial\varvec{\psi}}}{{\partial \varvec{W}}}} \right)_{jk} . $$

(53)

Since the update rule is essentially element-wise, it is sufficient to prove that each $F_{{w_{jk} }}$ is non-increasing under the update rule in Eq. (15).

Lemma 2

Function (54) is an auxiliary function of $F_{{w_{jk} }}$, which is the part of O_{LWNMF_RC} and only related to w_jk.

$$ G\left( {w,w_{jk}^{\left( t \right)} } \right) = F_{{w_{jk} }} \left( {w_{jk}^{\left( t \right)} } \right) + F^{\prime}_{{w_{jk} }} \left( {w_{jk}^{\left( t \right)} } \right)\left( {w - w_{jk}^{\left( t \right)} } \right) + \frac{{\left[ {\varvec{X}^{\text{T}} \varvec{XWV}^{\text{T}} \varvec{V} + \frac{1}{2}\lambda_{1}\varvec{\psi}} \right]_{jk} }}{{w_{jk}^{\left( t \right)} }}\left( {w - w_{jk}^{\left( t \right)} } \right)^{2} . $$

(54)

Proof

Since $G(w,w) = F_{w_{jk}}(w)$ is obvious, it only requires to prove that $G\left( {w,w_{jk}^{\left( t \right)} } \right) \ge F_{{w_{jk} }} \left( w \right)$. To do this, a comparison of Taylor series expansion of $F_{{w_{jk} }} \left( w \right)$ is made with Eq. (54):

$$ F_{{w_{jk} }} \left( w \right) = F_{{w_{jk} }} \left( {w_{jk}^{\left( t \right)} } \right) + F^{\prime}_{{w_{jk} }} \left( {w_{jk}^{\left( t \right)} } \right)\left( {w - w_{jk}^{\left( t \right)} } \right) + \frac{1}{2}F^{\prime\prime}_{{w_{jk} }} \left( {w_{jk}^{\left( t \right)} } \right)\left( {w - w_{jk}^{\left( t \right)} } \right)^{2} $$

(55)

and it can be found that: $G\left( {w,w_{jk}^{\left( t \right)} } \right) \ge F_{{w_{jk} }} \left( w \right)$ is equivalent to

$$ \frac{{\left[ {\varvec{X}^{\text{T}} \varvec{XWV}^{\text{T}} \varvec{V} + \frac{1}{2}\lambda_{1}\varvec{\psi}} \right]_{jk} }}{{w_{jk}^{\left( t \right)} }} \ge \frac{1}{2}F^{\prime\prime}_{{w_{jk} }} = \left( {\varvec{X}^{\text{T}} \varvec{X}} \right)_{jj} \left( {\varvec{V}^{{\rm T}} \varvec{V}} \right)_{kk} + \frac{1}{2}\lambda_{1} \left( {\frac{{\partial\varvec{\psi}}}{{\partial \varvec{W}}}} \right)_{jk} . $$

(56)

Meanwhile, the following equations hold:

$$ \begin{aligned} \left( {\varvec{X}^{\text{T}} \varvec{XWV}^{\text{T}} \varvec{V}} \right)_{jk} = \sum\limits_{l} {\left( {\varvec{X}^{\text{T}} \varvec{XW}} \right)_{jl} } \left( {\varvec{V}^{\text{T}} \varvec{V}} \right)_{lk} \ge \left( {\varvec{X}^{\text{T}} \varvec{XW}} \right)_{jk} \left( {\varvec{V}^{\text{T}} \varvec{V}} \right)_{kk} \\ \ge \sum\limits_{l} {\left( {\varvec{X}^{\text{T}} \varvec{X}} \right)_{jl} w_{lk} \left( {\varvec{V}^{\text{T}} \varvec{V}} \right)_{kk} } \ge w_{jk}^{\left( t \right)} \left( {\varvec{X}^{\text{T}} \varvec{X}} \right)_{jj} \left( {\varvec{V}^{\text{T}} \varvec{V}} \right)_{kk} \\ \end{aligned} $$

(57)

$$ \left(\varvec{\psi}\right)_{jk} \ge w_{jk}^{\left( t \right)} \left( {\frac{{\partial\varvec{\psi}}}{{\partial \varvec{W}}}} \right)_{jk} . $$

(58)

Therefore, Eq. (56) holds, from which $G\left( {w,w_{jk}^{\left( t \right)} } \right) \ge F_{{w_{jk} }} \left( w \right)$ holds.

Now, the objective function of Theorem 1 can be demonstrated to be non-increasing under the update rule in Eq. (15).

Proof

Substitute $G\left( {w,w_{jk}^{\left( t \right)} } \right)$ in Eq. (54) into Eq. (51), and the following update rule is obtained:

$$ w_{jk}^{{\left( {t + 1} \right)}} = \mathop {\arg \hbox{min} }\limits_{w} G\left( {w,w_{jk}^{\left( t \right)} } \right) = w_{jk}^{\left( t \right)} \frac{{\left( {2\varvec{X}^{\text{T}} \varvec{XV}} \right)_{jk} }}{{\left( {2\varvec{X}^{\text{T}} \varvec{XWV}^{\text{T}} \varvec{V} + \lambda_{1}\varvec{\psi}} \right)_{jk} }}. $$

(59)

Since Eq. (54) is an auxiliary function, $F_{w_{jk}}$ is non-increasing under this update rule.

Subsequently, the objective function is validated to be non-increasing under the update rule in Eq. (16).

Considering any element v_jk in V, $F_{{v_{jk} }}$ is used to denote the part of Eq. (3), which is only related to v_jk. It is easy to check that:

$$ F^{\prime}_{{v_{jk} }} = \left( {\frac{{\partial O_{LWNMF\_RC} }}{{\partial \varvec{V}}}} \right)_{jk} = \left( { - 2\varvec{X}^{\text{T}} \varvec{XW} + 2\varvec{VW}^{\text{T}} \varvec{X}^{\text{T}} \varvec{XW} + 2\lambda_{1} \varvec{Z}^{\text{T}} . * \varvec{V}. * \varvec{Z}^{\text{T}} + 2\lambda_{2} \varvec{V} - 2\lambda_{2} \varvec{B}} \right)_{jk} $$

(60)

$$ F^{\prime\prime}_{{v_{jk} }} = \left( {2\varvec{W}^{\text{T}} \varvec{X}^{\text{T}} \varvec{XW}} \right)_{kk} + \left( {2\lambda_{1} \varvec{Z}. * \varvec{Z}} \right)_{kj} + \left( {2\lambda_{2} \varvec{I}{\mathbf{ + }}2\lambda_{2} \left( {\varvec{B}^{ - } - \varvec{B}^{ + } } \right)} \right)_{jk} , $$

(61)

where I is an identity matrix.

Lemma 3

Function (62) is an auxiliary function for $F_{{v_{jk} }}$, which is the part of O_{LWNMF_RC} and only related to v_jk.

$$ G\left( {v,v_{jk}^{\left( t \right)} } \right) = F_{{v_{jk} }} \left( {v_{jk}^{\left( t \right)} } \right) + F^{\prime}_{{v_{jk} }} \left( {v_{jk}^{\left( t \right)} } \right)\left( {v - v_{jk}^{\left( t \right)} } \right) + \frac{{\left[ {\varvec{VW}^{\text{T}} \varvec{X}^{\text{T}} \varvec{XW} + \lambda_{1} \varvec{Z}^{\text{T}} . * \varvec{V}. * \varvec{Z}^{\text{T}} + \lambda_{2} \varvec{V} + \lambda_{2} \varvec{B}^{ - } } \right]_{jk} }}{{v_{jk}^{\left( t \right)} }}\left( {v - v_{jk}^{\left( t \right)} } \right)^{2} . $$

(62)

Proof

Since $G\left( {v,v} \right) = F_{{v_{jk} }} \left( v \right)$ is obvious, it only requires to show that $G\left( {v,v_{jk}^{\left( t \right)} } \right) \ge F_{{v_{jk} }} \left( v \right)$. To do this, a comparison of Taylor series expansion of $F_{{v_{jk} }} \left( v \right)$ is made with Eq. (62):

$$ F_{{v_{jk} }} \left( v \right) = F_{{v_{jk} }} \left( {v_{jk}^{\left( t \right)} } \right) + F^{\prime}_{{v_{jk} }} \left( {v_{jk}^{\left( t \right)} } \right)\left( {v - v_{jk}^{\left( t \right)} } \right) + \frac{1}{2}F^{\prime\prime}_{{v_{jk} }} \left( {v_{jk}^{\left( t \right)} } \right)\left( {v - v_{jk}^{\left( t \right)} } \right)^{2} $$

(63)

and it can be found that: $G\left( {v,v_{jk}^{\left( t \right)} } \right) \ge F_{{v_{jk} }} \left( v \right)$ is equivalent to

$$ \frac{{\left[ {\varvec{VW}^{\text{T}} \varvec{X}^{\text{T}} \varvec{XW} + \lambda_{1} \varvec{Z}^{\text{T}} . * \varvec{V}. * \varvec{Z}^{\text{T}} + \lambda_{2} \varvec{V} + \lambda_{2} \varvec{B}^{ - } } \right]_{jk} }}{{v_{jk}^{\left( t \right)} }} \ge \left( {\varvec{W}^{\text{T}} \varvec{X}^{\text{T}} \varvec{XW}} \right)_{kk} + \left( {\lambda_{1} \varvec{Z}. * \varvec{Z}} \right)_{kj} + \left( {\lambda_{2} \varvec{I}{\mathbf{ + }}\lambda_{2} \left( {\varvec{B}^{ - } - \varvec{B}^{ + } } \right)} \right)_{jk} $$

(64)

Meanwhile, the following equations hold:

$$ \left( {\varvec{VW}^{\text{T}} \varvec{X}^{\text{T}} \varvec{XW}} \right)_{jk} = \sum\limits_{l} {\left( \varvec{V} \right)_{jl} } \left( {\varvec{W}^{\text{T}} \varvec{X}^{\text{T}} \varvec{XW}} \right)_{lk} \ge v_{jk}^{\left( t \right)} \left( {\varvec{W}^{\text{T}} \varvec{X}^{\text{T}} \varvec{XW}} \right)_{kk} $$

(65)

$$ \lambda_{1} \left( {\varvec{Z}^{\text{T}} . * \varvec{V}. * \varvec{Z}^{\text{T}} } \right)_{jk} = \lambda_{1} \left( {\varvec{Z}^{\text{T}} . * \varvec{Z}^{\text{T}} } \right)_{jk} v_{jk}^{\left( t \right)} = \left( {\lambda_{1} \varvec{Z}. * \varvec{Z}} \right)_{kj} v_{jk}^{\left( t \right)} $$

(66)

$$ \left( {\lambda_{2} \varvec{V} + \lambda_{2} \varvec{B}^{ - } } \right)_{jk} \ge \left( {\lambda_{2} \varvec{I}{\mathbf{ + }}\lambda_{2} \varvec{B}^{ - } - \lambda_{2} \varvec{B}^{ + } } \right)_{jk} v_{jk}^{\left( t \right)} . $$

(67)

Thus, Eq. (64) holds and $G\left( {v,v_{jk}^{\left( t \right)} } \right) \ge F_{{v_{jk} }} \left( v \right)$.

Now, it can also be demonstrated that the objective function of Theorem 1 is non-increasing under the update rule in Eq. (16).

Proof

Substitute $G\left( {v,v_{jk}^{\left( t \right)} } \right)$ in Eq. (62) into Eq. (51), and the following update rule can be obtained:

$$ v_{jk}^{{\left( {t + 1} \right)}} = \mathop {\arg \hbox{min} }\limits_{v} G\left( {v,v_{jk}^{\left( t \right)} } \right) = v_{jk}^{\left( t \right)} \frac{{\left( {\varvec{X}^{\text{T}} \varvec{XW} + \lambda_{2} \varvec{B}^{ + } } \right)_{jk} }}{{\left( {\varvec{VW}^{\text{T}} \varvec{X}^{\text{T}} \varvec{XW} + \lambda_{1} \varvec{Z}^{\text{T}} . * \varvec{V}. * \varvec{Z}^{\text{T}} + \lambda_{2} \varvec{V} + \lambda_{2} \varvec{B}^{ - } } \right)_{jk} }}. $$

(68)

Since Eq. (62) is an auxiliary function, and $F_{{v_{jk} }}$ is non-increasing under this update rule. Therefore, Theorem 1 holds.

Appendix 2: Proof of Theorem 2

To prove Theorem 2, it is required to show the non-increasing property of the objective function in Eq. (19) under the update rules in Eqs. (31) and (40). Firstly, the objective function is proved to have non-increasing property under the update rule in Eq. (31). Then, it is also demonstrated to have non-increasing property under the update rule in Eq. (40). The proof will utilize the following auxiliary function, which is the same as that used in the EM algorithm.

According to Definition 1 and Lemma 1 in Appendix 1, it can also be demonstrated that the objective function of Theorem 2 has non-increasing property under the update rule in Eq. (31).

Considering any element w_jk in W, $\tilde{J}_{{w_{jk} }}$ is utilized to denote the part of Eq. (19), which is only related to w_jk. It is easy to check that:

$$ \tilde{J}^{\prime}_{{w_{jk} }} = \left( {\frac{{\partial O_{DMNMF\_SC} }}{{\partial \varvec{W}}}} \right)_{jk} = \left( { - 2\varvec{X}^{\text{T}} \varvec{XV} + 2\varvec{X}^{\text{T}} \varvec{XWV}^{\text{T}} \varvec{V} + 2\mu_{2} \varvec{X}^{\text{T}} \varvec{L}^{U} \varvec{XW}} \right)_{jk} $$

(69)

$$ \tilde{J}^{\prime\prime}_{{w_{jk} }} = 2\left( {\varvec{X}^{\text{T}} \varvec{X}} \right)_{jj} \left( {\varvec{V}^{\text{T}} \varvec{V}} \right)_{kk} + 2\mu_{2} \left( {{\mathbf{X}}^{\text{T}} {\mathbf{L}}^{U} {\mathbf{X}}} \right)_{jj} . $$

(70)

Lemma 4

Function (71) is an auxiliary function for $\tilde{J}_{{w_{jk} }}$, which is the part of O_{DWNMF_SC}, and only related to w_jk.

$$ G\left( {w,w_{jk}^{\left( t \right)} } \right) = \tilde{J}_{{w_{jk} }} \left( {w_{jk}^{\left( t \right)} } \right) + \tilde{J}^{\prime}_{{w_{jk} }} \left( {w_{jk}^{\left( t \right)} } \right)\left( {w - w_{jk}^{\left( t \right)} } \right) + \frac{{\left[ {{\mathbf{X}}^{\text{T}} {\mathbf{XWV}}^{\text{T}} {\mathbf{V}} + \mu_{2} {\mathbf{X}}^{\text{T}} {\mathbf{D}}^{U} {\mathbf{XW}}} \right]_{jk} }}{{w_{jk}^{\left( t \right)} }}\left( {w - w_{jk}^{\left( t \right)} } \right)^{2} . $$

(71)

Proof

Since $G\left( {w,w} \right) = \tilde{J}_{{w_{jk} }} \left( w \right)$ is obvious, it only requires to show that $G\left( {w,w_{jk}^{\left( t \right)} } \right) \ge \tilde{J}_{{w_{jk} }} \left( w \right)$. To do this, a comparison of Taylor series expansion of $\tilde{J}_{{w_{jk} }} \left( w \right)$ is made with Eq. (71):

$$ \tilde{J}_{{w_{jk} }} \left( w \right) = \tilde{J}_{{w_{jk} }} \left( {w_{jk}^{\left( t \right)} } \right) + \tilde{J}^{\prime}_{{w_{jk} }} \left( {w_{jk}^{\left( t \right)} } \right)\left( {w - w_{jk}^{\left( t \right)} } \right) + \frac{1}{2}\tilde{J}^{\prime\prime}_{{w_{jk} }} \left( {w_{jk}^{\left( t \right)} } \right)\left( {w - w_{jk}^{\left( t \right)} } \right)^{2} . $$

(72)

And it can be found that: $G\left( {w,w_{jk}^{\left( t \right)} } \right) \ge \tilde{J}_{{w_{jk} }} \left( w \right)$ is equivalent to

$$ \frac{{\left[ {\varvec{X}^{\text{T}} \varvec{XWV}^{\text{T}} \varvec{V} + \mu_{2} \varvec{X}^{\text{T}} \varvec{D}^{U} \varvec{XW}} \right]_{jk} }}{{w_{jk}^{\left( t \right)} }} \ge \frac{1}{2}\tilde{J}^{\prime\prime}_{{w_{jk} }} = \left( {\varvec{X}^{\text{T}} \varvec{X}} \right)_{jj} \left( {\varvec{V}^{\text{T}} \varvec{V}} \right)_{kk} + \mu_{2} \left( {\varvec{X}^{\text{T}} \varvec{L}^{U} \varvec{X}} \right)_{jj} . $$

(73)

Meanwhile, the following inequalities hold:

$$ \begin{aligned} \left( {\varvec{X}^{\text{T}} \varvec{XWV}^{\text{T}} \varvec{V}} \right)_{jk} & = \sum\limits_{l} {\left( {\varvec{X}^{\text{T}} \varvec{XW}} \right)_{jl} } \left( {\varvec{V}^{\text{T}} \varvec{V}} \right)_{lk} \ge \left( {\varvec{X}^{\text{T}} \varvec{XW}} \right)_{jk} \left( {\varvec{V}^{\text{T}} \varvec{V}} \right)_{kk} \\ & \ge \sum\limits_{l} {\left( {\varvec{X}^{\text{T}} \varvec{X}} \right)_{jl} w_{lk} } \left( {\varvec{V}^{\text{T}} \varvec{V}} \right)_{kk} \ge w_{jk}^{\left( t \right)} \left( {\varvec{X}^{\text{T}} \varvec{X}} \right)_{jj} \left( {\varvec{V}^{\text{T}} \varvec{V}} \right)_{kk} \\ \end{aligned} $$

(74)

$$ \begin{aligned} \mu_{2} \left( {\varvec{X}^{{^{\text{T}} }} \varvec{D}^{U} \varvec{XW}} \right)_{jk} & = \mu_{2} \sum\limits_{l} {\left( {\varvec{X}^{\text{T}} \varvec{D}^{U} \varvec{X}} \right)_{jl} \left( \varvec{W} \right)_{lk} } \ge \mu_{2} \left( {\varvec{X}^{\text{T}} \varvec{D}^{U} \varvec{X}} \right)_{jj} w_{jk}^{\left( t \right)} \\ & \ge \mu_{2} \left[ {\varvec{X}^{\text{T}} \left( {\varvec{D}^{U} - \varvec{A}^{U} } \right)\varvec{X}} \right]_{jj} w_{jk}^{\left( t \right)} = \mu_{2} \left( {\varvec{X}^{\text{T}} \varvec{L}^{U} \varvec{X}} \right)_{jj} w_{jk}^{\left( t \right)} . \\ \end{aligned} $$

(75)

Thus, Eq. (73) holds and $G\left( {w,w_{jk}^{\left( t \right)} } \right) \ge \tilde{J}_{{w_{jk} }} \left( w \right)$.

Now, it can be demonstrated that the objective function of Theorem 2 is non-increasing under the update rule of Eq. (31).

Proof

Replace $G\left( {w,w_{jk}^{\left( t \right)} } \right)$ in Eq. (51) by Eq. (71), and the following update rule is obtained:

$$ w_{jk}^{{\left( {t + 1} \right)}} = \mathop {\arg \hbox{min} }\limits_{w} G\left( {w,w_{jk}^{\left( t \right)} } \right) = w_{jk}^{\left( t \right)} \frac{{\left( {\varvec{X}^{\text{T}} \varvec{XV} + \mu_{2} \varvec{X}^{\text{T}} \varvec{A}^{U} \varvec{XW}} \right)_{jk} }}{{\left( {\varvec{X}^{\text{T}} \varvec{XWV}^{{\rm T}} \varvec{V} + \mu_{2} \varvec{X}^{\text{T}} \varvec{D}^{U} \varvec{XW}} \right)_{jk} }}. $$

(76)

Since Eq. (71) is an auxiliary function, $\tilde{J}_{{w_{jk} }}$ is non-increasing under this update rule.

Subsequently, the objective function is validated to be non-increasing under the update rule in Eq. (40).

Considering any element v_jk in V, $\tilde{J}_{{v_{jk} }}$ is used to denote the part of Eq. (19), which is only related to v_jk. It is easy to check that:

$$ \tilde{J}^{\prime}_{{v_{jk} }} = \left( {\frac{{\partial O_{DMNMF\_SC} }}{{\partial \varvec{V}}}} \right)_{jk} = \left( { - 2{\mathbf{X}}^{\text{T}} {\mathbf{XW}} + 2{\mathbf{VW}}^{\text{T}} {\mathbf{X}}^{\text{T}} {\mathbf{XW}} + 2\mu_{1} {\mathbf{L}}^{V} {\mathbf{V}} + \mu_{3} {\mathbf{MV}}} \right)_{jk} $$

(77)

$$ \tilde{J}^{\prime\prime}_{{v_{jk} }} = \left( {2{\mathbf{W}}^{\text{T}} {\mathbf{X}}^{\text{T}} {\mathbf{XW}}} \right)_{kk} + \left( {2\mu_{1} {\mathbf{L}}^{V} } \right){}_{jj} + \mu_{3} \left( {M_{jj} - v_{jk}^{2} \left( {M_{jj} } \right)^{3} } \right). $$

(78)

Lemma 5

Function (79) is an auxiliary function for $\tilde{J}_{{v_{jk} }}$, which is the part of O_{DWNMF_SC} and only related to v_jk.

$$ G\left( {v,v_{jk}^{\left( t \right)} } \right) = \tilde{J}_{{v_{jk} }} \left( {v_{jk}^{\left( t \right)} } \right) + \tilde{J}^{\prime}_{{v_{jk} }} \left( {v_{jk}^{\left( t \right)} } \right)\left( {v - v_{jk}^{\left( t \right)} } \right) + \frac{{\left[ {{\mathbf{VW}}^{\text{T}} {\mathbf{X}}^{\text{T}} {\mathbf{XW}} + \mu_{1} {\mathbf{D}}^{V} {\mathbf{V}} + \frac{{\mu_{3} }}{2}{\mathbf{MV}}} \right]_{jk} }}{{v_{jk}^{\left( t \right)} }}\left( {v - v_{jk}^{\left( t \right)} } \right)^{2} . $$

(79)

Proof

Since $G\left( {v,v} \right) = \tilde{J}_{{v_{jk} }} \left( v \right)$ is obvious, it only requires to show that $G\left( {v,v_{jk}^{\left( t \right)} } \right) \ge \tilde{J}_{{v_{jk} }} \left( v \right)$. To do this, a comparison of Taylor series expansion of $\tilde{J}_{{v_{jk} }} \left( v \right)$ is made with Eq. (79):

$$ \tilde{J}_{{v_{jk} }} \left( v \right) = \tilde{J}_{{v_{jk} }} \left( {v_{jk}^{\left( t \right)} } \right) + \tilde{J}^{\prime}_{{v_{jk} }} \left( {v_{jk}^{\left( t \right)} } \right)\left( {v - v_{jk}^{\left( t \right)} } \right) + \frac{1}{2}\tilde{J}^{\prime\prime}_{{v_{jk} }} \left( {v_{jk}^{\left( t \right)} } \right)\left( {v - v_{jk}^{\left( t \right)} } \right)^{2} $$

(80)

and it can be found that: $G\left( {v,v_{jk}^{\left( t \right)} } \right) \ge \tilde{J}_{{v_{jk} }} \left( v \right)$ is equivalent to

$$ \frac{{\left[ {\varvec{VW}^{\text{T}} \varvec{X}^{\text{T}} \varvec{XW} + \mu_{1} \varvec{D}^{V} \varvec{V} + \frac{1}{2}\mu_{3} \varvec{MV}} \right]_{jk} }}{{v_{jk}^{\left( t \right)} }} \ge \frac{1}{2}\tilde{J}^{\prime\prime}_{{v_{jk} }} = \left( {\varvec{W}^{\text{T}} \varvec{X}^{\text{T}} \varvec{XW}} \right)_{kk} + \mu_{1} \left( {\varvec{L}^{V} } \right){}_{jj} + \frac{{\mu_{3} }}{2}\left( {M_{jj} - v_{jk}^{2} \left( {M_{jj} } \right)^{3} } \right). $$

(81)

Meanwhile, the following inequalities hold:

$$ \left( {\varvec{VW}^{\text{T}} \varvec{X}^{\text{T}} \varvec{XW}} \right)_{jk} = \sum\limits_{l} {v_{jl} } \left( {\varvec{W}^{\text{T}} \varvec{X}^{\text{T}} \varvec{XW}} \right)_{lk} \ge v_{jk}^{\left( t \right)} \left( {\varvec{W}^{\text{T}} \varvec{X}^{\text{T}} \varvec{XW}} \right)_{kk} $$

(82)

$$ \mu_{1} \left( {\varvec{D}^{V} \varvec{V}} \right)_{jk} = \mu_{1} \sum\limits_{l} {\left( {\varvec{L}^{V} + \varvec{A}^{V} } \right)_{jl} } v_{lk} \ge \mu_{1} \left( {\varvec{L}^{V} } \right)_{jj} v_{jk}^{\left( t \right)} $$

(83)

$$ \frac{{\mu_{3} }}{2}\left( {\varvec{MV}} \right)_{jk} = \frac{{\mu_{3} }}{2}M_{jj} v_{jk} \ge \frac{{\mu_{3} }}{2}\left( {M_{jj} - v_{jk}^{2} \left( {M_{jj} } \right)^{3} } \right)v_{jk}^{\left( t \right)} . $$

(84)

Thus, Eq. (81) holds and $G\left( {v,v_{jk}^{\left( t \right)} } \right) \ge \tilde{J}_{{v_{jk} }} \left( v \right)$.

Now, it can also be demonstrated that the objective function of Theorem 2 is non-increasing under the update rule in Eq. (40).

Proof

Replace $G\left( {v,v_{jk}^{\left( t \right)} } \right)$ in Eq. (51) by Eq. (79) and the following update rule can be obtained:

$$ v_{jk}^{{\left( {t + 1} \right)}} = \mathop {\arg \hbox{min} }\limits_{v} G\left( {v,v_{jk}^{\left( t \right)} } \right) = v_{jk}^{\left( t \right)} \frac{{\left( {2\varvec{X}^{\text{T}} \varvec{XW} + 2\mu_{1} \varvec{A}^{V} \varvec{V}} \right)_{jk} }}{{\left( {2\varvec{VW}^{\text{T}} \varvec{X}^{\text{T}} \varvec{XW} + 2\mu_{1} \varvec{D}^{V} \varvec{V} + \mu_{3} \varvec{MV}} \right)_{jk} }}. $$

(85)

Since Eq. (79) is an auxiliary function, $\tilde{J}_{{v_{jk} }}$ is non-increasing under the update rule in Eq. (85). Thus, Theorem 2 holds.